Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 + 6x^4 + 13x^2 - 155$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 + 6x^4z^2 + 13x^2z^4 - 155z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 + 26x^4 + 53x^2 - 620$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(7440\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5 \cdot 31 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-476160\) | \(=\) | \( - 2^{10} \cdot 3 \cdot 5 \cdot 31 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(58556\) | \(=\) | \( 2^{2} \cdot 14639 \) |
\( I_4 \) | \(=\) | \(58997737\) | \(=\) | \( 23 \cdot 47 \cdot 54577 \) |
\( I_6 \) | \(=\) | \(1302611540179\) | \(=\) | \( 1249 \cdot 1042923571 \) |
\( I_{10} \) | \(=\) | \(59520\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 31 \) |
\( J_2 \) | \(=\) | \(58556\) | \(=\) | \( 2^{2} \cdot 14639 \) |
\( J_4 \) | \(=\) | \(103535056\) | \(=\) | \( 2^{4} \cdot 43 \cdot 61 \cdot 2467 \) |
\( J_6 \) | \(=\) | \(-53361108480\) | \(=\) | \( - 2^{9} \cdot 3 \cdot 5 \cdot 31 \cdot 224131 \) |
\( J_8 \) | \(=\) | \(-3461030222269504\) | \(=\) | \( - 2^{6} \cdot 199 \cdot 271751744839 \) |
\( J_{10} \) | \(=\) | \(476160\) | \(=\) | \( 2^{10} \cdot 3 \cdot 5 \cdot 31 \) |
\( g_1 \) | \(=\) | \(672290623153911817199/465\) | ||
\( g_2 \) | \(=\) | \(20300263373297500979/465\) | ||
\( g_3 \) | \(=\) | \(-384250761968408\) |
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: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 20z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-23xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 20z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-23xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(3x^2 - 20z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^3 - 45xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.4.3188367360000.5
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.450454 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.351363 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(10\) | \(3\) | \(1 - T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) | |
\(31\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 8 T + 31 T^{2} )\) |
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 |
\(3\) | 3.90.1 | 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 24.a
Elliptic curve isogeny class 310.b
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 \(\Q\).