Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -2x^4 + 2x^3 - 2x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -2x^4z^2 + 2x^3z^3 - 2x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 - 8x^4 + 10x^3 - 8x^2 + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(20172\) | \(=\) | \( 2^{2} \cdot 3 \cdot 41^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-161376\) | \(=\) | \( - 2^{5} \cdot 3 \cdot 41^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(332\) | \(=\) | \( 2^{2} \cdot 83 \) |
\( I_4 \) | \(=\) | \(-7751\) | \(=\) | \( - 23 \cdot 337 \) |
\( I_6 \) | \(=\) | \(-248141\) | \(=\) | \( -248141 \) |
\( I_{10} \) | \(=\) | \(20656128\) | \(=\) | \( 2^{12} \cdot 3 \cdot 41^{2} \) |
\( J_2 \) | \(=\) | \(83\) | \(=\) | \( 83 \) |
\( J_4 \) | \(=\) | \(610\) | \(=\) | \( 2 \cdot 5 \cdot 61 \) |
\( J_6 \) | \(=\) | \(-2676\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 223 \) |
\( J_8 \) | \(=\) | \(-148552\) | \(=\) | \( - 2^{3} \cdot 31 \cdot 599 \) |
\( J_{10} \) | \(=\) | \(161376\) | \(=\) | \( 2^{5} \cdot 3 \cdot 41^{2} \) |
\( g_1 \) | \(=\) | \(3939040643/161376\) | ||
\( g_2 \) | \(=\) | \(174395035/80688\) | ||
\( g_3 \) | \(=\) | \(-1536247/13448\) |
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: \(\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.449413\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.449413\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0.449413\) | \(\infty\) |
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.557715456.2
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.449413 \) |
Real period: | \( 10.82413 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.216128 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(5\) | \(4\) | \(( 1 - T )( 1 + T )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(41\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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 246.f
Elliptic curve isogeny class 82.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 \(\Q\).