Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = -x^6 - 3x^5 + 7x^4 + 2x^3 - 49x^2 + 41x - 9$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = -x^6 - 3x^5z + 7x^4z^2 + 2x^3z^3 - 49x^2z^4 + 41xz^5 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 - 12x^5 + 29x^4 + 8x^3 - 194x^2 + 164x - 35$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(784\) | \(=\) | \( 2^{4} \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-25088\) | \(=\) | \( - 2^{9} \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2740\) | \(=\) | \( 2^{2} \cdot 5 \cdot 137 \) |
\( I_4 \) | \(=\) | \(15382525\) | \(=\) | \( 5^{2} \cdot 521 \cdot 1181 \) |
\( I_6 \) | \(=\) | \(36170522453\) | \(=\) | \( 13 \cdot 23 \cdot 6211 \cdot 19477 \) |
\( I_{10} \) | \(=\) | \(3136\) | \(=\) | \( 2^{6} \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(2740\) | \(=\) | \( 2^{2} \cdot 5 \cdot 137 \) |
\( J_4 \) | \(=\) | \(-9942200\) | \(=\) | \( - 2^{3} \cdot 5^{2} \cdot 49711 \) |
\( J_6 \) | \(=\) | \(-24298750736\) | \(=\) | \( - 2^{4} \cdot 13 \cdot 61 \cdot 149 \cdot 12853 \) |
\( J_8 \) | \(=\) | \(-41356479464160\) | \(=\) | \( - 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 1531 \cdot 8039501 \) |
\( J_{10} \) | \(=\) | \(25088\) | \(=\) | \( 2^{9} \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(301635777856250/49\) | ||
\( g_2 \) | \(=\) | \(-399451653071875/49\) | ||
\( g_3 \) | \(=\) | \(-712598832131225/98\) |
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: \(0\)
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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz - 5z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(12xz^2 - 9z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz - 5z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(12xz^2 - 9z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 12xz - 5z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 24xz^2 - 17z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.3211264.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 0.626116 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.313058 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(9\) | \(1\) | \(1 + T\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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 56.b
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(4\) 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 \(\Q\).