Minimal equation
Minimal equation
Simplified equation
$y^2 + y = -4x^6 - 20x^5 - 19x^4 + 18x^3 + 5x^2 - 3x - 1$ | (homogenize, simplify) |
$y^2 + z^3y = -4x^6 - 20x^5z - 19x^4z^2 + 18x^3z^3 + 5x^2z^4 - 3xz^5 - z^6$ | (dehomogenize, simplify) |
$y^2 = -16x^6 - 80x^5 - 76x^4 + 72x^3 + 20x^2 - 12x - 3$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -3, 5, 18, -19, -20, -4]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -3, 5, 18, -19, -20, -4], R![1]);
sage: X = HyperellipticCurve(R([-3, -12, 20, 72, -76, -80, -16]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(2976\) | \(=\) | \( 2^{5} \cdot 3 \cdot 31 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(2976,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-761856\) | \(=\) | \( - 2^{13} \cdot 3 \cdot 31 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(5144\) | \(=\) | \( 2^{3} \cdot 643 \) |
\( I_4 \) | \(=\) | \(1850140\) | \(=\) | \( 2^{2} \cdot 5 \cdot 92507 \) |
\( I_6 \) | \(=\) | \(2332652300\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 11 \cdot 47 \cdot 45119 \) |
\( I_{10} \) | \(=\) | \(-2976\) | \(=\) | \( - 2^{5} \cdot 3 \cdot 31 \) |
\( J_2 \) | \(=\) | \(10288\) | \(=\) | \( 2^{4} \cdot 643 \) |
\( J_4 \) | \(=\) | \(-523584\) | \(=\) | \( - 2^{6} \cdot 3^{4} \cdot 101 \) |
\( J_6 \) | \(=\) | \(32317696\) | \(=\) | \( 2^{8} \cdot 126241 \) |
\( J_8 \) | \(=\) | \(14586062848\) | \(=\) | \( 2^{11} \cdot 7^{2} \cdot 145349 \) |
\( J_{10} \) | \(=\) | \(-761856\) | \(=\) | \( - 2^{13} \cdot 3 \cdot 31 \) |
\( g_1 \) | \(=\) | \(-14069051982264704/93\) | ||
\( g_2 \) | \(=\) | \(23198934303648/31\) | ||
\( g_3 \) | \(=\) | \(-417553721672/93\) |
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz + 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.13284864.4
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.756854 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.756854 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(13\) | \(1\) | \(1\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(31\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 8 T + 31 T^{2} )\) |
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);