Minimal equation
Minimal equation
Simplified equation
$y^2 = -x^6 - 2x^4 - 2x^2 - 1$ | (homogenize, simplify) |
$y^2 = -x^6 - 2x^4z^2 - 2x^2z^4 - z^6$ | (dehomogenize, simplify) |
$y^2 = -x^6 - 2x^4 - 2x^2 - 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2304\) | \(=\) | \( 2^{8} \cdot 3^{2} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(2304,2),R![1]>*])); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-147456\) | \(=\) | \( - 2^{14} \cdot 3^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(152\) | \(=\) | \( 2^{3} \cdot 19 \) |
\( I_4 \) | \(=\) | \(109\) | \(=\) | \( 109 \) |
\( I_6 \) | \(=\) | \(5469\) | \(=\) | \( 3 \cdot 1823 \) |
\( I_{10} \) | \(=\) | \(18\) | \(=\) | \( 2 \cdot 3^{2} \) |
\( J_2 \) | \(=\) | \(608\) | \(=\) | \( 2^{5} \cdot 19 \) |
\( J_4 \) | \(=\) | \(14240\) | \(=\) | \( 2^{5} \cdot 5 \cdot 89 \) |
\( J_6 \) | \(=\) | \(405504\) | \(=\) | \( 2^{12} \cdot 3^{2} \cdot 11 \) |
\( J_8 \) | \(=\) | \(10942208\) | \(=\) | \( 2^{8} \cdot 42743 \) |
\( J_{10} \) | \(=\) | \(147456\) | \(=\) | \( 2^{14} \cdot 3^{2} \) |
\( g_1 \) | \(=\) | \(5071050752/9\) | ||
\( g_2 \) | \(=\) | \(195344320/9\) | ||
\( g_3 \) | \(=\) | \(1016576\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\R$, $\Q_{2}$, and $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\zeta_{12})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.683508 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.710438 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(14\) | \(2\) | \(1\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.180.7 | yes |
\(3\) | 3.2160.25 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_1$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 48.a
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{}) \otimes \Q\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).