Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^6 - 2x^5 + 2x^3 - x^2$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^6 - 2x^5z + 2x^3z^3 - x^2z^4$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 8x^5 + 8x^3 - 3x^2 + 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(976\) | \(=\) | \( 2^{4} \cdot 61 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(976,2),R![1, -1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(999424\) | \(=\) | \( 2^{14} \cdot 61 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(152\) | \(=\) | \( 2^{3} \cdot 19 \) |
\( I_4 \) | \(=\) | \(1012\) | \(=\) | \( 2^{2} \cdot 11 \cdot 23 \) |
\( I_6 \) | \(=\) | \(68714\) | \(=\) | \( 2 \cdot 17 \cdot 43 \cdot 47 \) |
\( I_{10} \) | \(=\) | \(-124928\) | \(=\) | \( - 2^{11} \cdot 61 \) |
\( J_2 \) | \(=\) | \(152\) | \(=\) | \( 2^{3} \cdot 19 \) |
\( J_4 \) | \(=\) | \(288\) | \(=\) | \( 2^{5} \cdot 3^{2} \) |
\( J_6 \) | \(=\) | \(-24464\) | \(=\) | \( - 2^{4} \cdot 11 \cdot 139 \) |
\( J_8 \) | \(=\) | \(-950368\) | \(=\) | \( - 2^{5} \cdot 17 \cdot 1747 \) |
\( J_{10} \) | \(=\) | \(-999424\) | \(=\) | \( - 2^{14} \cdot 61 \) |
\( g_1 \) | \(=\) | \(-4952198/61\) | ||
\( g_2 \) | \(=\) | \(-61731/61\) | ||
\( g_3 \) | \(=\) | \(551969/976\) |
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
Number of rational Weierstrass points: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{29}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(29\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(29\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(29\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 12.90036 \) |
Tamagawa product: | \( 29 \) |
Torsion order: | \( 29 \) |
Leading coefficient: | \( 0.444840 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(14\) | \(29\) | \(1 - T\) | |
\(61\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 61 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | no |
\(29\) | not computed | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
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 \(\Q\).
Additional information
The Jacobian of this curve has a rational 29-torsion point. This is the largest known prime factor occurring in the torsion of the Mordell-Weil group of any abelian surface over $\mathbb{Q}$. This example was discovered by Franck Leprévost in 1995: see [MR:1413580].