Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^5 + 15x^4 + 20x^3 - 297x^2 + 94x - 8$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^5z + 15x^4z^2 + 20x^3z^3 - 297x^2z^4 + 94xz^5 - 8z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 60x^4 + 80x^3 - 1187x^2 + 378x - 31$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(810\) | \(=\) | \( 2 \cdot 3^{4} \cdot 5 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-196830\) | \(=\) | \( - 2 \cdot 3^{9} \cdot 5 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(103200\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 43 \) |
\( I_4 \) | \(=\) | \(92148840\) | \(=\) | \( 2^{3} \cdot 3^{4} \cdot 5 \cdot 7 \cdot 17 \cdot 239 \) |
\( I_6 \) | \(=\) | \(2874875039973\) | \(=\) | \( 3^{2} \cdot 17 \cdot 18790032941 \) |
\( I_{10} \) | \(=\) | \(-3240\) | \(=\) | \( - 2^{3} \cdot 3^{4} \cdot 5 \) |
\( J_2 \) | \(=\) | \(154800\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 43 \) |
\( J_4 \) | \(=\) | \(860236740\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 97 \cdot 1493 \) |
\( J_6 \) | \(=\) | \(5905731060081\) | \(=\) | \( 3^{4} \cdot 31 \cdot 2351943871 \) |
\( J_8 \) | \(=\) | \(43549979813677800\) | \(=\) | \( 2^{3} \cdot 3^{10} \cdot 5^{2} \cdot 7753 \cdot 475637 \) |
\( J_{10} \) | \(=\) | \(-196830\) | \(=\) | \( - 2 \cdot 3^{9} \cdot 5 \) |
\( g_1 \) | \(=\) | \(-451609936896000000000\) | ||
\( g_2 \) | \(=\) | \(-16212110811776000000\) | ||
\( g_3 \) | \(=\) | \(-2156977131869584000/3\) |
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: \(\Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 6xz - 31z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 6xz - 31z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 6xz - 31z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
2-torsion field: 6.2.186624000.3
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 0.328982 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.328982 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(3\) | \(4\) | \(9\) | \(4\) | \(1 - T\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 5 T^{2} )\) |
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 |
\(3\) | 3.640.4 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
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 30.a
Elliptic curve isogeny class 27.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(3\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(27\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-3}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |