Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = 2x^6 - 25x^4 + 88x^2 - 105$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = 2x^6 - 25x^4z^2 + 88x^2z^4 - 105z^6$ | (dehomogenize, simplify) |
$y^2 = 9x^6 - 98x^4 + 353x^2 - 420$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(10080\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \) | 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
\( I_2 \) | \(=\) | \(182588\) | \(=\) | \( 2^{2} \cdot 7 \cdot 6521 \) |
\( I_4 \) | \(=\) | \(82357\) | \(=\) | \( 11 \cdot 7487 \) |
\( I_6 \) | \(=\) | \(5005132713\) | \(=\) | \( 3^{2} \cdot 7 \cdot 1013 \cdot 78427 \) |
\( I_{10} \) | \(=\) | \(30240\) | \(=\) | \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \) |
\( J_2 \) | \(=\) | \(182588\) | \(=\) | \( 2^{2} \cdot 7 \cdot 6521 \) |
\( J_4 \) | \(=\) | \(1389044168\) | \(=\) | \( 2^{3} \cdot 151 \cdot 761 \cdot 1511 \) |
\( J_6 \) | \(=\) | \(14089048001280\) | \(=\) | \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 3253 \cdot 17903 \) |
\( J_8 \) | \(=\) | \(160761848950725104\) | \(=\) | \( 2^{4} \cdot 1213 \cdot 8283277460363 \) |
\( J_{10} \) | \(=\) | \(241920\) | \(=\) | \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
\( g_1 \) | \(=\) | \(113246073358644668236004/135\) | ||
\( g_2 \) | \(=\) | \(4718399886030325759138/135\) | ||
\( g_3 \) | \(=\) | \(1941575745370456496\) |
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: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) - (1 : 1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 - 11z^3\) | \(0.532646\) | \(\infty\) |
\(D_0 - (1 : -2 : 0) - (1 : 1 : 0)\) | \(9x^2 - 35z^2\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(-22xz^2\) | \(0\) | \(2\) |
\(D_0 - (1 : -2 : 0) - (1 : 1 : 0)\) | \(3x^2 - 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(-7xz^2 + z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) - (1 : 1 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 - 11z^3\) | \(0.532646\) | \(\infty\) |
\(D_0 - (1 : -2 : 0) - (1 : 1 : 0)\) | \(9x^2 - 35z^2\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(-22xz^2\) | \(0\) | \(2\) |
\(D_0 - (1 : -2 : 0) - (1 : 1 : 0)\) | \(3x^2 - 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(-7xz^2 + z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 0 : 1) - (1 : 3 : 0)\) | \(z (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3x^3 + xz^2 - 22z^3\) | \(0.532646\) | \(\infty\) |
\(D_0 - (1 : -3 : 0) - (1 : 3 : 0)\) | \(9x^2 - 35z^2\) | \(=\) | \(0,\) | \(9y\) | \(=\) | \(x^3 - 43xz^2\) | \(0\) | \(2\) |
\(D_0 - (1 : -3 : 0) - (1 : 3 : 0)\) | \(3x^2 - 11z^2\) | \(=\) | \(0,\) | \(3y\) | \(=\) | \(x^3 - 13xz^2 + 2z^3\) | \(0\) | \(4\) |
2-torsion field: \(\Q(\sqrt{3}, \sqrt{35})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.532646 \) |
Real period: | \( 12.08548 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.804661 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(8\) | \(4\) | \(1 + T\) | |
\(3\) | \(2\) | \(3\) | \(2\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | 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 210.a
Elliptic curve isogeny class 48.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 \(\Q\).