Minimal equation
Minimal equation
Simplified equation
$y^2 + y = 6x^6 - 8x^4 + 4x^2 - 1$ | (homogenize, simplify) |
$y^2 + z^3y = 6x^6 - 8x^4z^2 + 4x^2z^4 - z^6$ | (dehomogenize, simplify) |
$y^2 = 24x^6 - 32x^4 + 16x^2 - 3$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(278784\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(557568\) | \(=\) | \( 2^{9} \cdot 3^{2} \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1592\) | \(=\) | \( 2^{3} \cdot 199 \) |
\( I_4 \) | \(=\) | \(1189\) | \(=\) | \( 29 \cdot 41 \) |
\( I_6 \) | \(=\) | \(630369\) | \(=\) | \( 3^{3} \cdot 37 \cdot 631 \) |
\( I_{10} \) | \(=\) | \(2178\) | \(=\) | \( 2 \cdot 3^{2} \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(3184\) | \(=\) | \( 2^{4} \cdot 199 \) |
\( J_4 \) | \(=\) | \(419240\) | \(=\) | \( 2^{3} \cdot 5 \cdot 47 \cdot 223 \) |
\( J_6 \) | \(=\) | \(73041408\) | \(=\) | \( 2^{9} \cdot 3^{2} \cdot 11^{2} \cdot 131 \) |
\( J_8 \) | \(=\) | \(14200416368\) | \(=\) | \( 2^{4} \cdot 887526023 \) |
\( J_{10} \) | \(=\) | \(557568\) | \(=\) | \( 2^{9} \cdot 3^{2} \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(639139022845952/1089\) | ||
\( g_2 \) | \(=\) | \(26430898598080/1089\) | ||
\( g_3 \) | \(=\) | \(1328059136\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\Q_{2}$ and $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.4857532416.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.710507 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 2.855253 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(9\) | \(1\) | \(1\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.90.5 | yes |
\(3\) | 3.1080.10 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_1)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
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 132.b
Elliptic curve isogeny class 2112.u
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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-2}) \) with defining polynomial \(x^{2} + 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |