Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -x^6 - 8x^4 - 24x^2 - 23$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -x^6 - 8x^4z^2 - 24x^2z^4 - 23z^6$ | (dehomogenize, simplify) |
$y^2 = -3x^6 - 30x^4 - 95x^2 - 92$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2484\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 23 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-9936\) | \(=\) | \( - 2^{4} \cdot 3^{3} \cdot 23 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(27960\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 233 \) |
\( I_4 \) | \(=\) | \(133920\) | \(=\) | \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 31 \) |
\( I_6 \) | \(=\) | \(1232036820\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 59 \cdot 16573 \) |
\( I_{10} \) | \(=\) | \(39744\) | \(=\) | \( 2^{6} \cdot 3^{3} \cdot 23 \) |
\( J_2 \) | \(=\) | \(13980\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 233 \) |
\( J_4 \) | \(=\) | \(8121030\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 270701 \) |
\( J_6 \) | \(=\) | \(6274451520\) | \(=\) | \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 23 \cdot 94723 \) |
\( J_8 \) | \(=\) | \(5441425997175\) | \(=\) | \( 3^{2} \cdot 5^{2} \cdot 7 \cdot 3454873649 \) |
\( J_{10} \) | \(=\) | \(9936\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 23 \) |
\( g_1 \) | \(=\) | \(1236095741507400000/23\) | ||
\( g_2 \) | \(=\) | \(51362822628555000/23\) | ||
\( g_3 \) | \(=\) | \(123418006728000\) |
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 $\R$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 3xz^2 + 2z^3\) | \(0\) | \(6\) |
2-torsion field: 8.0.1579585536.10
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.060188 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.843364 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(3\) | \(1 - T + 2 T^{2}\) | |
\(3\) | \(3\) | \(3\) | \(1\) | \(1 - T\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 23 T^{2} )\) |
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 |
\(3\) | 3.720.4 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
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 36.a
Elliptic curve isogeny class 69.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(8\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-3}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |