Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = -21x^6 - 88x^4 - 123x^2 - 57$ | (homogenize, simplify) |
$y^2 + xz^2y = -21x^6 - 88x^4z^2 - 123x^2z^4 - 57z^6$ | (dehomogenize, simplify) |
$y^2 = -84x^6 - 352x^4 - 491x^2 - 228$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(76608\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-76608\) | \(=\) | \( - 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(920224\) | \(=\) | \( 2^{5} \cdot 149 \cdot 193 \) |
\( I_4 \) | \(=\) | \(53821\) | \(=\) | \( 107 \cdot 503 \) |
\( I_6 \) | \(=\) | \(16505652732\) | \(=\) | \( 2^{2} \cdot 3 \cdot 9437 \cdot 145753 \) |
\( I_{10} \) | \(=\) | \(9576\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \) |
\( J_2 \) | \(=\) | \(920224\) | \(=\) | \( 2^{5} \cdot 149 \cdot 193 \) |
\( J_4 \) | \(=\) | \(35283806210\) | \(=\) | \( 2 \cdot 5 \cdot 1039 \cdot 3395939 \) |
\( J_6 \) | \(=\) | \(1803829961380608\) | \(=\) | \( 2^{8} \cdot 3^{2} \cdot 7 \cdot 19 \cdot 5886558719 \) |
\( J_8 \) | \(=\) | \(103745160429168513023\) | \(=\) | \( 1069 \cdot 121291 \cdot 755147 \cdot 1059571 \) |
\( J_{10} \) | \(=\) | \(76608\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 19 \) |
\( g_1 \) | \(=\) | \(10310691783200514787538108416/1197\) | ||
\( g_2 \) | \(=\) | \(429611720754327142357775360/1197\) | ||
\( g_3 \) | \(=\) | \(19939239196668773298176\) |
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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(14x^2 + 19z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(14x^2 + 19z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(14x^2 + 19z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 4z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.1661007193767936.315
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 1 \) |
Real period: | \( 3.786797 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.893398 \) |
Analytic order of Ш: | \( 8 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 - T + 2 T^{2}\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 4 T + 19 T^{2} )\) |
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 57.c
Elliptic curve isogeny class 1344.h
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 \(\Q\).