Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -5x^6 + 11x^5 - 20x^4 + 20x^3 - 20x^2 + 11x - 5$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -5x^6 + 11x^5z - 20x^4z^2 + 20x^3z^3 - 20x^2z^4 + 11xz^5 - 5z^6$ | (dehomogenize, simplify) |
$y^2 = -20x^6 + 44x^5 - 79x^4 + 82x^3 - 79x^2 + 44x - 20$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(644\) | \(=\) | \( 2^{2} \cdot 7 \cdot 23 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-2576\) | \(=\) | \( - 2^{4} \cdot 7 \cdot 23 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(39036\) | \(=\) | \( 2^{2} \cdot 3 \cdot 3253 \) |
\( I_4 \) | \(=\) | \(4124865\) | \(=\) | \( 3 \cdot 5 \cdot 73 \cdot 3767 \) |
\( I_6 \) | \(=\) | \(50880984159\) | \(=\) | \( 3 \cdot 73 \cdot 643 \cdot 361327 \) |
\( I_{10} \) | \(=\) | \(329728\) | \(=\) | \( 2^{11} \cdot 7 \cdot 23 \) |
\( J_2 \) | \(=\) | \(9759\) | \(=\) | \( 3 \cdot 3253 \) |
\( J_4 \) | \(=\) | \(3796384\) | \(=\) | \( 2^{5} \cdot 31 \cdot 43 \cdot 89 \) |
\( J_6 \) | \(=\) | \(1910683600\) | \(=\) | \( 2^{4} \cdot 5^{2} \cdot 7 \cdot 23 \cdot 29669 \) |
\( J_8 \) | \(=\) | \(1058457444236\) | \(=\) | \( 2^{2} \cdot 269891 \cdot 980449 \) |
\( J_{10} \) | \(=\) | \(2576\) | \(=\) | \( 2^{4} \cdot 7 \cdot 23 \) |
\( g_1 \) | \(=\) | \(88516980336138032799/2576\) | ||
\( g_2 \) | \(=\) | \(220529201888022246/161\) | ||
\( g_3 \) | \(=\) | \(70640465629725\) |
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);
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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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(6\) |
2-torsion field: 8.0.1698758656.7
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 3.928431 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.218246 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(1\) | \(( 1 + T )^{2}\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 23 T^{2} )\) |
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
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 14.a
Elliptic curve isogeny class 46.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 \(\Q\).