Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -x^6 - 24x^4 - 192x^2 - 510$ | (homogenize, simplify) |
$y^2 + xz^2y = -x^6 - 24x^4z^2 - 192x^2z^4 - 510z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 - 96x^4 - 767x^2 - 2040$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(32640\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-32640\) | \(=\) | \( - 2^{7} \cdot 3 \cdot 5 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(392064\) | \(=\) | \( 2^{7} \cdot 3 \cdot 1021 \) |
\( I_4 \) | \(=\) | \(9213\) | \(=\) | \( 3 \cdot 37 \cdot 83 \) |
\( I_6 \) | \(=\) | \(1204027536\) | \(=\) | \( 2^{4} \cdot 3 \cdot 163 \cdot 153889 \) |
\( I_{10} \) | \(=\) | \(4080\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5 \cdot 17 \) |
\( J_2 \) | \(=\) | \(392064\) | \(=\) | \( 2^{7} \cdot 3 \cdot 1021 \) |
\( J_4 \) | \(=\) | \(6404751362\) | \(=\) | \( 2 \cdot 331 \cdot 9674851 \) |
\( J_6 \) | \(=\) | \(139503756771840\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \cdot 17 \cdot 59 \cdot 18110221 \) |
\( J_8 \) | \(=\) | \(3418390221488455679\) | \(=\) | \( 179 \cdot 13933 \cdot 1370641791097 \) |
\( J_{10} \) | \(=\) | \(32640\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 17 \) |
\( g_1 \) | \(=\) | \(24124238194999818543169536/85\) | ||
\( g_2 \) | \(=\) | \(1005175627519469245857792/85\) | ||
\( g_3 \) | \(=\) | \(656976274279877246976\) |
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 $\R$ and $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 + 17z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 + 17z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 8z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 + 17z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.277102632960000.103
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 1 \) |
Real period: | \( 4.794347 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 2.397173 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(7\) | \(1\) | \(1 - T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 17 T^{2} )\) |
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 |
\(3\) | 3.270.2 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
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 64.a
Elliptic curve isogeny class 510.f
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(4\) in \(\Z \times \Z [\sqrt{-1}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-1}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |