Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -82x^6 + 97x^4 - 18x^2 - 7$ | (homogenize, simplify) |
$y^2 + xz^2y = -82x^6 + 97x^4z^2 - 18x^2z^4 - 7z^6$ | (dehomogenize, simplify) |
$y^2 = -328x^6 + 388x^4 - 71x^2 - 28$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(330624\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 7 \cdot 41 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-330624\) | \(=\) | \( - 2^{7} \cdot 3^{2} \cdot 7 \cdot 41 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(220424\) | \(=\) | \( 2^{3} \cdot 59 \cdot 467 \) |
\( I_4 \) | \(=\) | \(3036230395\) | \(=\) | \( 5 \cdot 11 \cdot 103 \cdot 107 \cdot 5009 \) |
\( I_6 \) | \(=\) | \(278853472846809\) | \(=\) | \( 3^{2} \cdot 7 \cdot 17 \cdot 31 \cdot 59 \cdot 829 \cdot 171719 \) |
\( I_{10} \) | \(=\) | \(41328\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 7 \cdot 41 \) |
\( J_2 \) | \(=\) | \(220424\) | \(=\) | \( 2^{3} \cdot 59 \cdot 467 \) |
\( J_4 \) | \(=\) | \(293894\) | \(=\) | \( 2 \cdot 23 \cdot 6389 \) |
\( J_6 \) | \(=\) | \(-99142143883776\) | \(=\) | \( - 2^{9} \cdot 3^{2} \cdot 7 \cdot 41 \cdot 59 \cdot 1270609 \) |
\( J_8 \) | \(=\) | \(-5463327002452781065\) | \(=\) | \( - 5 \cdot 71 \cdot 223 \cdot 69011899228861 \) |
\( J_{10} \) | \(=\) | \(330624\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 7 \cdot 41 \) |
\( g_1 \) | \(=\) | \(4065223489663723989246208/2583\) | ||
\( g_2 \) | \(=\) | \(24589935397587408152/2583\) | ||
\( g_3 \) | \(=\) | \(-14569400726250285824\) |
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 $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(41x^2 - 28z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(41x^2 - 28z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(41x^2 - 28z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.4.144063024343547904.5
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 0.168180 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.513625 \) |
Analytic order of Ш: | \( 36 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(7\) | \(1\) | \(1 + T + 2 T^{2}\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 7 T^{2} )\) | |
\(41\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 41 T^{2} )\) |
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 15744.v
Elliptic curve isogeny class 21.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 \(\Q\).