Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = -5x^6 - 34x^5 - 47x^4 + 39x^3 - 6x - 1$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = -5x^6 - 34x^5z - 47x^4z^2 + 39x^3z^3 - 6xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = -20x^6 - 136x^5 - 187x^4 + 158x^3 + 3x^2 - 22x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -6, 0, 39, -47, -34, -5]), R([1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -6, 0, 39, -47, -34, -5], R![1, 1, 1]);
sage: X = HyperellipticCurve(R([-3, -22, 3, 158, -187, -136, -20]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(143039\) | \(=\) | \( 13 \cdot 11003 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-143039\) | \(=\) | \( - 13 \cdot 11003 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(132020\) | \(=\) | \( 2^{2} \cdot 5 \cdot 7 \cdot 23 \cdot 41 \) |
\( I_4 \) | \(=\) | \(1851612817\) | \(=\) | \( 17 \cdot 2657 \cdot 40993 \) |
\( I_6 \) | \(=\) | \(54920704394825\) | \(=\) | \( 5^{2} \cdot 29 \cdot 75752695717 \) |
\( I_{10} \) | \(=\) | \(-18308992\) | \(=\) | \( - 2^{7} \cdot 13 \cdot 11003 \) |
\( J_2 \) | \(=\) | \(33005\) | \(=\) | \( 5 \cdot 7 \cdot 23 \cdot 41 \) |
\( J_4 \) | \(=\) | \(-31761783\) | \(=\) | \( - 3^{2} \cdot 3529087 \) |
\( J_6 \) | \(=\) | \(27758139425\) | \(=\) | \( 5^{2} \cdot 5879 \cdot 188863 \) |
\( J_8 \) | \(=\) | \(-23163366904241\) | \(=\) | \( - 523 \cdot 44289420467 \) |
\( J_{10} \) | \(=\) | \(-143039\) | \(=\) | \( - 13 \cdot 11003 \) |
\( g_1 \) | \(=\) | \(-39165050010611353128125/143039\) | ||
\( g_2 \) | \(=\) | \(1141942103010688147875/143039\) | ||
\( g_3 \) | \(=\) | \(-30237774713788735625/143039\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\) (conditional)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 16xz + 3z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(12xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 16xz + 3z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(12xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 16xz + 3z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 25xz^2 - z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.265982021773.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) (lower bound) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1.0 \) |
Real period: | \( 0.380625 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.522500 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 13 T^{2} )\) | |
\(11003\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 136 T + 11003 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.15.1 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);