Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^3z^3 - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 3x^4 + 3x^2 - 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(576\) | \(=\) | \( 2^{6} \cdot 3^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-576\) | \(=\) | \( - 2^{6} \cdot 3^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(68\) | \(=\) | \( 2^{2} \cdot 17 \) |
\( I_4 \) | \(=\) | \(124\) | \(=\) | \( 2^{2} \cdot 31 \) |
\( I_6 \) | \(=\) | \(2616\) | \(=\) | \( 2^{3} \cdot 3 \cdot 109 \) |
\( I_{10} \) | \(=\) | \(72\) | \(=\) | \( 2^{3} \cdot 3^{2} \) |
\( J_2 \) | \(=\) | \(68\) | \(=\) | \( 2^{2} \cdot 17 \) |
\( J_4 \) | \(=\) | \(110\) | \(=\) | \( 2 \cdot 5 \cdot 11 \) |
\( J_6 \) | \(=\) | \(-36\) | \(=\) | \( - 2^{2} \cdot 3^{2} \) |
\( J_8 \) | \(=\) | \(-3637\) | \(=\) | \( -3637 \) |
\( J_{10} \) | \(=\) | \(576\) | \(=\) | \( 2^{6} \cdot 3^{2} \) |
\( g_1 \) | \(=\) | \(22717712/9\) | ||
\( g_2 \) | \(=\) | \(540430/9\) | ||
\( g_3 \) | \(=\) | \(-289\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{10}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z + xz^2 - z^3\) | \(0\) | \(10\) |
2-torsion field: \(\Q(\zeta_{12})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 22.39625 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 10 \) |
Leading coefficient: | \( 0.223962 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 + 2 T + 2 T^{2}\) | |
\(3\) | \(2\) | \(2\) | \(1\) | \(1 + T^{2}\) |
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? |
---|---|---|
\(2\) | 2.180.4 | yes |
\(3\) | 3.1080.16 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_2$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 2.2.8.1-9.1-a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |