Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -20x^6 + 13x^5 - 61x^4 + 25x^3 - 61x^2 + 13x - 20$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -20x^6 + 13x^5z - 61x^4z^2 + 25x^3z^3 - 61x^2z^4 + 13xz^5 - 20z^6$ | (dehomogenize, simplify) |
$y^2 = -80x^6 + 52x^5 - 243x^4 + 102x^3 - 243x^2 + 52x - 80$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(585750\) | \(=\) | \( 2 \cdot 3 \cdot 5^{3} \cdot 11 \cdot 71 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-585750\) | \(=\) | \( - 2 \cdot 3 \cdot 5^{3} \cdot 11 \cdot 71 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1155100\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 11551 \) |
\( I_4 \) | \(=\) | \(57987985\) | \(=\) | \( 5 \cdot 11 \cdot 1054327 \) |
\( I_6 \) | \(=\) | \(22180182105575\) | \(=\) | \( 5^{2} \cdot 13 \cdot 59^{2} \cdot 223 \cdot 87917 \) |
\( I_{10} \) | \(=\) | \(74976000\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5^{3} \cdot 11 \cdot 71 \) |
\( J_2 \) | \(=\) | \(288775\) | \(=\) | \( 5^{2} \cdot 11551 \) |
\( J_4 \) | \(=\) | \(3472208860\) | \(=\) | \( 2^{2} \cdot 5 \cdot 17 \cdot 29 \cdot 157 \cdot 2243 \) |
\( J_6 \) | \(=\) | \(55629357672900\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 71 \cdot 237427903 \) |
\( J_8 \) | \(=\) | \(1002033348632299475\) | \(=\) | \( 5^{2} \cdot 797 \cdot 50290255891207 \) |
\( J_{10} \) | \(=\) | \(585750\) | \(=\) | \( 2 \cdot 3 \cdot 5^{3} \cdot 11 \cdot 71 \) |
\( g_1 \) | \(=\) | \(16065267067698561152421875/4686\) | ||
\( g_2 \) | \(=\) | \(334460326849742509866250/2343\) | ||
\( g_3 \) | \(=\) | \(7919740162986175750\) |
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);
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Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\R$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 - 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(-37xz^2 + 21z^3\) | \(0.348037\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(-7xz^2 + 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 - 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(-37xz^2 + 21z^3\) | \(0.348037\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(-7xz^2 + 5z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 - 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(x^2z - 73xz^2 + 42z^3\) | \(0.348037\) | \(\infty\) |
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(10y\) | \(=\) | \(x^2z - 13xz^2 + 10z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.3162037824000000.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.348037 \) |
Real period: | \( 2.099589 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.461472 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(5\) | \(3\) | \(3\) | \(1\) | \(1 + T\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 11 T^{2} )\) | |
\(71\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 71 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.45.1 | 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 550.c
Elliptic curve isogeny class 1065.c
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 \(\Q\).