Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 + 6x^4 - 17x^2 + 11$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 + 6x^4z^2 - 17x^2z^4 + 11z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 + 26x^4 - 67x^2 + 44$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(1584\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(684288\) | \(=\) | \( 2^{8} \cdot 3^{5} \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(7444\) | \(=\) | \( 2^{2} \cdot 1861 \) |
\( I_4 \) | \(=\) | \(76621\) | \(=\) | \( 193 \cdot 397 \) |
\( I_6 \) | \(=\) | \(183223627\) | \(=\) | \( 67 \cdot 313 \cdot 8737 \) |
\( I_{10} \) | \(=\) | \(85536\) | \(=\) | \( 2^{5} \cdot 3^{5} \cdot 11 \) |
\( J_2 \) | \(=\) | \(7444\) | \(=\) | \( 2^{2} \cdot 1861 \) |
\( J_4 \) | \(=\) | \(2257800\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5^{2} \cdot 53 \cdot 71 \) |
\( J_6 \) | \(=\) | \(897608448\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 11 \cdot 107 \cdot 331 \) |
\( J_8 \) | \(=\) | \(396034111728\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 103 \cdot 227 \cdot 39209 \) |
\( J_{10} \) | \(=\) | \(684288\) | \(=\) | \( 2^{8} \cdot 3^{5} \cdot 11 \) |
\( g_1 \) | \(=\) | \(89287745446261204/2673\) | ||
\( g_2 \) | \(=\) | \(1212671977685150/891\) | ||
\( g_3 \) | \(=\) | \(1962567037712/27\) |
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: \(4\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - D_\infty\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((-2 : 5 : 1) + (-1 : 1 : 1) - D_\infty\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 - 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\((-1 : 1 : 1) + (1 : -1 : 1) - D_\infty\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\((-2 : 5 : 1) + (-1 : 1 : 1) - D_\infty\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 - 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 9xz^2\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - D_\infty\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - xz^2\) | \(0\) | \(2\) |
\((-2 : 0 : 1) + (-1 : 0 : 1) - D_\infty\) | \((x + z) (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 7xz^2 - 6z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{33}) \)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 4.753790 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.594223 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(4\) | \(8\) | \(4\) | \(1 - T\) | |
\(3\) | \(2\) | \(5\) | \(2\) | \(( 1 + T )^{2}\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 11 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.360.2 | 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 66.b
Elliptic curve isogeny class 24.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 \(\Q\).