Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x + 1)y = -5x^6 + 15x^5 - 38x^4 + 50x^3 - 57x^2 + 33x - 16$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2 + z^3)y = -5x^6 + 15x^5z - 38x^4z^2 + 50x^3z^3 - 57x^2z^4 + 33xz^5 - 16z^6$ | (dehomogenize, simplify) |
$y^2 = -20x^6 + 60x^5 - 151x^4 + 202x^3 - 225x^2 + 134x - 63$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(8370\) | \(=\) | \( 2 \cdot 3^{3} \cdot 5 \cdot 31 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-225990\) | \(=\) | \( - 2 \cdot 3^{6} \cdot 5 \cdot 31 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(46596\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 353 \) |
\( I_4 \) | \(=\) | \(239073\) | \(=\) | \( 3 \cdot 79691 \) |
\( I_6 \) | \(=\) | \(3674852529\) | \(=\) | \( 3 \cdot 19 \cdot 751 \cdot 85847 \) |
\( I_{10} \) | \(=\) | \(119040\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5 \cdot 31 \) |
\( J_2 \) | \(=\) | \(34947\) | \(=\) | \( 3^{2} \cdot 11 \cdot 353 \) |
\( J_4 \) | \(=\) | \(50797548\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 1411043 \) |
\( J_6 \) | \(=\) | \(98289730260\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 5 \cdot 31 \cdot 1249 \cdot 1567 \) |
\( J_8 \) | \(=\) | \(213635080145979\) | \(=\) | \( 3^{4} \cdot 64399 \cdot 40955141 \) |
\( J_{10} \) | \(=\) | \(225990\) | \(=\) | \( 2 \cdot 3^{6} \cdot 5 \cdot 31 \) |
\( g_1 \) | \(=\) | \(71502622649365111083/310\) | ||
\( g_2 \) | \(=\) | \(1487013548016809538/155\) | ||
\( g_3 \) | \(=\) | \(531176338621566\) |
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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 5xz + 8z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(-5xz^2 + 3z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 5xz + 8z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(-5xz^2 + 3z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2 + z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(5x^2 - 5xz + 8z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(x^2z - 9xz^2 + 7z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.191501314560000.152
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 3.621258 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.452657 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(3\) | \(6\) | \(4\) | \(1 + T\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 5 T^{2} )\) | |
\(31\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 31 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | 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 15.a
Elliptic curve isogeny class 558.c
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 \(\Q\).