Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -10x^6 + 81x^4 - 227x^2 + 210$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -10x^6 + 81x^4z^2 - 227x^2z^4 + 210z^6$ | (dehomogenize, simplify) |
$y^2 = -39x^6 + 326x^4 - 907x^2 + 840$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(131040\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(131040\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(3148328\) | \(=\) | \( 2^{3} \cdot 393541 \) |
\( I_4 \) | \(=\) | \(708952\) | \(=\) | \( 2^{3} \cdot 23 \cdot 3853 \) |
\( I_6 \) | \(=\) | \(743814934788\) | \(=\) | \( 2^{2} \cdot 3 \cdot 97 \cdot 9431 \cdot 67757 \) |
\( I_{10} \) | \(=\) | \(524160\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \) |
\( J_2 \) | \(=\) | \(1574164\) | \(=\) | \( 2^{2} \cdot 393541 \) |
\( J_4 \) | \(=\) | \(103249560962\) | \(=\) | \( 2 \cdot 17 \cdot 83 \cdot 5927 \cdot 6173 \) |
\( J_6 \) | \(=\) | \(9029520569946240\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31 \cdot 555698369 \) |
\( J_8 \) | \(=\) | \(888368594905774644479\) | \(=\) | \( 888368594905774644479 \) |
\( J_{10} \) | \(=\) | \(131040\) | \(=\) | \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \) |
\( g_1 \) | \(=\) | \(302064649214662608101539958432/4095\) | ||
\( g_2 \) | \(=\) | \(12586012647194024913614166004/4095\) | ||
\( g_3 \) | \(=\) | \(170750018582492394877376\) |
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 everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 21z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-7xz^2\) | \(0.532646\) | \(\infty\) |
\(D_0 - D_\infty\) | \(13x^2 - 35z^2\) | \(=\) | \(0,\) | \(13y\) | \(=\) | \(-24xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 21z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-7xz^2\) | \(0.532646\) | \(\infty\) |
\(D_0 - D_\infty\) | \(13x^2 - 35z^2\) | \(=\) | \(0,\) | \(13y\) | \(=\) | \(-24xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(8x^2 - 21z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 - 13xz^2\) | \(0.532646\) | \(\infty\) |
\(D_0 - D_\infty\) | \(13x^2 - 35z^2\) | \(=\) | \(0,\) | \(13y\) | \(=\) | \(x^3 - 47xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.8.227515262607360000.28
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.532646 \) |
Real period: | \( 10.74591 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.430943 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(5\) | \(1\) | \(1 + T\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 13 T^{2} )\) |
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 210.a
Elliptic curve isogeny class 624.d
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 \(\Q\).