Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -x^6 - 5x^5 + 3x^4 + 56x^3 - 7x^2 - 195x + 131$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -x^6 - 5x^5z + 3x^4z^2 + 56x^3z^3 - 7x^2z^4 - 195xz^5 + 131z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 - 20x^5 + 12x^4 + 226x^3 - 28x^2 - 780x + 525$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2730\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(13650\) | \(=\) | \( 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(656916\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13 \cdot 4211 \) |
\( I_4 \) | \(=\) | \(183993\) | \(=\) | \( 3 \cdot 61331 \) |
\( I_6 \) | \(=\) | \(40270870029\) | \(=\) | \( 3 \cdot 13 \cdot 827 \cdot 1248593 \) |
\( I_{10} \) | \(=\) | \(1747200\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \) |
\( J_2 \) | \(=\) | \(164229\) | \(=\) | \( 3 \cdot 13 \cdot 4211 \) |
\( J_4 \) | \(=\) | \(1123790852\) | \(=\) | \( 2^{2} \cdot 280947713 \) |
\( J_6 \) | \(=\) | \(10253140797900\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13 \cdot 34142993 \) |
\( J_8 \) | \(=\) | \(105239295264858299\) | \(=\) | \( 79 \cdot 181 \cdot 25609 \cdot 287395289 \) |
\( J_{10} \) | \(=\) | \(13650\) | \(=\) | \( 2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \) |
\( g_1 \) | \(=\) | \(3063265468298882029687491/350\) | ||
\( g_2 \) | \(=\) | \(63817595233091546052726/175\) | ||
\( g_3 \) | \(=\) | \(20259278131640062086\) |
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 $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 4z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-11xz^2 + 13z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-6xz^2 + 4z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-11xz^2 + 13z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3xz - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 12xz^2 + 9z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 22xz^2 + 27z^3\) | \(0\) | \(2\) |
2-torsion field: 8.8.22751526260736.3
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 7.805946 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.975743 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 7 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 13 T^{2} )\) |
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 |
\(3\) | 3.90.1 | no |
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 39.a
Elliptic curve isogeny class 70.a
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 \(\Q\).