Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 + 15x^4 - 75x^2 - 56$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 + 15x^4z^2 - 75x^2z^4 - 56z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 + 62x^4 - 299x^2 - 224$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(336\) | \(=\) | \( 2^{4} \cdot 3 \cdot 7 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(336,2),R![1, 1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-172032\) | \(=\) | \( - 2^{13} \cdot 3 \cdot 7 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(16916\) | \(=\) | \( 2^{2} \cdot 4229 \) |
\( I_4 \) | \(=\) | \(151117825\) | \(=\) | \( 5^{2} \cdot 6044713 \) |
\( I_6 \) | \(=\) | \(232872423961\) | \(=\) | \( 397 \cdot 586580413 \) |
\( I_{10} \) | \(=\) | \(-21504\) | \(=\) | \( - 2^{10} \cdot 3 \cdot 7 \) |
\( J_2 \) | \(=\) | \(16916\) | \(=\) | \( 2^{2} \cdot 4229 \) |
\( J_4 \) | \(=\) | \(-88822256\) | \(=\) | \( - 2^{4} \cdot 103 \cdot 53897 \) |
\( J_6 \) | \(=\) | \(277597802496\) | \(=\) | \( 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \) |
\( J_8 \) | \(=\) | \(-798387183476800\) | \(=\) | \( - 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \) |
\( J_{10} \) | \(=\) | \(-172032\) | \(=\) | \( - 2^{13} \cdot 3 \cdot 7 \) |
\( g_1 \) | \(=\) | \(-1352659309173012149/168\) | ||
\( g_2 \) | \(=\) | \(419870026410625699/168\) | ||
\( g_3 \) | \(=\) | \(-461744933079368\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 32z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-35xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 32z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-35xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 32z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^3 - 69xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.4.260112384.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 0.356066 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.178033 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(13\) | \(1\) | \(1 + T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 7 T^{2} )\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.45.1 | yes |
\(3\) | 3.720.5 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 24.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
Additional information
The Jacobian of this curve is conjectured to have the smallest conductor among all genus 2 curves whose analytic order of Sha is not a square.