Minimal equation
Minimal equation
Simplified equation
$y^2 = 6x^6 - 13x^5 + 27x^4 - 28x^3 + 27x^2 - 13x + 6$ | (homogenize, simplify) |
$y^2 = 6x^6 - 13x^5z + 27x^4z^2 - 28x^3z^3 + 27x^2z^4 - 13xz^5 + 6z^6$ | (dehomogenize, simplify) |
$y^2 = 6x^6 - 13x^5 + 27x^4 - 28x^3 + 27x^2 - 13x + 6$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2880\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 5 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(2880,2),R![1]>*])); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-368640\) | \(=\) | \( - 2^{13} \cdot 3^{2} \cdot 5 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(8840\) | \(=\) | \( 2^{3} \cdot 5 \cdot 13 \cdot 17 \) |
\( I_4 \) | \(=\) | \(2488\) | \(=\) | \( 2^{3} \cdot 311 \) |
\( I_6 \) | \(=\) | \(7315860\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 121931 \) |
\( I_{10} \) | \(=\) | \(1440\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \) |
\( J_2 \) | \(=\) | \(17680\) | \(=\) | \( 2^{4} \cdot 5 \cdot 13 \cdot 17 \) |
\( J_4 \) | \(=\) | \(13017632\) | \(=\) | \( 2^{5} \cdot 23^{2} \cdot 769 \) |
\( J_6 \) | \(=\) | \(12773283840\) | \(=\) | \( 2^{11} \cdot 3^{2} \cdot 5 \cdot 138599 \) |
\( J_8 \) | \(=\) | \(14093228850944\) | \(=\) | \( 2^{8} \cdot 1453 \cdot 37888283 \) |
\( J_{10} \) | \(=\) | \(368640\) | \(=\) | \( 2^{13} \cdot 3^{2} \cdot 5 \) |
\( g_1 \) | \(=\) | \(42174637208080000/9\) | ||
\( g_2 \) | \(=\) | \(1756381379464400/9\) | ||
\( g_3 \) | \(=\) | \(10830902014400\) |
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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 4xz + 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(xz^2 + 5z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 4xz + 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(xz^2 + 5z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 4xz + 5z^2\) | \(=\) | \(0,\) | \(25y\) | \(=\) | \(1/2xz^2 + 5/2z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.207360000.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 5.369336 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.671167 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(13\) | \(2\) | \(1\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) |
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 |
\(3\) | 3.90.1 | 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 24.a
Elliptic curve isogeny class 120.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 \(\Q\).