Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = 47x^6 + 155x^4 + 170x^2 + 62$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = 47x^6 + 155x^4z^2 + 170x^2z^4 + 62z^6$ | (dehomogenize, simplify) |
$y^2 = 189x^6 + 622x^4 + 681x^2 + 248$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(20832\) | \(=\) | \( 2^{5} \cdot 3 \cdot 7 \cdot 31 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-187488\) | \(=\) | \( - 2^{5} \cdot 3^{3} \cdot 7 \cdot 31 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(4506648\) | \(=\) | \( 2^{3} \cdot 3 \cdot 19 \cdot 9883 \) |
\( I_4 \) | \(=\) | \(3006744\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \cdot 23 \cdot 419 \) |
\( I_6 \) | \(=\) | \(4515054549276\) | \(=\) | \( 2^{2} \cdot 3 \cdot 376254545773 \) |
\( I_{10} \) | \(=\) | \(749952\) | \(=\) | \( 2^{7} \cdot 3^{3} \cdot 7 \cdot 31 \) |
\( J_2 \) | \(=\) | \(2253324\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 9883 \) |
\( J_4 \) | \(=\) | \(211560709250\) | \(=\) | \( 2 \cdot 5^{3} \cdot 11 \cdot 3491 \cdot 22037 \) |
\( J_6 \) | \(=\) | \(26484031592742528\) | \(=\) | \( 2^{7} \cdot 3^{3} \cdot 7 \cdot 31 \cdot 15727 \cdot 2245457 \) |
\( J_8 \) | \(=\) | \(3729792576580482150143\) | \(=\) | \( 163 \cdot 22882163046506025461 \) |
\( J_{10} \) | \(=\) | \(187488\) | \(=\) | \( 2^{5} \cdot 3^{3} \cdot 7 \cdot 31 \) |
\( g_1 \) | \(=\) | \(67236402653316392567461245216/217\) | ||
\( g_2 \) | \(=\) | \(2801510141282484875195210500/217\) | ||
\( g_3 \) | \(=\) | \(717229106418825429829056\) |
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);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(27x^2 + 31z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(2xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(7x^2 + 8z^2\) | \(=\) | \(0,\) | \(14y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(27x^2 + 31z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(2xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(7x^2 + 8z^2\) | \(=\) | \(0,\) | \(14y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(27x^2 + 31z^2\) | \(=\) | \(0,\) | \(27y\) | \(=\) | \(x^3 + 5xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(7x^2 + 8z^2\) | \(=\) | \(0,\) | \(14y\) | \(=\) | \(x^3 + 3xz^2\) | \(0\) | \(2\) |
2-torsion field: 8.0.11770743200219136.209
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 2.657054 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.992790 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(5\) | \(5\) | \(1\) | \(1 - T\) | |
\(3\) | \(1\) | \(3\) | \(3\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(31\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 8 T + 31 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | 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 336.e
Elliptic curve isogeny class 62.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 \(\Q\).