This is a model for the quotient of the modular curve $X_0(129)$ by the action of the group of order 4 generated by the Atkin-Lehner involutions $w_3$ and $w_{43}$. This quotient is also denoted $X_0^*(129)$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - 3x^5 + x^4 + 3x^3 - x^2 - x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - 3x^5z + x^4z^2 + 3x^3z^3 - x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 12x^5 + 4x^4 + 12x^3 - 4x^2 - 4x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(5547\) | \(=\) | \( 3 \cdot 43^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(16641\) | \(=\) | \( 3^{2} \cdot 43^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(520\) | \(=\) | \( 2^{3} \cdot 5 \cdot 13 \) |
\( I_4 \) | \(=\) | \(6292\) | \(=\) | \( 2^{2} \cdot 11^{2} \cdot 13 \) |
\( I_6 \) | \(=\) | \(896816\) | \(=\) | \( 2^{4} \cdot 23 \cdot 2437 \) |
\( I_{10} \) | \(=\) | \(66564\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 43^{2} \) |
\( J_2 \) | \(=\) | \(260\) | \(=\) | \( 2^{2} \cdot 5 \cdot 13 \) |
\( J_4 \) | \(=\) | \(1768\) | \(=\) | \( 2^{3} \cdot 13 \cdot 17 \) |
\( J_6 \) | \(=\) | \(16776\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 233 \) |
\( J_8 \) | \(=\) | \(308984\) | \(=\) | \( 2^{3} \cdot 13 \cdot 2971 \) |
\( J_{10} \) | \(=\) | \(16641\) | \(=\) | \( 3^{2} \cdot 43^{2} \) |
\( g_1 \) | \(=\) | \(1188137600000/16641\) | ||
\( g_2 \) | \(=\) | \(31074368000/16641\) | ||
\( g_3 \) | \(=\) | \(126006400/1849\) |
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
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((2 : 1 : 1)\) | \((2 : -2 : 1)\) | \((-5 : 20 : 7)\) | \((12 : 20 : 7)\) |
\((-5 : -363 : 7)\) | \((12 : -363 : 7)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((-1 : -2 : 1)\) | \((2 : 1 : 1)\) | \((2 : -2 : 1)\) | \((-5 : 20 : 7)\) | \((12 : 20 : 7)\) |
\((-5 : -363 : 7)\) | \((12 : -363 : 7)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) | \((2 : -3 : 1)\) | \((2 : 3 : 1)\) | \((-5 : -383 : 7)\) | \((-5 : 383 : 7)\) |
\((12 : -383 : 7)\) | \((12 : 383 : 7)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.081387\) | \(\infty\) |
\((-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.081387\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0.081387\) | \(\infty\) |
\((-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - z^3\) | \(0.081387\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 1 : 1) + (2 : -3 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 + z^3\) | \(0.081387\) | \(\infty\) |
\((-1 : -3 : 1) + (1 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2xz^2 - z^3\) | \(0.081387\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.006279 \) |
Real period: | \( 24.46037 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.307177 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(43\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.30.2 | no |
\(3\) | 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 129.a
Elliptic curve isogeny class 43.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 \(\Q\).
Additional information
The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Bars--González--Xarles using 2-cover descent and elliptic Chabauty and Edixhoven--Lido using geometric quadratic Chabauty.