Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -x^6 + 16x^5 - 80x^4 + 128x^3 - 64x^2 - 2x + 2$ | (homogenize, simplify) |
$y^2 + xz^2y = -x^6 + 16x^5z - 80x^4z^2 + 128x^3z^3 - 64x^2z^4 - 2xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 64x^5 - 320x^4 + 512x^3 - 255x^2 - 8x + 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -2, -64, 128, -80, 16, -1]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -2, -64, 128, -80, 16, -1], R![0, 1]);
sage: X = HyperellipticCurve(R([8, -8, -255, 512, -320, 64, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(434048\) | \(=\) | \( 2^{7} \cdot 3391 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(434048\) | \(=\) | \( 2^{7} \cdot 3391 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(31808\) | \(=\) | \( 2^{6} \cdot 7 \cdot 71 \) |
\( I_4 \) | \(=\) | \(56868925\) | \(=\) | \( 5^{2} \cdot 467 \cdot 4871 \) |
\( I_6 \) | \(=\) | \(460055176656\) | \(=\) | \( 2^{4} \cdot 3 \cdot 53 \cdot 6581 \cdot 27479 \) |
\( I_{10} \) | \(=\) | \(54256\) | \(=\) | \( 2^{4} \cdot 3391 \) |
\( J_2 \) | \(=\) | \(31808\) | \(=\) | \( 2^{6} \cdot 7 \cdot 71 \) |
\( J_4 \) | \(=\) | \(4243586\) | \(=\) | \( 2 \cdot 2121793 \) |
\( J_6 \) | \(=\) | \(535790144\) | \(=\) | \( 2^{6} \cdot 53 \cdot 191 \cdot 827 \) |
\( J_8 \) | \(=\) | \(-241402309761\) | \(=\) | \( - 3 \cdot 1051 \cdot 76562737 \) |
\( J_{10} \) | \(=\) | \(434048\) | \(=\) | \( 2^{7} \cdot 3391 \) |
\( g_1 \) | \(=\) | \(254373487305614688256/3391\) | ||
\( g_2 \) | \(=\) | \(1066920663724396544/3391\) | ||
\( g_3 \) | \(=\) | \(4235039605737472/3391\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
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 except over $\Q_{2}$.
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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - 8xz + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - 8xz + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - 8xz + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 6.6.1471856768.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.909244 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 3.636976 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(7\) | \(1\) | \(1 - T\) | |
\(3391\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 64 T + 3391 T^{2} )\) |
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);