Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = x^4 + 2x^2 + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = x^4z^2 + 2x^2z^4 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 6x^4 + 9x^2 + 8$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(46656\) | \(=\) | \( 2^{6} \cdot 3^{6} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-93312\) | \(=\) | \( - 2^{7} \cdot 3^{6} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(116\) | \(=\) | \( 2^{2} \cdot 29 \) |
\( I_4 \) | \(=\) | \(27\) | \(=\) | \( 3^{3} \) |
\( I_6 \) | \(=\) | \(504\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 7 \) |
\( I_{10} \) | \(=\) | \(48\) | \(=\) | \( 2^{4} \cdot 3 \) |
\( J_2 \) | \(=\) | \(348\) | \(=\) | \( 2^{2} \cdot 3 \cdot 29 \) |
\( J_4 \) | \(=\) | \(4884\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 37 \) |
\( J_6 \) | \(=\) | \(101120\) | \(=\) | \( 2^{8} \cdot 5 \cdot 79 \) |
\( J_8 \) | \(=\) | \(2834076\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 33739 \) |
\( J_{10} \) | \(=\) | \(93312\) | \(=\) | \( 2^{7} \cdot 3^{6} \) |
\( g_1 \) | \(=\) | \(164089192/3\) | ||
\( g_2 \) | \(=\) | \(19852646/9\) | ||
\( g_3 \) | \(=\) | \(10630240/81\) |
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 everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{3}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.634777\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.634777\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - 2z^3\) | \(0.634777\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2\) | \(0\) | \(3\) |
2-torsion field: 6.0.1492992.6
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.634777 \) |
Real period: | \( 10.65951 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 2.255474 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(7\) | \(3\) | \(1 - T\) | |
\(3\) | \(6\) | \(6\) | \(1\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.60.2 | no |
\(3\) | 3.6480.8 | yes |
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 864.e
Elliptic curve isogeny class 54.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 \(\Q\).