Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -4x^6 - 45x^4 - 176x^2 - 231$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -4x^6 - 45x^4z^2 - 176x^2z^4 - 231z^6$ | (dehomogenize, simplify) |
$y^2 = -15x^6 - 178x^4 - 703x^2 - 924$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(110880\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-887040\) | \(=\) | \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(666068\) | \(=\) | \( 2^{2} \cdot 13 \cdot 12809 \) |
\( I_4 \) | \(=\) | \(38797\) | \(=\) | \( 11 \cdot 3527 \) |
\( I_6 \) | \(=\) | \(8613261963\) | \(=\) | \( 3^{2} \cdot 19 \cdot 2903 \cdot 17351 \) |
\( I_{10} \) | \(=\) | \(110880\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
\( J_2 \) | \(=\) | \(666068\) | \(=\) | \( 2^{2} \cdot 13 \cdot 12809 \) |
\( J_4 \) | \(=\) | \(18485248328\) | \(=\) | \( 2^{3} \cdot 43 \cdot 6869 \cdot 7823 \) |
\( J_6 \) | \(=\) | \(684022107605760\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 24091 \cdot 32009 \) |
\( J_8 \) | \(=\) | \(28475207855231639024\) | \(=\) | \( 2^{4} \cdot 826663 \cdot 2152873046153 \) |
\( J_{10} \) | \(=\) | \(887040\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \) |
\( g_1 \) | \(=\) | \(512097765844517969700609428/3465\) | ||
\( g_2 \) | \(=\) | \(21337377054806526458909866/3465\) | ||
\( g_3 \) | \(=\) | \(342108663769973403056\) |
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 $\R$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 + 2z^3\) | \(0.891621\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(1.006808\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(4xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 + 2z^3\) | \(0.891621\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2\) | \(1.006808\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(3xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(4xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 5xz^2 + 4z^3\) | \(0.891621\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 5xz^2\) | \(1.006808\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 + 7xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(x^3 + 9xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.455583411360000.109
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.897691 \) |
Real period: | \( 3.441709 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.544796 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(8\) | \(4\) | \(1 + T\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 11 T^{2} )\) |
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 240.a
Elliptic curve isogeny class 462.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 \(\Q\).