Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = 6x^6 + 7x^5 + 19x^4 + 13x^3 + 19x^2 + 7x + 6$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = 6x^6 + 7x^5z + 19x^4z^2 + 13x^3z^3 + 19x^2z^4 + 7xz^5 + 6z^6$ | (dehomogenize, simplify) |
$y^2 = 24x^6 + 28x^5 + 77x^4 + 54x^3 + 77x^2 + 28x + 24$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(16146\) | \(=\) | \( 2 \cdot 3^{3} \cdot 13 \cdot 23 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-16146\) | \(=\) | \( - 2 \cdot 3^{3} \cdot 13 \cdot 23 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(92124\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 853 \) |
\( I_4 \) | \(=\) | \(2258865\) | \(=\) | \( 3^{2} \cdot 5 \cdot 7 \cdot 71 \cdot 101 \) |
\( I_6 \) | \(=\) | \(68236671159\) | \(=\) | \( 3^{4} \cdot 103 \cdot 8178913 \) |
\( I_{10} \) | \(=\) | \(2066688\) | \(=\) | \( 2^{8} \cdot 3^{3} \cdot 13 \cdot 23 \) |
\( J_2 \) | \(=\) | \(23031\) | \(=\) | \( 3^{3} \cdot 853 \) |
\( J_4 \) | \(=\) | \(22007004\) | \(=\) | \( 2^{2} \cdot 3 \cdot 157 \cdot 11681 \) |
\( J_6 \) | \(=\) | \(27932784516\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \cdot 23 \cdot 389 \cdot 953 \) |
\( J_8 \) | \(=\) | \(39752933782995\) | \(=\) | \( 3^{2} \cdot 5 \cdot 883398528511 \) |
\( J_{10} \) | \(=\) | \(16146\) | \(=\) | \( 2 \cdot 3^{3} \cdot 13 \cdot 23 \) |
\( g_1 \) | \(=\) | \(239993905095208279413/598\) | ||
\( g_2 \) | \(=\) | \(4978580669663460966/299\) | ||
\( g_3 \) | \(=\) | \(917645361271506\) |
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 everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.277159\) | \(\infty\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.277159\) | \(\infty\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(-xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + 3xz^2 + 2z^3\) | \(0.277159\) | \(\infty\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^2z - xz^2 + 6z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.4271199289344.9
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 0.277159 \) |
Real period: | \( 4.616489 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.279504 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(3\) | \(3\) | \(1\) | \(1 - T\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 6 T + 13 T^{2} )\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 23 T^{2} )\) |
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 |
\(3\) | 3.90.1 | 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 69.a
Elliptic curve isogeny class 234.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 \(\Q\).