Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = -112x^6 + 93x^4 - 2x^2 - 9$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = -112x^6 + 93x^4z^2 - 2x^2z^4 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -448x^6 + 373x^4 - 6x^2 - 35$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(15680\) | \(=\) | \( 2^{6} \cdot 5 \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-250880\) | \(=\) | \( - 2^{10} \cdot 5 \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(465924\) | \(=\) | \( 2^{2} \cdot 3 \cdot 41 \cdot 947 \) |
\( I_4 \) | \(=\) | \(5137879035\) | \(=\) | \( 3 \cdot 5 \cdot 23 \cdot 14892403 \) |
\( I_6 \) | \(=\) | \(920713062008316\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 43 \cdot 773 \cdot 121490273 \) |
\( I_{10} \) | \(=\) | \(31360\) | \(=\) | \( 2^{7} \cdot 5 \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(465924\) | \(=\) | \( 2^{2} \cdot 3 \cdot 41 \cdot 947 \) |
\( J_4 \) | \(=\) | \(5619962884\) | \(=\) | \( 2^{2} \cdot 17 \cdot 82646513 \) |
\( J_6 \) | \(=\) | \(-140971599964160\) | \(=\) | \( - 2^{11} \cdot 5 \cdot 7^{2} \cdot 2503 \cdot 112247 \) |
\( J_8 \) | \(=\) | \(-24316508639809720324\) | \(=\) | \( - 2^{2} \cdot 19 \cdot 319954061050127899 \) |
\( J_{10} \) | \(=\) | \(250880\) | \(=\) | \( 2^{10} \cdot 5 \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(21442501652207789032006401/245\) | ||
\( g_2 \) | \(=\) | \(2220438389769740808604161/980\) | ||
\( g_3 \) | \(=\) | \(-121982000461178368032\) |
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);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(64x^2 - 35z^2\) | \(=\) | \(0,\) | \(128y\) | \(=\) | \(-99z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(64x^2 - 35z^2\) | \(=\) | \(0,\) | \(128y\) | \(=\) | \(-99z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(64x^2 - 35z^2\) | \(=\) | \(0,\) | \(128y\) | \(=\) | \(x^2z - 197z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.2007040000.5
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.147456 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.589825 \) |
Analytic order of Ш: | \( 16 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(10\) | \(1\) | \(1 + T\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 5 T^{2} )\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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 1120.j
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 \(\Q\).