Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = x^4 - x^3 + 2x^2 + 2$ | (homogenize, simplify) |
$y^2 + x^3y = x^4z^2 - x^3z^3 + 2x^2z^4 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^4 - 4x^3 + 8x^2 + 8$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(27200\) | \(=\) | \( 2^{6} \cdot 5^{2} \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-680000\) | \(=\) | \( - 2^{6} \cdot 5^{4} \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(292\) | \(=\) | \( 2^{2} \cdot 73 \) |
\( I_4 \) | \(=\) | \(1084\) | \(=\) | \( 2^{2} \cdot 271 \) |
\( I_6 \) | \(=\) | \(64760\) | \(=\) | \( 2^{3} \cdot 5 \cdot 1619 \) |
\( I_{10} \) | \(=\) | \(85000\) | \(=\) | \( 2^{3} \cdot 5^{4} \cdot 17 \) |
\( J_2 \) | \(=\) | \(292\) | \(=\) | \( 2^{2} \cdot 73 \) |
\( J_4 \) | \(=\) | \(2830\) | \(=\) | \( 2 \cdot 5 \cdot 283 \) |
\( J_6 \) | \(=\) | \(58684\) | \(=\) | \( 2^{2} \cdot 17 \cdot 863 \) |
\( J_8 \) | \(=\) | \(2281707\) | \(=\) | \( 3^{2} \cdot 59 \cdot 4297 \) |
\( J_{10} \) | \(=\) | \(680000\) | \(=\) | \( 2^{6} \cdot 5^{4} \cdot 17 \) |
\( g_1 \) | \(=\) | \(33169145488/10625\) | ||
\( g_2 \) | \(=\) | \(220183622/2125\) | ||
\( g_3 \) | \(=\) | \(4598927/625\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 3 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 2z^3\) | \(0.090887\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + 2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 3 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 2z^3\) | \(0.090887\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 + 2z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 5 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 4z^3\) | \(0.090887\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2xz^2 + 4z^3\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.090887 \) |
Real period: | \( 9.796349 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.890367 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(6\) | \(1\) | \(1 + 2 T + 2 T^{2}\) | |
\(5\) | \(2\) | \(4\) | \(4\) | \(( 1 - T )( 1 + T )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 17 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.1 | yes |
\(3\) | 3.45.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 2.2.8.1-425.2-d
Elliptic curve isogeny class 2.2.8.1-425.1-d
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)
Of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |