Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = 24x^6 + 95x^4 + 125x^2 + 55$ | (homogenize, simplify) |
$y^2 + xz^2y = 24x^6 + 95x^4z^2 + 125x^2z^4 + 55z^6$ | (dehomogenize, simplify) |
$y^2 = 96x^6 + 380x^4 + 501x^2 + 220$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(5280\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-84480\) | \(=\) | \( - 2^{9} \cdot 3 \cdot 5 \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1014360\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 79 \cdot 107 \) |
\( I_4 \) | \(=\) | \(22497\) | \(=\) | \( 3 \cdot 7499 \) |
\( I_6 \) | \(=\) | \(7606321185\) | \(=\) | \( 3 \cdot 5 \cdot 507088079 \) |
\( I_{10} \) | \(=\) | \(10560\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
\( J_2 \) | \(=\) | \(1014360\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 79 \cdot 107 \) |
\( J_4 \) | \(=\) | \(42871910402\) | \(=\) | \( 2 \cdot 29 \cdot 467 \cdot 1063 \cdot 1489 \) |
\( J_6 \) | \(=\) | \(2415973367470080\) | \(=\) | \( 2^{11} \cdot 3 \cdot 5 \cdot 11 \cdot 103 \cdot 5813 \cdot 11941 \) |
\( J_8 \) | \(=\) | \(153166510877458636799\) | \(=\) | \( 153166510877458636799 \) |
\( J_{10} \) | \(=\) | \(84480\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \cdot 11 \) |
\( g_1 \) | \(=\) | \(139829677203278295877320000/11\) | ||
\( g_2 \) | \(=\) | \(5826234511928725040734650/11\) | ||
\( g_3 \) | \(=\) | \(29425406243910243321600\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | 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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(8x^2 + 11z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(8x^2 + 11z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(8x^2 + 11z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.48575324160000.293
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 4.119112 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.029778 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(9\) | \(1\) | \(1 - T + 2 T^{2}\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 5 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 11 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | yes |
\(3\) | 3.90.1 | 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 55.a
Elliptic curve isogeny class 96.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 \(\Q\).