Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = 3x^5 + 28x^4 + 72x^3 + 28x^2 + 3x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = 3x^5z + 28x^4z^2 + 72x^3z^3 + 28x^2z^4 + 3xz^5$ | (dehomogenize, simplify) |
$y^2 = 12x^5 + 113x^4 + 290x^3 + 113x^2 + 12x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(990\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(240570\) | \(=\) | \( 2 \cdot 3^{7} \cdot 5 \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(153028\) | \(=\) | \( 2^{2} \cdot 67 \cdot 571 \) |
\( I_4 \) | \(=\) | \(6848257\) | \(=\) | \( 13 \cdot 263 \cdot 2003 \) |
\( I_6 \) | \(=\) | \(343366646113\) | \(=\) | \( 31 \cdot 11076343423 \) |
\( I_{10} \) | \(=\) | \(30792960\) | \(=\) | \( 2^{8} \cdot 3^{7} \cdot 5 \cdot 11 \) |
\( J_2 \) | \(=\) | \(38257\) | \(=\) | \( 67 \cdot 571 \) |
\( J_4 \) | \(=\) | \(60697908\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 37 \cdot 45569 \) |
\( J_6 \) | \(=\) | \(127876480380\) | \(=\) | \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 11 \cdot 2392003 \) |
\( J_8 \) | \(=\) | \(301983618580299\) | \(=\) | \( 3^{4} \cdot 19^{2} \cdot 20117 \cdot 513367 \) |
\( J_{10} \) | \(=\) | \(240570\) | \(=\) | \( 2 \cdot 3^{7} \cdot 5 \cdot 11 \) |
\( g_1 \) | \(=\) | \(81951056110393451083057/240570\) | ||
\( g_2 \) | \(=\) | \(188813894774599018858/13365\) | ||
\( g_3 \) | \(=\) | \(7001861848004294/9\) |
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: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + 19xz + 4z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(15xz^2 + 4z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + 14xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(11xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + 19xz + 4z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(15xz^2 + 4z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + 14xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(11xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(4x^2 + 19xz + 4z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z + 31xz^2 + 8z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(3x^2 + 14xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^2z + 23xz^2 + 6z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{10}, \sqrt{33})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 3.087468 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.385933 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(2\) | \(7\) | \(2\) | \(( 1 + T )^{2}\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 5 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 11 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | 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 15.a
Elliptic curve isogeny class 66.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 \(\Q\).