Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = -51x^6 - 315x^4 - 104x^2 - 9$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = -51x^6 - 315x^4z^2 - 104x^2z^4 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -204x^6 - 1259x^4 - 414x^2 - 35$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(114240\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-114240\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1256652\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 67 \cdot 521 \) |
\( I_4 \) | \(=\) | \(52204481283\) | \(=\) | \( 3 \cdot 29 \cdot 600051509 \) |
\( I_6 \) | \(=\) | \(17891676439496172\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 151 \cdot 3847 \cdot 855557891 \) |
\( I_{10} \) | \(=\) | \(14280\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
\( J_2 \) | \(=\) | \(1256652\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 67 \cdot 521 \) |
\( J_4 \) | \(=\) | \(30995939524\) | \(=\) | \( 2^{2} \cdot 11 \cdot 704453171 \) |
\( J_6 \) | \(=\) | \(838652756430720\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17 \cdot 7789 \cdot 42841 \) |
\( J_8 \) | \(=\) | \(23286599174677950716\) | \(=\) | \( 2^{2} \cdot 11 \cdot 1217233 \cdot 434790126733 \) |
\( J_{10} \) | \(=\) | \(114240\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
\( g_1 \) | \(=\) | \(16322019979578992366124730896/595\) | ||
\( g_2 \) | \(=\) | \(320367650677941067851565476/595\) | ||
\( g_3 \) | \(=\) | \(11592952003636922822112\) |
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: \(\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(13xz^2\) | \(5.366837\) | \(\infty\) |
\(D_0 - D_\infty\) | \(29x^2 + 5z^2\) | \(=\) | \(0,\) | \(29y\) | \(=\) | \(-xz^2 - 12z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(13xz^2\) | \(5.366837\) | \(\infty\) |
\(D_0 - D_\infty\) | \(29x^2 + 5z^2\) | \(=\) | \(0,\) | \(29y\) | \(=\) | \(-xz^2 - 12z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 26xz^2 + z^3\) | \(5.366837\) | \(\infty\) |
\(D_0 - D_\infty\) | \(29x^2 + 5z^2\) | \(=\) | \(0,\) | \(29y\) | \(=\) | \(x^2z - 2xz^2 - 23z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.665323421736960000.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 5.366837 \) |
Real period: | \( 0.350940 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 3.766882 \) |
Analytic order of Ш: | \( 32 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 - T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 17 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.5 | 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 1120.i
Elliptic curve isogeny class 102.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 \(\Q\).