Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -14x^6 + 33x^5 - 61x^4 + 69x^3 - 61x^2 + 33x - 14$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -14x^6 + 33x^5z - 61x^4z^2 + 69x^3z^3 - 61x^2z^4 + 33xz^5 - 14z^6$ | (dehomogenize, simplify) |
$y^2 = -55x^6 + 132x^5 - 244x^4 + 278x^3 - 244x^2 + 132x - 55$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(482790\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-482790\) | \(=\) | \( - 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(258956\) | \(=\) | \( 2^{2} \cdot 41 \cdot 1579 \) |
\( I_4 \) | \(=\) | \(284225353\) | \(=\) | \( 284225353 \) |
\( I_6 \) | \(=\) | \(22936728525979\) | \(=\) | \( 3546173 \cdot 6468023 \) |
\( I_{10} \) | \(=\) | \(61797120\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19 \) |
\( J_2 \) | \(=\) | \(64739\) | \(=\) | \( 41 \cdot 1579 \) |
\( J_4 \) | \(=\) | \(162788032\) | \(=\) | \( 2^{6} \cdot 11 \cdot 79 \cdot 2927 \) |
\( J_6 \) | \(=\) | \(522481586340\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \cdot 65473883 \) |
\( J_8 \) | \(=\) | \(1831248013908059\) | \(=\) | \( 1282069 \cdot 1428353711 \) |
\( J_{10} \) | \(=\) | \(482790\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2} \cdot 19 \) |
\( g_1 \) | \(=\) | \(1137181886990894545487699/482790\) | ||
\( g_2 \) | \(=\) | \(2007695068464697012064/21945\) | ||
\( g_3 \) | \(=\) | \(548820173942387686/121\) |
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 except over $\R$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 3xz + 8z^2\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(11xz^2 - 8z^3\) | \(0.513485\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(21xz^2 - 15z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 3xz + 8z^2\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(11xz^2 - 8z^3\) | \(0.513485\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(21xz^2 - 15z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 3xz + 8z^2\) | \(=\) | \(0,\) | \(32y\) | \(=\) | \(x^3 + 22xz^2 - 15z^3\) | \(0.513485\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(x^3 + 42xz^2 - 29z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.34369956362649600.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.513485 \) |
Real period: | \( 1.616909 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.660517 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 7 T^{2} )\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 19 T^{2} )\) |
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 3135.b
Elliptic curve isogeny class 154.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 \(\Q\).