Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -x^6 - x^4 - x^2 - 1$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -x^6 - x^4z^2 - x^2z^4 - z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 - 4x^4 + 2x^3 - 4x^2 - 3$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2178\) | \(=\) | \( 2 \cdot 3^{2} \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-13068\) | \(=\) | \( - 2^{2} \cdot 3^{3} \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1196\) | \(=\) | \( 2^{2} \cdot 13 \cdot 23 \) |
\( I_4 \) | \(=\) | \(22441\) | \(=\) | \( 22441 \) |
\( I_6 \) | \(=\) | \(8056043\) | \(=\) | \( 8056043 \) |
\( I_{10} \) | \(=\) | \(1672704\) | \(=\) | \( 2^{9} \cdot 3^{3} \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(299\) | \(=\) | \( 13 \cdot 23 \) |
\( J_4 \) | \(=\) | \(2790\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 31 \) |
\( J_6 \) | \(=\) | \(27648\) | \(=\) | \( 2^{10} \cdot 3^{3} \) |
\( J_8 \) | \(=\) | \(120663\) | \(=\) | \( 3^{3} \cdot 41 \cdot 109 \) |
\( J_{10} \) | \(=\) | \(13068\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(2389769101499/13068\) | ||
\( g_2 \) | \(=\) | \(4143289345/726\) | ||
\( g_3 \) | \(=\) | \(22886656/121\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\R$ and $\Q_{3}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(4\) |
2-torsion field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 6.589693 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 0.823711 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(2\) | \(3\) | \(1\) | \(( 1 + T )^{2}\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.180.4 | 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 33.a
Elliptic curve isogeny class 66.b
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 \(\Q\).