Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -2x^5 - 14x^4 + 77x^2 - 93x + 31$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -2x^5z - 14x^4z^2 + 77x^2z^4 - 93xz^5 + 31z^6$ | (dehomogenize, simplify) |
$y^2 = -8x^5 - 55x^4 + 2x^3 + 309x^2 - 372x + 124$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2604\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 31 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(62496\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 7 \cdot 31 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(195492\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 1481 \) |
\( I_4 \) | \(=\) | \(4003809\) | \(=\) | \( 3 \cdot 1334603 \) |
\( I_6 \) | \(=\) | \(258242732529\) | \(=\) | \( 3 \cdot 109 \cdot 789733127 \) |
\( I_{10} \) | \(=\) | \(7999488\) | \(=\) | \( 2^{12} \cdot 3^{2} \cdot 7 \cdot 31 \) |
\( J_2 \) | \(=\) | \(48873\) | \(=\) | \( 3 \cdot 11 \cdot 1481 \) |
\( J_4 \) | \(=\) | \(99356930\) | \(=\) | \( 2 \cdot 5 \cdot 89 \cdot 111637 \) |
\( J_6 \) | \(=\) | \(268901672004\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \cdot 31 \cdot 73 \cdot 27737 \) |
\( J_8 \) | \(=\) | \(817557969206648\) | \(=\) | \( 2^{3} \cdot 239 \cdot 427593080129 \) |
\( J_{10} \) | \(=\) | \(62496\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 7 \cdot 31 \) |
\( g_1 \) | \(=\) | \(30981502983804063186177/6944\) | ||
\( g_2 \) | \(=\) | \(644366055215080402545/3472\) | ||
\( g_3 \) | \(=\) | \(82218446158078593/8\) |
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);
|
Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{8}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 8xz - 9z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(4xz^2 - 5z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 8xz - 9z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(4xz^2 - 5z^3\) | \(0\) | \(8\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2 - 2z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 8xz - 9z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z + 9xz^2 - 10z^3\) | \(0\) | \(8\) |
2-torsion field: \(\Q(\sqrt{2}, \sqrt{217})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 15.41347 \) |
Tamagawa product: | \( 16 \) |
Torsion order: | \( 16 \) |
Leading coefficient: | \( 0.963342 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(5\) | \(8\) | \(( 1 - T )^{2}\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(31\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 31 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | 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 62.a
Elliptic curve isogeny class 42.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 \(\Q\).