Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = 15x^6 + 28x^5 + 62x^4 + 59x^3 + 62x^2 + 28x + 15$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = 15x^6 + 28x^5z + 62x^4z^2 + 59x^3z^3 + 62x^2z^4 + 28xz^5 + 15z^6$ | (dehomogenize, simplify) |
$y^2 = 60x^6 + 112x^5 + 249x^4 + 238x^3 + 249x^2 + 112x + 60$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(1170\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-10530\) | \(=\) | \( - 2 \cdot 3^{4} \cdot 5 \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(507196\) | \(=\) | \( 2^{2} \cdot 23 \cdot 37 \cdot 149 \) |
\( I_4 \) | \(=\) | \(192673\) | \(=\) | \( 13 \cdot 14821 \) |
\( I_6 \) | \(=\) | \(32552199279\) | \(=\) | \( 3^{2} \cdot 3616911031 \) |
\( I_{10} \) | \(=\) | \(1347840\) | \(=\) | \( 2^{8} \cdot 3^{4} \cdot 5 \cdot 13 \) |
\( J_2 \) | \(=\) | \(126799\) | \(=\) | \( 23 \cdot 37 \cdot 149 \) |
\( J_4 \) | \(=\) | \(669908072\) | \(=\) | \( 2^{3} \cdot 991 \cdot 84499 \) |
\( J_6 \) | \(=\) | \(4718980180980\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 5 \cdot 13 \cdot 224073133 \) |
\( J_8 \) | \(=\) | \(37396285759331459\) | \(=\) | \( 71 \cdot 11505961 \cdot 45776989 \) |
\( J_{10} \) | \(=\) | \(10530\) | \(=\) | \( 2 \cdot 3^{4} \cdot 5 \cdot 13 \) |
\( g_1 \) | \(=\) | \(32777750301275239538233999/10530\) | ||
\( g_2 \) | \(=\) | \(682861614668954802420364/5265\) | ||
\( g_3 \) | \(=\) | \(7205289570406928666\) |
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: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(10x^2 + 7xz + 10z^2\) | \(=\) | \(0,\) | \(20y\) | \(=\) | \(-3xz^2 + 10z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(17x^2 + 11xz + 17z^2\) | \(=\) | \(0,\) | \(289y\) | \(=\) | \(-65xz^2 + 136z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(10x^2 + 7xz + 10z^2\) | \(=\) | \(0,\) | \(20y\) | \(=\) | \(-3xz^2 + 10z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(17x^2 + 11xz + 17z^2\) | \(=\) | \(0,\) | \(289y\) | \(=\) | \(-65xz^2 + 136z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(10x^2 + 7xz + 10z^2\) | \(=\) | \(0,\) | \(20y\) | \(=\) | \(x^2z - 5xz^2 + 20z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(17x^2 + 11xz + 17z^2\) | \(=\) | \(0,\) | \(289y\) | \(=\) | \(x^2z - 129xz^2 + 272z^3\) | \(0\) | \(6\) |
2-torsion field: 8.0.5922408960000.12
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 5.542029 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 12 \) |
Leading coefficient: | \( 0.461835 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(2\) | \(4\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 13 T^{2} )\) |
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 |
\(3\) | 3.720.4 | yes |
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 30.a
Elliptic curve isogeny class 39.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 \(\Q\).