dan shanks’ cuffqi algorithm resurrected · 2018-11-02 · dan shanks’ cuffqi algorithm...
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Dan Shanks’ CUFFQI AlgorithmResurrected
Renate Scheidler
Celebrating 75 Years of Mathematics of ComputationICERM (Providence, RI)
November 1, 2018
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What is CUFFQI?
Short for Cubic Fields From Quadratic Infrastructure
Invented by Dan Shanks (1987)Editor for Math. Comp. 1959-1996
Made practical and implemented by Gilbert Fung (1990)
Unpublished (to appear as Chapter 4 in Cubic Fields With Geometryby S. Hambleton & H. C. Williams, Springer Monograph 2018/19)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 2 / 32
![Page 3: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/3.jpg)
What is CUFFQI?
Short for Cubic Fields From Quadratic Infrastructure
Invented by Dan Shanks (1987)Editor for Math. Comp. 1959-1996
Made practical and implemented by Gilbert Fung (1990)
Unpublished (to appear as Chapter 4 in Cubic Fields With Geometryby S. Hambleton & H. C. Williams, Springer Monograph 2018/19)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 2 / 32
![Page 4: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/4.jpg)
What is CUFFQI?
Short for Cubic Fields From Quadratic Infrastructure
Invented by Dan Shanks (1987)Editor for Math. Comp. 1959-1996
Made practical and implemented by Gilbert Fung (1990)
Unpublished (to appear as Chapter 4 in Cubic Fields With Geometryby S. Hambleton & H. C. Williams, Springer Monograph 2018/19)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 2 / 32
![Page 5: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/5.jpg)
What is CUFFQI?
Short for Cubic Fields From Quadratic Infrastructure
Invented by Dan Shanks (1987)Editor for Math. Comp. 1959-1996
Made practical and implemented by Gilbert Fung (1990)
Unpublished (to appear as Chapter 4 in Cubic Fields With Geometryby S. Hambleton & H. C. Williams, Springer Monograph 2018/19)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 2 / 32
![Page 6: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/6.jpg)
Cubic Fields
A cubic field of discriminant D has a generating polynomials of the form
f (x) = x3 − 3N(λ)1/3x + Tr(λ)
λ is an algebraic integer in Q(√−3D)
Norm and trace are taken in Q(√−3D)/Q
N(λ) ∈ Z3
(Berwick 1925)
Roots of f (x) (Cardano 1545):
ζ iλ1/3 + ζ−iλ1/3
(i = 0, 1, 2)
where ζ is a primitive cube root of unity
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 3 / 32
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Cubic Fields
A cubic field of discriminant D has a generating polynomials of the form
f (x) = x3 − 3N(λ)1/3x + Tr(λ)
λ is an algebraic integer in Q(√−3D)
Norm and trace are taken in Q(√−3D)/Q
N(λ) ∈ Z3
(Berwick 1925)
Roots of f (x) (Cardano 1545):
ζ iλ1/3 + ζ−iλ1/3
(i = 0, 1, 2)
where ζ is a primitive cube root of unity
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 3 / 32
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Example: D = 44806173
Naively(take λ to be the fundamental unit of Q(
√−3 · 44806173)
):
f (x) = x3 − 3x + 9631353811877867340405658366
Using CUFFQI (all 13 cubic fields with D = 44806173):
f1(x) = x3 − 61x2 + 697x − 330
f2(x) = x3 − 279x2 + 441x − 170
f3(x) = x3 − 63x2 + 423x − 8
f4(x) = x3 − 69x2 + 435x − 216
f5(x) = x3 − 63x2 + 603x − 494
f6(x) = x3 − 83x2 + 297x − 54
f7(x) = x3 − 63x2 + 837x − 494
f8(x) = x3 − 257x2 + 477x − 216
f9(x) = x3 − 87x2 + 273x − 36
f10(x) = x3 − 62x2 + 546x − 261
f11(x) = x3 − 60x2 + 660x − 97
f12(x) = x3 − 165x2 + 273x − 90
f13(x) = x3 − 127x2 + 185x − 62
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 4 / 32
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Example: D = 44806173
Naively(
take λ to be the fundamental unit of Q(√−3 · 44806173)
):
f (x) = x3 − 3x + 9631353811877867340405658366
Using CUFFQI (all 13 cubic fields with D = 44806173):
f1(x) = x3 − 61x2 + 697x − 330
f2(x) = x3 − 279x2 + 441x − 170
f3(x) = x3 − 63x2 + 423x − 8
f4(x) = x3 − 69x2 + 435x − 216
f5(x) = x3 − 63x2 + 603x − 494
f6(x) = x3 − 83x2 + 297x − 54
f7(x) = x3 − 63x2 + 837x − 494
f8(x) = x3 − 257x2 + 477x − 216
f9(x) = x3 − 87x2 + 273x − 36
f10(x) = x3 − 62x2 + 546x − 261
f11(x) = x3 − 60x2 + 660x − 97
f12(x) = x3 − 165x2 + 273x − 90
f13(x) = x3 − 127x2 + 185x − 62Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 4 / 32
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Cubic Field Construction
Problem with Berwick construction: polynomial coefficients can be HUGE!(E.g. Tr(ε) ≈ ε ≈ exp(
√|D|) for the fundamental unit ε ∈ Q(
√−3D)
)CUFFQI to the rescue!
Goal: for a a given fundamental discriminant D, produce all the cubicfields of discriminant D a la Berwick via generating polynomials with smallcoefficients
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 5 / 32
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Cubic Field Construction
Problem with Berwick construction: polynomial coefficients can be HUGE!(E.g. Tr(ε) ≈ ε ≈ exp(
√|D|) for the fundamental unit ε ∈ Q(
√−3D)
)CUFFQI to the rescue!
Goal: for a a given fundamental discriminant D, produce all the cubicfields of discriminant D a la Berwick via generating polynomials with smallcoefficients
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 5 / 32
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The Berwick MapThere is a map from the set of unordered triples of conjugate cubic fields
{ K, K′, K′′ } disc(K) = D
to the set of unordered pairs of 3-torsion ideal classes
{ [ a ] , [ a ] }
in OQ(√−3D) via
x3 − 3N(λ)1/3x + Tr(λ) 7−→ { [a], [a] } where a3 = (λ)
For D > 0:
bijection onto non-principal ideal classesnothing maps to the principal class
For D < 0:
3-to-1 onto non-principal ideal classes1-to-1 onto to the principal class
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 6 / 32
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The Berwick MapThere is a map from the set of unordered triples of conjugate cubic fields
{ K, K′, K′′ } disc(K) = D
to the set of unordered pairs of 3-torsion ideal classes
{ [ a ] , [ a ] }
in OQ(√−3D) via
x3 − 3N(λ)1/3x + Tr(λ) 7−→ { [a], [a] } where a3 = (λ)
For D > 0:
bijection onto non-principal ideal classesnothing maps to the principal class
For D < 0:
3-to-1 onto non-principal ideal classes1-to-1 onto to the principal class
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 6 / 32
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Some Counting
Put
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)
Number of cubic fields of discriminant D (Hasse 1929):3r − 1
2
Number of cubic fields produced by the Berwick map:
For D > 0:3s − 1
2
For D < 0: 3 · 3s − 1
2+ 1 =
3s+1 − 1
2
Connection between r and s (Scholz 1932):
|r − s| ≤ 1
If r 6= s, then the imaginary quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 7 / 32
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Some Counting
Put
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Number of cubic fields of discriminant D (Hasse 1929):
3r − 1
2
Number of cubic fields produced by the Berwick map:
For D > 0:3s − 1
2
For D < 0: 3 · 3s − 1
2+ 1 =
3s+1 − 1
2
Connection between r and s (Scholz 1932):
|r − s| ≤ 1
If r 6= s, then the imaginary quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 7 / 32
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Some Counting
Put
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Number of cubic fields of discriminant D (Hasse 1929):
3r − 1
2
Number of cubic fields produced by the Berwick map:
For D > 0:3s − 1
2
For D < 0: 3 · 3s − 1
2+ 1 =
3s+1 − 1
2
Connection between r and s (Scholz 1932):
|r − s| ≤ 1
If r 6= s, then the imaginary quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 7 / 32
![Page 17: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/17.jpg)
Some Counting
Put
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Number of cubic fields of discriminant D (Hasse 1929):
3r − 1
2
Number of cubic fields produced by the Berwick map:
For D > 0:3s − 1
2
For D < 0: 3 · 3s − 1
2+ 1 =
3s+1 − 1
2
Connection between r and s (Scholz 1932):
|r − s| ≤ 1
If r 6= s, then the imaginary quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 7 / 32
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More Counting
Case D > 0:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case D < 0:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the complete collection of cubic fields of discriminant
9D if 3 | D , 81D if 3 - D
In the , cases there are no fields of these discriminants
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 8 / 32
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More Counting
Case D > 0:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case D < 0:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the complete collection of cubic fields of discriminant
9D if 3 | D , 81D if 3 - D
In the , cases there are no fields of these discriminants
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 8 / 32
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More Counting
Case D > 0:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case D < 0:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the complete collection of cubic fields of discriminant
9D if 3 | D , 81D if 3 - D
In the , cases there are no fields of these discriminants
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 8 / 32
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More Counting
Case D > 0:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case D < 0:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the complete collection of cubic fields of discriminant
9D if 3 | D , 81D if 3 - D
In the , cases there are no fields of these discriminantsRenate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 8 / 32
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Berwick Construction Algorithm
Input: D and a basis of Cl(Q(√−3D)[3](
For D < 0, also the regulator R of Q(√−3D)
)Output: generating polynomials of all cubic fields of discriminant D
Algorithm:
For each basis class C of Cl(Q(√−3D)[3], collect generators λ of
one ideal in C whose cube has a small generator when D > 0
three ideals in C whose cube has a small generator when D < 0
Collect a small element λ (/∈ Z) in some principal ideal when D < 0
For each λ collected
compute f (x) = x3 − 3N(λ)1/3x + Tr(λ)
if disc(f ) = D, output f (x)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 9 / 32
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Reduced Ideals
An ideal a in OQ(√−3D) is reduced if no non-zero element α ∈ a satisfies
|α| < N(a) and |α| < N(a)
If a is reduced, then
N(a) <
{√|D ′|/3 when D ′ < 0√D ′ when D ′ > 0
where D ′ = −D/3 when 3 | D and D ′ = −3D when 3 - D.
If a is reduced and a3 = (λ), then
N(λ) <
{(|D ′|/3)3/2 when D ′ < 0
(D ′)3/2 when D ′ > 0
Hence, to get λ of small norm, use reduced ideals (exist in every idealclass)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 10 / 32
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Reduced Ideals
An ideal a in OQ(√−3D) is reduced if no non-zero element α ∈ a satisfies
|α| < N(a) and |α| < N(a)
If a is reduced, then
N(a) <
{√|D ′|/3 when D ′ < 0√D ′ when D ′ > 0
where D ′ = −D/3 when 3 | D and D ′ = −3D when 3 - D.
If a is reduced and a3 = (λ), then
N(λ) <
{(|D ′|/3)3/2 when D ′ < 0
(D ′)3/2 when D ′ > 0
Hence, to get λ of small norm, use reduced ideals (exist in every idealclass)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 10 / 32
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Reduced Ideals
An ideal a in OQ(√−3D) is reduced if no non-zero element α ∈ a satisfies
|α| < N(a) and |α| < N(a)
If a is reduced, then
N(a) <
{√|D ′|/3 when D ′ < 0√D ′ when D ′ > 0
where D ′ = −D/3 when 3 | D and D ′ = −3D when 3 - D.
If a is reduced and a3 = (λ), then
N(λ) <
{(|D ′|/3)3/2 when D ′ < 0
(D ′)3/2 when D ′ > 0
Hence, to get λ of small norm, use reduced ideals (exist in every idealclass)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 10 / 32
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Generators λ of Small Trace, D ′ < 0
Here, the reduced ideal a is unique.
Write λ =A + B
√D ′
2(A,B ∈ Z). Then
4N(λ) = A2 − B2D ′ = A2 + B2|D ′|
N(λ) < (|D ′|/3)3/2 implies
|Tr(λ)| = |A| < 1
2
(|D ′|
3
)3/4
Happily, the reduced ideal also yields a small trace!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 11 / 32
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Generators λ of Small Trace, D ′ < 0
Here, the reduced ideal a is unique.
Write λ =A + B
√D ′
2(A,B ∈ Z). Then
4N(λ) = A2 − B2D ′ = A2 + B2|D ′|
N(λ) < (|D ′|/3)3/2 implies
|Tr(λ)| = |A| < 1
2
(|D ′|
3
)3/4
Happily, the reduced ideal also yields a small trace!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 11 / 32
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Generators λ of Small Trace, D ′ < 0
Here, the reduced ideal a is unique.
Write λ =A + B
√D ′
2(A,B ∈ Z). Then
4N(λ) = A2 − B2D ′ = A2 + B2|D ′|
N(λ) < (|D ′|/3)3/2 implies
|Tr(λ)| = |A| < 1
2
(|D ′|
3
)3/4
Happily, the reduced ideal also yields a small trace!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 11 / 32
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Generators λ of Small Trace, D ′ < 0
Here, the reduced ideal a is unique.
Write λ =A + B
√D ′
2(A,B ∈ Z). Then
4N(λ) = A2 − B2D ′ = A2 + B2|D ′|
N(λ) < (|D ′|/3)3/2 implies
|Tr(λ)| = |A| < 1
2
(|D ′|
3
)3/4
Happily, the reduced ideal also yields a small trace!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 11 / 32
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Infrastructures, D ′ > 0
For any ideal class C, the infrastructure of the C is the collection of allreduced ideals in C (Shanks 1972)
Infrastructures are finite.
Can move from one infrastructure ideal a to its neighbour ρ(a) viaone step in a simple continued fraction expansion
Infrastructure ideals are discretely spaced on a circle ofcircumference R, the regulator of Q(
√D ′)
For any point P on the circle, there is a unique reduced ideal closestto P (efficiently computable)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 12 / 32
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Infrastructures, D ′ > 0
For any ideal class C, the infrastructure of the C is the collection of allreduced ideals in C (Shanks 1972)
Infrastructures are finite.
Can move from one infrastructure ideal a to its neighbour ρ(a) viaone step in a simple continued fraction expansion
Infrastructure ideals are discretely spaced on a circle ofcircumference R, the regulator of Q(
√D ′)
For any point P on the circle, there is a unique reduced ideal closestto P (efficiently computable)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 12 / 32
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Infrastructures, D ′ > 0
r ρ(r)ρ2(r)
ρ3(r)
ρi(r)
ρn-1(r)r
a
ρ(a)P
Infrastructure of C = [r] a is closest to P
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Suitable Reduced Ideals, D ′ < 0
λ ∈ OQ(√D′) is small if
1 < λ < (D ′)3/2 , |N(λ)| < (D ′)3/2
The following reduced ideals have cubes with small generators (Shanks):
For the principal ideal class, the reduced ideal closest to
R
3+
log(D ′)
4
For any non-principal ideal class C, the three reduced ideals closest to
d ,R
3+ d ,
2R
3+ d
where 0 < d < R/3 and z can be explicitly computed
(z depends on the representative of C)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 14 / 32
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Suitable Reduced Ideals, D ′ < 0
λ ∈ OQ(√D′) is small if
1 < λ < (D ′)3/2 , |N(λ)| < (D ′)3/2
The following reduced ideals have cubes with small generators (Shanks):
For the principal ideal class, the reduced ideal closest to
R
3+
log(D ′)
4
For any non-principal ideal class C, the three reduced ideals closest to
d ,R
3+ d ,
2R
3+ d
where 0 < d < R/3 and z can be explicitly computed
(z depends on the representative of C)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 14 / 32
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Suitable Reduced Ideals, D ′ < 0
λ ∈ OQ(√D′) is small if
1 < λ < (D ′)3/2 , |N(λ)| < (D ′)3/2
The following reduced ideals have cubes with small generators (Shanks):
For the principal ideal class, the reduced ideal closest to
R
3+
log(D ′)
4
For any non-principal ideal class C, the three reduced ideals closest to
d ,R
3+ d ,
2R
3+ d
where 0 < d < R/3 and z can be explicitly computed
(z depends on the representative of C)
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 14 / 32
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Suitable Reduced Ideals, D ′ < 0
(1)
a0
a3
r
a2
a1
Principal infrastructure Non-principal infrastructures
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Finding Small Generators, D ′ > 0
Shanks’ strategy for finding λ (or λ):
Search the infrastructures of [a] and of [ a ] simultaneouslyto find λ or λ
The two infrastructures are mirror images of each other
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 16 / 32
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Fung’s Work
In his 1990 PhD dissertation, Fung
translated CUFFQI from Shanksian into a form suitable forcomputation
implemented CUFFQI in Fortran on an Amdahl 5870 mainframecomputer
produced a number of examples, including the
36 − 1
2= 364
cubic fields of the 19-digit discriminant
D = −3161659186633662283
in under 3 CPU minutes
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 17 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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CUFFFQI — Function Fields
Jacobson, Lee, S. and Williams, Int. J. Number Theory 11 (2015)
Dictionary:
Q→ Fq(t), q a prime power, gcd(q, 6) = 1
Z→ Fq[x ]
D → D(t) ∈ Fq[t] square-free
K = Fq(t, x), [K : Fq(t)] = 3
minimal polynomial f (x) = x3 − 3N(λ)1/3x + Tr(λ) ∈ Fq[t, x ]
R→ Fq((x−1))
C→ Fq2((x−1)) or Fq((x−1/2))
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 18 / 32
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Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
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Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
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Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
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Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
![Page 51: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/51.jpg)
Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
![Page 52: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/52.jpg)
Problems
Infinite place of Fq(t) is archimedian — can decompose in any way
f (x) need not have a root in Fq((x−1))
Fq(t,√−3D) = Fq(t,
√D) if q ≡ 1 (mod 3)
Extra fields? Fq(t,√D) = Fq(t,
√9D) = Fq(t,
√81D)
Hasse count is wrong
There are three types of quadratic fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 19 / 32
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Quadratic Function Fields
Let D(t) ∈ Fq[t] be squarefree
Let sgn(D) ∈ F∗q denote the leading coefficient of D(t).
Fq(t,√D) is
imaginary if deg(D) is odd
infinite place of Fq(t) ramifies
real if deg(D) is even and sgn(D) is a square in Fq
infinite place of Fq(t) splits
unusual if deg(D) is even and sgn(D) is a non-square in Fq
infinite place of Fq(t) is inert – no number field analogue!
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Quadratic Function Fields
Let D(t) ∈ Fq[t] be squarefree
Let sgn(D) ∈ F∗q denote the leading coefficient of D(t).
Fq(t,√D) is
imaginary if deg(D) is odd
infinite place of Fq(t) ramifies
real if deg(D) is even and sgn(D) is a square in Fq
infinite place of Fq(t) splits
unusual if deg(D) is even and sgn(D) is a non-square in Fq
infinite place of Fq(t) is inert – no number field analogue!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 20 / 32
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Quadratic Function Fields
Let D(t) ∈ Fq[t] be squarefree
Let sgn(D) ∈ F∗q denote the leading coefficient of D(t).
Fq(t,√D) is
imaginary if deg(D) is odd
infinite place of Fq(t) ramifies
real if deg(D) is even and sgn(D) is a square in Fq
infinite place of Fq(t) splits
unusual if deg(D) is even and sgn(D) is a non-square in Fq
infinite place of Fq(t) is inert – no number field analogue!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 20 / 32
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Quadratic Function Fields
Let D(t) ∈ Fq[t] be squarefree
Let sgn(D) ∈ F∗q denote the leading coefficient of D(t).
Fq(t,√D) is
imaginary if deg(D) is odd
infinite place of Fq(t) ramifies
real if deg(D) is even and sgn(D) is a square in Fq
infinite place of Fq(t) splits
unusual if deg(D) is even and sgn(D) is a non-square in Fq
infinite place of Fq(t) is inert – no number field analogue!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 20 / 32
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Decomposition at Infinity in K
Let K be a cubic extension of Fq(t) of square-free discriminant D ∈ Fq[t]
Let ∞ denote the place at infinity in Fq(t).
deg(D) odd: ∞ = pq2 in K
deg(D) even:
q ≡ 1 (mod 3):
sgn(D) = 2: ∞ = pqr or p3 or p in Ksgn(D) 6= 2: ∞ = pq in K
q ≡ −1 (mod 3):
sgn(D) = 2: ∞ = pqr or p in Ksgn(D) 6= 2: ∞ = pq or p3 in K
Hasse count does not include the red cases.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 21 / 32
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Decomposition at Infinity in K
Let K be a cubic extension of Fq(t) of square-free discriminant D ∈ Fq[t]
Let ∞ denote the place at infinity in Fq(t).
deg(D) odd: ∞ = pq2 in K
deg(D) even:
q ≡ 1 (mod 3):
sgn(D) = 2: ∞ = pqr or p3 or p in Ksgn(D) 6= 2: ∞ = pq in K
q ≡ −1 (mod 3):
sgn(D) = 2: ∞ = pqr or p in Ksgn(D) 6= 2: ∞ = pq or p3 in K
Hasse count does not include the red cases.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 21 / 32
![Page 59: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/59.jpg)
Decomposition at Infinity in K
Let K be a cubic extension of Fq(t) of square-free discriminant D ∈ Fq[t]
Let ∞ denote the place at infinity in Fq(t).
deg(D) odd: ∞ = pq2 in K
deg(D) even:
q ≡ 1 (mod 3):
sgn(D) = 2: ∞ = pqr or p3 or p in Ksgn(D) 6= 2: ∞ = pq in K
q ≡ −1 (mod 3):
sgn(D) = 2: ∞ = pqr or p in Ksgn(D) 6= 2: ∞ = pq or p3 in K
Hasse count does not include the red cases.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 21 / 32
![Page 60: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/60.jpg)
Decomposition at Infinity in K
Let K be a cubic extension of Fq(t) of square-free discriminant D ∈ Fq[t]
Let ∞ denote the place at infinity in Fq(t).
deg(D) odd: ∞ = pq2 in K
deg(D) even:
q ≡ 1 (mod 3):
sgn(D) = 2: ∞ = pqr or p3 or p in Ksgn(D) 6= 2: ∞ = pq in K
q ≡ −1 (mod 3):
sgn(D) = 2: ∞ = pqr or p in Ksgn(D) 6= 2: ∞ = pq or p3 in K
Hasse count does not include the red cases.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 21 / 32
![Page 61: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/61.jpg)
Decomposition at Infinity in K
Let K be a cubic extension of Fq(t) of square-free discriminant D ∈ Fq[t]
Let ∞ denote the place at infinity in Fq(t).
deg(D) odd: ∞ = pq2 in K
deg(D) even:
q ≡ 1 (mod 3):
sgn(D) = 2: ∞ = pqr or p3 or p in Ksgn(D) 6= 2: ∞ = pq in K
q ≡ −1 (mod 3):
sgn(D) = 2: ∞ = pqr or p in Ksgn(D) 6= 2: ∞ = pq or p3 in K
Hasse count does not include the red cases.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 21 / 32
![Page 62: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/62.jpg)
The Berwick Map
As before, triples of conjugate cubic function fields are mapped onto pairsof 3-torsion ideal classes in Fq[t,
√D].
For Fq(t,√−3D) imaginary or unusual:
bijection onto non-principal ideal classesnothing maps to the principal class
For Fq(t,√−3D) real:
3-to-1 onto non-principal ideal classes1-to-1 onto to the principal class
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 22 / 32
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The Berwick Map
As before, triples of conjugate cubic function fields are mapped onto pairsof 3-torsion ideal classes in Fq[t,
√D].
For Fq(t,√−3D) imaginary or unusual:
bijection onto non-principal ideal classesnothing maps to the principal class
For Fq(t,√−3D) real:
3-to-1 onto non-principal ideal classes1-to-1 onto to the principal class
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 22 / 32
![Page 64: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/64.jpg)
Some CountingPut
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Same field unless deg(D) even and q ≡ −1 (mod 3)
Number of cubic fields of discriminant D with at least two infinite places:
3r − 1
2
Number of cubic fields produced by the Berwick map:
For Fq(t,√−3D) imaginary or unusual:
3s − 1
2
For Fq(t,√−3D) real:
3s+1 − 1
2
Connection between r and s (Lee 2007):
|r − s| ≤ 1If r 6= s, then the unusual quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 23 / 32
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Some CountingPut
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Same field unless deg(D) even and q ≡ −1 (mod 3)
Number of cubic fields of discriminant D with at least two infinite places:
3r − 1
2
Number of cubic fields produced by the Berwick map:
For Fq(t,√−3D) imaginary or unusual:
3s − 1
2
For Fq(t,√−3D) real:
3s+1 − 1
2
Connection between r and s (Lee 2007):
|r − s| ≤ 1If r 6= s, then the unusual quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 23 / 32
![Page 66: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/66.jpg)
Some CountingPut
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Same field unless deg(D) even and q ≡ −1 (mod 3)
Number of cubic fields of discriminant D with at least two infinite places:
3r − 1
2
Number of cubic fields produced by the Berwick map:
For Fq(t,√−3D) imaginary or unusual:
3s − 1
2
For Fq(t,√−3D) real:
3s+1 − 1
2
Connection between r and s (Lee 2007):
|r − s| ≤ 1If r 6= s, then the unusual quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 23 / 32
![Page 67: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/67.jpg)
Some CountingPut
r = 3-rank(Cl(Q(
√D))
s = 3-rank(Cl(Q(
√−3D)
)Same field unless deg(D) even and q ≡ −1 (mod 3)
Number of cubic fields of discriminant D with at least two infinite places:
3r − 1
2
Number of cubic fields produced by the Berwick map:
For Fq(t,√−3D) imaginary or unusual:
3s − 1
2
For Fq(t,√−3D) real:
3s+1 − 1
2
Connection between r and s (Lee 2007):
|r − s| ≤ 1If r 6= s, then the unusual quadratic field has the bigger 3-rank
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 23 / 32
![Page 68: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/68.jpg)
More Counting
If Fq(t,√D) = Fq(t,
√−3D) (imaginary or real), then r = s ,
Case Fq(t,√−3D) unusual, Fq(t,
√D) real:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case Fq(t,√−3D) real, Fq(t,
√D) unusual:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the fields with one infinite place that are missing fromthe Hasse count. In the , cases, there are no such fields.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 24 / 32
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More Counting
If Fq(t,√D) = Fq(t,
√−3D) (imaginary or real), then r = s ,
Case Fq(t,√−3D) unusual, Fq(t,
√D) real:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case Fq(t,√−3D) real, Fq(t,
√D) unusual:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the fields with one infinite place that are missing fromthe Hasse count. In the , cases, there are no such fields.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 24 / 32
![Page 70: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/70.jpg)
More Counting
If Fq(t,√D) = Fq(t,
√−3D) (imaginary or real), then r = s ,
Case Fq(t,√−3D) unusual, Fq(t,
√D) real:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case Fq(t,√−3D) real, Fq(t,
√D) unusual:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the fields with one infinite place that are missing fromthe Hasse count. In the , cases, there are no such fields.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 24 / 32
![Page 71: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/71.jpg)
More Counting
If Fq(t,√D) = Fq(t,
√−3D) (imaginary or real), then r = s ,
Case Fq(t,√−3D) unusual, Fq(t,
√D) real:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case Fq(t,√−3D) real, Fq(t,
√D) unusual:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the fields with one infinite place that are missing fromthe Hasse count. In the , cases, there are no such fields.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 24 / 32
![Page 72: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/72.jpg)
More Counting
If Fq(t,√D) = Fq(t,
√−3D) (imaginary or real), then r = s ,
Case Fq(t,√−3D) unusual, Fq(t,
√D) real:
r = s:3s − 1
2=
3r − 1
2,
r = s − 1:3s − 1
2=
3r − 1
2+ 3r /
Case Fq(t,√−3D) real, Fq(t,
√D) unusual:
r = s:3s+1 − 1
2=
3r − 1
2+ 3r /
r = s + 1:3s+1 − 1
2=
3r − 1
2,
So what are these extra 3r cubic fields?
Answer: they are the fields with one infinite place that are missing fromthe Hasse count. In the , cases, there are no such fields.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 24 / 32
![Page 73: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/73.jpg)
Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋
An ideal a in Fq(t,√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Reduced Ideals
The genus of Fq(t,√−3D) is
⌊deg(D)− 1
2
⌋An ideal a in Fq(t,
√−3D) is reduced if deg(N(a)) ≤ g
Equivalent: |N(a)| <√|D| where | · | = qdeg(·)
Every ideal class of Fq[t,√−3D] contains
a unique reduced ideal when Fq(t,√−3D) is imaginary
either a unique reduced ideal or q + 1 “almost” reduced ideals(degree g + 1) when Fq(t,
√−3D) is unusual (Artin 1924)
many reduced ideals when Fq(t,√−3D) is real.
(Almost) reduced ideals produce λ with small norm: |N(λ)| ≤ |D|3/2
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 25 / 32
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Generators λ of Small Trace
Suppose Fq(t,√−3D) is imaginary or unusual
Write λ = A + B√−3D (A,B ∈ Fq[t]). Then
N(λ) = A2 + 3B2D
If deg(D) is odd, or deg(D) is even and sgn(−3D) 6= 2, then there is nocancellation of leading coefficients on the right hand side.
|N(λ)| ≤ |D|3/2 implies
|Tr(λ)| = |A| ≤ |N(λ)|1/2 ≤ |D|3/4
Yields again a small trace.
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Generators λ of Small Trace
Suppose Fq(t,√−3D) is imaginary or unusual
Write λ = A + B√−3D (A,B ∈ Fq[t]). Then
N(λ) = A2 + 3B2D
If deg(D) is odd, or deg(D) is even and sgn(−3D) 6= 2, then there is nocancellation of leading coefficients on the right hand side.
|N(λ)| ≤ |D|3/2 implies
|Tr(λ)| = |A| ≤ |N(λ)|1/2 ≤ |D|3/4
Yields again a small trace.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 26 / 32
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Generators λ of Small Trace
Suppose Fq(t,√−3D) is imaginary or unusual
Write λ = A + B√−3D (A,B ∈ Fq[t]). Then
N(λ) = A2 + 3B2D
If deg(D) is odd, or deg(D) is even and sgn(−3D) 6= 2, then there is nocancellation of leading coefficients on the right hand side.
|N(λ)| ≤ |D|3/2 implies
|Tr(λ)| = |A| ≤ |N(λ)|1/2 ≤ |D|3/4
Yields again a small trace.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 26 / 32
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Generators λ of Small Trace
Suppose Fq(t,√−3D) is imaginary or unusual
Write λ = A + B√−3D (A,B ∈ Fq[t]). Then
N(λ) = A2 + 3B2D
If deg(D) is odd, or deg(D) is even and sgn(−3D) 6= 2, then there is nocancellation of leading coefficients on the right hand side.
|N(λ)| ≤ |D|3/2 implies
|Tr(λ)| = |A| ≤ |N(λ)|1/2 ≤ |D|3/4
Yields again a small trace.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 26 / 32
![Page 84: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/84.jpg)
Generators λ of Small Trace
Suppose Fq(t,√−3D) is imaginary or unusual
Write λ = A + B√−3D (A,B ∈ Fq[t]). Then
N(λ) = A2 + 3B2D
If deg(D) is odd, or deg(D) is even and sgn(−3D) 6= 2, then there is nocancellation of leading coefficients on the right hand side.
|N(λ)| ≤ |D|3/2 implies
|Tr(λ)| = |A| ≤ |N(λ)|1/2 ≤ |D|3/4
Yields again a small trace.
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 26 / 32
![Page 85: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/85.jpg)
Generators λ of Small Trace (Cont’d)
Suppose Fq(t,√−3D) is real
Same infrastructure framework (Stein 1992)
Can also use arithmetic in the divisor class group of Fq(t,√−3D) via
balanced divisors (Galbraith, Harrison, Mireles Morales 2008)
λ small: deg(Tr(λ)) ≤ 3g + 1, deg(N(λ)) ≤ 3g
Principal class: take reduced ideal closest to dR/3 + g/2cNon-principal classes: take ideals closest to d , R/3 + d , 2R/3 + dwhere −g/2 ≤ d < R/3− g/2 and d can be explicitly computedusing integer arithmetic only!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 27 / 32
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Generators λ of Small Trace (Cont’d)
Suppose Fq(t,√−3D) is real
Same infrastructure framework (Stein 1992)
Can also use arithmetic in the divisor class group of Fq(t,√−3D) via
balanced divisors (Galbraith, Harrison, Mireles Morales 2008)
λ small: deg(Tr(λ)) ≤ 3g + 1, deg(N(λ)) ≤ 3g
Principal class: take reduced ideal closest to dR/3 + g/2cNon-principal classes: take ideals closest to d , R/3 + d , 2R/3 + dwhere −g/2 ≤ d < R/3− g/2 and d can be explicitly computedusing integer arithmetic only!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 27 / 32
![Page 87: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/87.jpg)
Generators λ of Small Trace (Cont’d)
Suppose Fq(t,√−3D) is real
Same infrastructure framework (Stein 1992)
Can also use arithmetic in the divisor class group of Fq(t,√−3D) via
balanced divisors (Galbraith, Harrison, Mireles Morales 2008)
λ small: deg(Tr(λ)) ≤ 3g + 1, deg(N(λ)) ≤ 3g
Principal class: take reduced ideal closest to dR/3 + g/2cNon-principal classes: take ideals closest to d , R/3 + d , 2R/3 + dwhere −g/2 ≤ d < R/3− g/2 and d can be explicitly computedusing integer arithmetic only!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 27 / 32
![Page 88: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/88.jpg)
Generators λ of Small Trace (Cont’d)
Suppose Fq(t,√−3D) is real
Same infrastructure framework (Stein 1992)
Can also use arithmetic in the divisor class group of Fq(t,√−3D) via
balanced divisors (Galbraith, Harrison, Mireles Morales 2008)
λ small: deg(Tr(λ)) ≤ 3g + 1, deg(N(λ)) ≤ 3g
Principal class: take reduced ideal closest to dR/3 + g/2cNon-principal classes: take ideals closest to d , R/3 + d , 2R/3 + dwhere −g/2 ≤ d < R/3− g/2 and d can be explicitly computedusing integer arithmetic only!
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 27 / 32
![Page 89: Dan Shanks’ CUFFQI Algorithm Resurrected · 2018-11-02 · Dan Shanks’ CUFFQI Algorithm Resurrected Renate Scheidler rscheidl@ucalgary.ca Celebrating 75 Years of Mathematics of](https://reader033.vdocument.in/reader033/viewer/2022041611/5e37a9b56559331798290632/html5/thumbnails/89.jpg)
Example — Different 3-Rank
q = 11, D(x) = 7x10 + x7 + 3x6 + 2x5 + 7x4 + 8x3 + 4x2 + 2x
r = 3, s = 2 ⇒ (33 − 1)/2 = 13 fields, all with ∞ = pq in K.
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 5t3 + 10t + 4 4t6 + t5 + t3 + 9t2 + 6t + 4
2 10t4 + 9t3 + t2 + 5t + 9 10t6 + 8t5 + 5t3 + 5t2 + 5t + 3
3 6t4 + 4t3 + 10t + 4 5t6 + 4t5 + 3t4 + 5t3 + 3t2 + t + 7
4 9t4 + 4t3 + 6t2 + 5t + 1 t6 + 4t5 + 8t4 + 9t3 + 4t2 + 7t + 5
5 4t4 + 7t3 + 10t2 + 5t + 4 6t6 + 6t5 + 4t4 + 4t3 + 8t2 + 10t + 4
6 9t3 + 4t2 + 8t + 9 t6 + 3t5 + 3t3 + 6t + 3
7 t4 + 3t3 + 9t + 3 t6 + 2t5 + 2t4 + 3t3 + 6t2 + 3t + 2
8 t4 + 8t3 + 6t2 + 3t + 1 t6 + 9t5 + 7t4 + 4t3 + 6t2 + 3t + 6
9 7t4 + 4t3 + 9t2 + 6t 9t6 + 10t5 + 10t4 + 9t3 + 6t2
10 6t4 + 4t3 + 5t2 + 9t + 4 5t6 + 10t4 + 2t3 + 5t2 + 8t + 7
11 3t4 + 5t3 + 4t2 + 6t + 9 8t6 + 10t5 + 4t4 + 4t3 + 8t2 + 2t + 3
12 5t4 + 6t2 + 8t + 9 2t6 + 10t5 + 3t4 + t3 + t2 + 10t + 3
13 4t4 + 3t3 + 5t2 + 10t + 9 8t6 + 5t4 + 3t3 + 9t2 + t + 3
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Example — Different 3-Rank
q = 11, D(x) = 7x10 + x7 + 3x6 + 2x5 + 7x4 + 8x3 + 4x2 + 2x
r = 3, s = 2 ⇒ (33 − 1)/2 = 13 fields, all with ∞ = pq in K.
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 5t3 + 10t + 4 4t6 + t5 + t3 + 9t2 + 6t + 4
2 10t4 + 9t3 + t2 + 5t + 9 10t6 + 8t5 + 5t3 + 5t2 + 5t + 3
3 6t4 + 4t3 + 10t + 4 5t6 + 4t5 + 3t4 + 5t3 + 3t2 + t + 7
4 9t4 + 4t3 + 6t2 + 5t + 1 t6 + 4t5 + 8t4 + 9t3 + 4t2 + 7t + 5
5 4t4 + 7t3 + 10t2 + 5t + 4 6t6 + 6t5 + 4t4 + 4t3 + 8t2 + 10t + 4
6 9t3 + 4t2 + 8t + 9 t6 + 3t5 + 3t3 + 6t + 3
7 t4 + 3t3 + 9t + 3 t6 + 2t5 + 2t4 + 3t3 + 6t2 + 3t + 2
8 t4 + 8t3 + 6t2 + 3t + 1 t6 + 9t5 + 7t4 + 4t3 + 6t2 + 3t + 6
9 7t4 + 4t3 + 9t2 + 6t 9t6 + 10t5 + 10t4 + 9t3 + 6t2
10 6t4 + 4t3 + 5t2 + 9t + 4 5t6 + 10t4 + 2t3 + 5t2 + 8t + 7
11 3t4 + 5t3 + 4t2 + 6t + 9 8t6 + 10t5 + 4t4 + 4t3 + 8t2 + 2t + 3
12 5t4 + 6t2 + 8t + 9 2t6 + 10t5 + 3t4 + t3 + t2 + 10t + 3
13 4t4 + 3t3 + 5t2 + 10t + 9 8t6 + 5t4 + 3t3 + 9t2 + t + 3
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Example — Different 3-Rank
q = 11, D(x) = 7x10 + x7 + 3x6 + 2x5 + 7x4 + 8x3 + 4x2 + 2x
r = 3, s = 2 ⇒ (33 − 1)/2 = 13 fields, all with ∞ = pq in K.
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 5t3 + 10t + 4 4t6 + t5 + t3 + 9t2 + 6t + 4
2 10t4 + 9t3 + t2 + 5t + 9 10t6 + 8t5 + 5t3 + 5t2 + 5t + 3
3 6t4 + 4t3 + 10t + 4 5t6 + 4t5 + 3t4 + 5t3 + 3t2 + t + 7
4 9t4 + 4t3 + 6t2 + 5t + 1 t6 + 4t5 + 8t4 + 9t3 + 4t2 + 7t + 5
5 4t4 + 7t3 + 10t2 + 5t + 4 6t6 + 6t5 + 4t4 + 4t3 + 8t2 + 10t + 4
6 9t3 + 4t2 + 8t + 9 t6 + 3t5 + 3t3 + 6t + 3
7 t4 + 3t3 + 9t + 3 t6 + 2t5 + 2t4 + 3t3 + 6t2 + 3t + 2
8 t4 + 8t3 + 6t2 + 3t + 1 t6 + 9t5 + 7t4 + 4t3 + 6t2 + 3t + 6
9 7t4 + 4t3 + 9t2 + 6t 9t6 + 10t5 + 10t4 + 9t3 + 6t2
10 6t4 + 4t3 + 5t2 + 9t + 4 5t6 + 10t4 + 2t3 + 5t2 + 8t + 7
11 3t4 + 5t3 + 4t2 + 6t + 9 8t6 + 10t5 + 4t4 + 4t3 + 8t2 + 2t + 3
12 5t4 + 6t2 + 8t + 9 2t6 + 10t5 + 3t4 + t3 + t2 + 10t + 3
13 4t4 + 3t3 + 5t2 + 10t + 9 8t6 + 5t4 + 3t3 + 9t2 + t + 3
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Example — Same 3-Rank
q = 11, D(x) = 2x8 + x6 + 5x4 + 6x2 + 7
r = s = 2⇒{
(32 − 1)/2 = 4 fields with ∞ = pq in K32 = 9 fields with ∞ = p3 in K
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 9t2 + 6 t6 + 7t4 + 6t2
2 7t3 + 7t + 8 6t6 + 7t5 + 8t4 + 5t3 + 4t2 + 4
3 9t3 + 3t2 + 8t + 1 2t6 + 6t5 + 6t4 + t3 + 5t + 5
4 9t3 + 2t2 + 8t + 4 4t6 + 6t5 + 4t3 + 3t2 + t + 5
5 4t3 + 4t2 + 6t + 2 10t5 + 4t4 + 8t3 + 10t
6 5t2 + 8t + 5 2t5 + 6t3 + 2t + 10
7 10t3 + 5t2 + 5t + 1 8t5 + 6t4 + 6t3 + 9t2 + t + 6
8 5t2 + 3t + 5 9t5 + 5t3 + 9t + 10
9 t3 + 5t2 + 6t + 1 8t5 + 5t4 + 6t3 + 2t2 + t + 5
10 7t3 + 4t2 + 5t + 2 10t5 + 7t4 + 8t3 + 10t
11 5t2 + 1 10t4 + 2t2 + 1
12 3t2 + 4 10t4 + 6t2 + 6
13 3t2 10t4 + 6t2 + 3
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Example — Same 3-Rank
q = 11, D(x) = 2x8 + x6 + 5x4 + 6x2 + 7
r = s = 2⇒{
(32 − 1)/2 = 4 fields with ∞ = pq in K32 = 9 fields with ∞ = p3 in K
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 9t2 + 6 t6 + 7t4 + 6t2
2 7t3 + 7t + 8 6t6 + 7t5 + 8t4 + 5t3 + 4t2 + 4
3 9t3 + 3t2 + 8t + 1 2t6 + 6t5 + 6t4 + t3 + 5t + 5
4 9t3 + 2t2 + 8t + 4 4t6 + 6t5 + 4t3 + 3t2 + t + 5
5 4t3 + 4t2 + 6t + 2 10t5 + 4t4 + 8t3 + 10t
6 5t2 + 8t + 5 2t5 + 6t3 + 2t + 10
7 10t3 + 5t2 + 5t + 1 8t5 + 6t4 + 6t3 + 9t2 + t + 6
8 5t2 + 3t + 5 9t5 + 5t3 + 9t + 10
9 t3 + 5t2 + 6t + 1 8t5 + 5t4 + 6t3 + 2t2 + t + 5
10 7t3 + 4t2 + 5t + 2 10t5 + 7t4 + 8t3 + 10t
11 5t2 + 1 10t4 + 2t2 + 1
12 3t2 + 4 10t4 + 6t2 + 6
13 3t2 10t4 + 6t2 + 3
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 29 / 32
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Example — Same 3-Rank
q = 11, D(x) = 2x8 + x6 + 5x4 + 6x2 + 7
r = s = 2⇒{
(32 − 1)/2 = 4 fields with ∞ = pq in K32 = 9 fields with ∞ = p3 in K
f (x) = x3 − S(t)x + T (t) with
# S(t) T (t)
1 9t2 + 6 t6 + 7t4 + 6t2
2 7t3 + 7t + 8 6t6 + 7t5 + 8t4 + 5t3 + 4t2 + 4
3 9t3 + 3t2 + 8t + 1 2t6 + 6t5 + 6t4 + t3 + 5t + 5
4 9t3 + 2t2 + 8t + 4 4t6 + 6t5 + 4t3 + 3t2 + t + 5
5 4t3 + 4t2 + 6t + 2 10t5 + 4t4 + 8t3 + 10t
6 5t2 + 8t + 5 2t5 + 6t3 + 2t + 10
7 10t3 + 5t2 + 5t + 1 8t5 + 6t4 + 6t3 + 9t2 + t + 6
8 5t2 + 3t + 5 9t5 + 5t3 + 9t + 10
9 t3 + 5t2 + 6t + 1 8t5 + 5t4 + 6t3 + 2t2 + t + 5
10 7t3 + 4t2 + 5t + 2 10t5 + 7t4 + 8t3 + 10t
11 5t2 + 1 10t4 + 2t2 + 1
12 3t2 + 4 10t4 + 6t2 + 6
13 3t2 10t4 + 6t2 + 3
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 29 / 32
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BIG Example — Same 3-Rank
q = 125, D = 2x12 + 3x9 + x3 + 1
r = s = 5 ⇒{
(35 − 1)/2 = 121 fields with ∞ = pq in K35 = 243 fields with ∞ = p3 in K
364 fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 30 / 32
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BIG Example — Same 3-Rank
q = 125, D = 2x12 + 3x9 + x3 + 1
r = s = 5 ⇒{
(35 − 1)/2 = 121 fields with ∞ = pq in K35 = 243 fields with ∞ = p3 in K
364 fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 30 / 32
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BIG Example — Same 3-Rank
q = 125, D = 2x12 + 3x9 + x3 + 1
r = s = 5 ⇒{
(35 − 1)/2 = 121 fields with ∞ = pq in K35 = 243 fields with ∞ = p3 in K
364 fields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 30 / 32
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Concluding Remarks
CUFFQI’s run time dominated is dominated by 3-torsion andregulator computation
CUFFQI can be extended to non-fundamental discriminants via basiccass field theory and Kummer theory
I Number Fields: Cohen, Advanced Topics in ComputationalNumber Theory, Ch. 5
I Function Fields: Weir, S & Howe, ANTS-X, 2012 (Dihedraldegree p extensions)
Ideas can be extended to higher degree fields with quadratic resolventfields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 31 / 32
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Concluding Remarks
CUFFQI’s run time dominated is dominated by 3-torsion andregulator computation
CUFFQI can be extended to non-fundamental discriminants via basiccass field theory and Kummer theory
I Number Fields: Cohen, Advanced Topics in ComputationalNumber Theory, Ch. 5
I Function Fields: Weir, S & Howe, ANTS-X, 2012 (Dihedraldegree p extensions)
Ideas can be extended to higher degree fields with quadratic resolventfields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 31 / 32
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Concluding Remarks
CUFFQI’s run time dominated is dominated by 3-torsion andregulator computation
CUFFQI can be extended to non-fundamental discriminants via basiccass field theory and Kummer theory
I Number Fields: Cohen, Advanced Topics in ComputationalNumber Theory, Ch. 5
I Function Fields: Weir, S & Howe, ANTS-X, 2012 (Dihedraldegree p extensions)
Ideas can be extended to higher degree fields with quadratic resolventfields
Renate Scheidler (Calgary) CUFFQI Resurrected ICERM Nov. 1, 2018 31 / 32
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Thank You — Questions?
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