Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -x^6 + 3x^5 - 13x^4 + 20x^3 - 41x^2 + 31x - 32$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -x^6 + 3x^5z - 13x^4z^2 + 20x^3z^3 - 41x^2z^4 + 31xz^5 - 32z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 12x^5 - 51x^4 + 82x^3 - 163x^2 + 124x - 128$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(459510\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 53 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-459510\) | \(=\) | \( - 2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 53 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(78012\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 11 \cdot 197 \) |
\( I_4 \) | \(=\) | \(178113\) | \(=\) | \( 3 \cdot 13 \cdot 4567 \) |
\( I_6 \) | \(=\) | \(4622163327\) | \(=\) | \( 3^{2} \cdot 673 \cdot 763111 \) |
\( I_{10} \) | \(=\) | \(58817280\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5 \cdot 17^{2} \cdot 53 \) |
\( J_2 \) | \(=\) | \(19503\) | \(=\) | \( 3^{2} \cdot 11 \cdot 197 \) |
\( J_4 \) | \(=\) | \(15841204\) | \(=\) | \( 2^{2} \cdot 23 \cdot 233 \cdot 739 \) |
\( J_6 \) | \(=\) | \(17147994180\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 17^{2} \cdot 47 \cdot 53 \cdot 397 \) |
\( J_8 \) | \(=\) | \(20873396580731\) | \(=\) | \( 39359 \cdot 530333509 \) |
\( J_{10} \) | \(=\) | \(459510\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 17^{2} \cdot 53 \) |
\( g_1 \) | \(=\) | \(940558579042975132581/153170\) | ||
\( g_2 \) | \(=\) | \(19585794735118436418/76585\) | ||
\( g_3 \) | \(=\) | \(14194536041862\) |
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 $\R$ and $\Q_{7}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + 2z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + 2z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2 + 4z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.760137984360000.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 4.595700 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 2.297850 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) | |
\(17\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(53\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 6 T + 53 T^{2} )\) |
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 17.a
Elliptic curve isogeny class 27030.n
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 \(\Q\).