Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^6 + x^5 + 4x^4 + 6x^3 + 6x^2 + 9x + 5$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^6 + x^5z + 4x^4z^2 + 6x^3z^3 + 6x^2z^4 + 9xz^5 + 5z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^6 + 4x^5 + 17x^4 + 26x^3 + 25x^2 + 36x + 20$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2028\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(64896\) | \(=\) | \( 2^{7} \cdot 3 \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(8092\) | \(=\) | \( 2^{2} \cdot 7 \cdot 17^{2} \) |
\( I_4 \) | \(=\) | \(4165489\) | \(=\) | \( 4165489 \) |
\( I_6 \) | \(=\) | \(8401719159\) | \(=\) | \( 3^{2} \cdot 53 \cdot 661 \cdot 26647 \) |
\( I_{10} \) | \(=\) | \(-8306688\) | \(=\) | \( - 2^{14} \cdot 3 \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(2023\) | \(=\) | \( 7 \cdot 17^{2} \) |
\( J_4 \) | \(=\) | \(-3040\) | \(=\) | \( - 2^{5} \cdot 5 \cdot 19 \) |
\( J_6 \) | \(=\) | \(6464\) | \(=\) | \( 2^{6} \cdot 101 \) |
\( J_8 \) | \(=\) | \(958768\) | \(=\) | \( 2^{4} \cdot 31 \cdot 1933 \) |
\( J_{10} \) | \(=\) | \(-64896\) | \(=\) | \( - 2^{7} \cdot 3 \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(-33882809484846343/64896\) | ||
\( g_2 \) | \(=\) | \(786522685865/2028\) | ||
\( g_3 \) | \(=\) | \(-413345429/1014\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(1\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 2 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 + x^2z + xz^2 - 2z^3\) | \(0\) | \(6\) |
2-torsion field: 6.2.2336256.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 6.730298 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 0.373905 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(7\) | \(2\) | \(( 1 + T )^{2}\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 3 T^{2} )\) | |
\(13\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.60.1 | 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 26.a
Elliptic curve isogeny class 78.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(5\) 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 \(\Q\).