This is a model for the modular curve $X_0(26)$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 2x^5 + 2x^4 + 4x^3 + 2x^2 + 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 2x^5z + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 + 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 8x^5 + 8x^4 + 18x^3 + 8x^2 + 8x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(676\) | \(=\) | \( 2^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(562432\) | \(=\) | \( 2^{8} \cdot 13^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1620\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 5 \) |
\( I_4 \) | \(=\) | \(52953\) | \(=\) | \( 3 \cdot 19 \cdot 929 \) |
\( I_6 \) | \(=\) | \(29527389\) | \(=\) | \( 3^{3} \cdot 251 \cdot 4357 \) |
\( I_{10} \) | \(=\) | \(71991296\) | \(=\) | \( 2^{15} \cdot 13^{3} \) |
\( J_2 \) | \(=\) | \(405\) | \(=\) | \( 3^{4} \cdot 5 \) |
\( J_4 \) | \(=\) | \(4628\) | \(=\) | \( 2^{2} \cdot 13 \cdot 89 \) |
\( J_6 \) | \(=\) | \(-8112\) | \(=\) | \( - 2^{4} \cdot 3 \cdot 13^{2} \) |
\( J_8 \) | \(=\) | \(-6175936\) | \(=\) | \( - 2^{6} \cdot 13^{2} \cdot 571 \) |
\( J_{10} \) | \(=\) | \(562432\) | \(=\) | \( 2^{8} \cdot 13^{3} \) |
\( g_1 \) | \(=\) | \(10896201253125/562432\) | ||
\( g_2 \) | \(=\) | \(5912281125/10816\) | ||
\( g_3 \) | \(=\) | \(-492075/208\) |
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 everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{21}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + xz + 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(xz^2 - 3z^3\) | \(0\) | \(21\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + xz + 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(xz^2 - 3z^3\) | \(0\) | \(21\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 + xz + 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 + 2xz^2 - 5z^3\) | \(0\) | \(21\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 6.723259 \) |
Tamagawa product: | \( 21 \) |
Torsion order: | \( 21 \) |
Leading coefficient: | \( 0.320155 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(8\) | \(7\) | \(( 1 - T )( 1 + T )\) | |
\(13\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.60.2 | no |
\(3\) | 3.2160.20 | yes |
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 26.a
Elliptic curve isogeny class 26.b
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).