Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = -135x^6 - 96x^4 - 23x^2 - 2$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = -135x^6 - 96x^4z^2 - 23x^2z^4 - 2z^6$ | (dehomogenize, simplify) |
$y^2 = -540x^6 - 383x^4 - 90x^2 - 7$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3360\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \cdot 7 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-241920\) | \(=\) | \( - 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(182340\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 1013 \) |
\( I_4 \) | \(=\) | \(50613\) | \(=\) | \( 3 \cdot 16871 \) |
\( I_6 \) | \(=\) | \(3073006935\) | \(=\) | \( 3^{2} \cdot 5 \cdot 127 \cdot 537709 \) |
\( I_{10} \) | \(=\) | \(30240\) | \(=\) | \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \) |
\( J_2 \) | \(=\) | \(182340\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 1013 \) |
\( J_4 \) | \(=\) | \(1385294408\) | \(=\) | \( 2^{3} \cdot 19 \cdot 9113779 \) |
\( J_6 \) | \(=\) | \(14032351630080\) | \(=\) | \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 58004099 \) |
\( J_8 \) | \(=\) | \(159904599848179184\) | \(=\) | \( 2^{4} \cdot 97 \cdot 7612427 \cdot 13534621 \) |
\( J_{10} \) | \(=\) | \(241920\) | \(=\) | \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
\( g_1 \) | \(=\) | \(5832248478791381977500/7\) | ||
\( g_2 \) | \(=\) | \(243004434356588125950/7\) | ||
\( g_3 \) | \(=\) | \(1928513067842084400\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | 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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(-3z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(45x^2 + 11z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(-2z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(-3z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(45x^2 + 11z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(-2z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + z^2\) | \(=\) | \(0,\) | \(8y\) | \(=\) | \(x^2z - 5z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(45x^2 + 11z^2\) | \(=\) | \(0,\) | \(5y\) | \(=\) | \(x^2z - 3z^3\) | \(0\) | \(6\) |
2-torsion field: 8.0.31116960000.8
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 3.848390 \) |
Tamagawa product: | \( 12 \) |
Torsion order: | \( 12 \) |
Leading coefficient: | \( 0.641398 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(8\) | \(4\) | \(1 + T\) | |
\(3\) | \(1\) | \(3\) | \(3\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) |
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 |
\(3\) | 3.720.4 | 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 30.a
Elliptic curve isogeny class 112.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 \(\Q\).