Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 + 9x^4 - 40x^2 + 55$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 + 9x^4z^2 - 40x^2z^4 + 55z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 + 38x^4 - 159x^2 + 220$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(1320\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(2640\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(63768\) | \(=\) | \( 2^{3} \cdot 3 \cdot 2657 \) |
\( I_4 \) | \(=\) | \(10392\) | \(=\) | \( 2^{3} \cdot 3 \cdot 433 \) |
\( I_6 \) | \(=\) | \(220729308\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 968111 \) |
\( I_{10} \) | \(=\) | \(10560\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \) |
\( J_2 \) | \(=\) | \(31884\) | \(=\) | \( 2^{2} \cdot 3 \cdot 2657 \) |
\( J_4 \) | \(=\) | \(42356162\) | \(=\) | \( 2 \cdot 21178081 \) |
\( J_6 \) | \(=\) | \(75020763840\) | \(=\) | \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 31 \cdot 101 \cdot 2269 \) |
\( J_8 \) | \(=\) | \(149479393726079\) | \(=\) | \( 16349 \cdot 9143029771 \) |
\( J_{10} \) | \(=\) | \(2640\) | \(=\) | \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
\( g_1 \) | \(=\) | \(686471900571962215488/55\) | ||
\( g_2 \) | \(=\) | \(28601826290311163976/55\) | ||
\( g_3 \) | \(=\) | \(28888377841215936\) |
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: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : 5 : 1) + (2 : -5 : 1) - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-5xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \((x - 2z) (x + 2z)\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 9xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2 + 2z^3\) | \(0\) | \(4\) |
2-torsion field: \(\Q(\sqrt{5}, \sqrt{33})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 17.74674 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.554585 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 - T + 2 T^{2}\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 11 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | 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 24.a
Elliptic curve isogeny class 55.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 \(\Q\).