This is a model for the modular curve $X_0(50)$. This is the largest level $N \in \mathbb{N}$ such that $X_0(N)$ has genus 2.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 + 2x^3 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z + 2x^3z^3 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 10x^3 + 4x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2500\) | \(=\) | \( 2^{2} \cdot 5^{4} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(50000\) | \(=\) | \( 2^{4} \cdot 5^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(100\) | \(=\) | \( 2^{2} \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(625\) | \(=\) | \( 5^{4} \) |
\( I_6 \) | \(=\) | \(21385\) | \(=\) | \( 5 \cdot 7 \cdot 13 \cdot 47 \) |
\( I_{10} \) | \(=\) | \(2048\) | \(=\) | \( 2^{11} \) |
\( J_2 \) | \(=\) | \(125\) | \(=\) | \( 5^{3} \) |
\( J_4 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_6 \) | \(=\) | \(-10000\) | \(=\) | \( - 2^{4} \cdot 5^{4} \) |
\( J_8 \) | \(=\) | \(-312500\) | \(=\) | \( - 2^{2} \cdot 5^{7} \) |
\( J_{10} \) | \(=\) | \(50000\) | \(=\) | \( 2^{4} \cdot 5^{5} \) |
\( g_1 \) | \(=\) | \(9765625/16\) | ||
\( g_2 \) | \(=\) | \(0\) | ||
\( g_3 \) | \(=\) | \(-3125\) |
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: \(\Z/{15}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(15\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2z^3\) | \(0\) | \(15\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + 5z^3\) | \(0\) | \(15\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 10.23546 \) |
Tamagawa product: | \( 15 \) |
Torsion order: | \( 15 \) |
Leading coefficient: | \( 0.682364 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(4\) | \(5\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(4\) | \(5\) | \(3\) | \(1\) |
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? |
---|---|---|
\(2\) | 2.60.2 | no |
\(3\) | 3.8640.9 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_1)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
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 50.a
Elliptic curve isogeny class 50.b
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) 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 Eichler order of index \(5\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |