This is a model for the modular curve $X_1(13)$. The integer $13$ is the smallest $N \in \mathbb{N}$ such that $X_1(N)$ has genus $2$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^5 + x^4$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + x^4z^2$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 6x^4 + 2x^3 + x^2 + 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(169\) | \(=\) | \( 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-169\) | \(=\) | \( - 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(4\) | \(=\) | \( 2^{2} \) |
\( I_4 \) | \(=\) | \(793\) | \(=\) | \( 13 \cdot 61 \) |
\( I_6 \) | \(=\) | \(3757\) | \(=\) | \( 13 \cdot 17^{2} \) |
\( I_{10} \) | \(=\) | \(-21632\) | \(=\) | \( - 2^{7} \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(1\) | \(=\) | \( 1 \) |
\( J_4 \) | \(=\) | \(-33\) | \(=\) | \( - 3 \cdot 11 \) |
\( J_6 \) | \(=\) | \(-43\) | \(=\) | \( -43 \) |
\( J_8 \) | \(=\) | \(-283\) | \(=\) | \( -283 \) |
\( J_{10} \) | \(=\) | \(-169\) | \(=\) | \( - 13^{2} \) |
\( g_1 \) | \(=\) | \(-1/169\) | ||
\( g_2 \) | \(=\) | \(33/169\) | ||
\( g_3 \) | \(=\) | \(43/169\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{19}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(19\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(19\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0\) | \(19\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(0\) |
Regulator: | \( 1 \) |
Real period: | \( 32.66703 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 19 \) |
Leading coefficient: | \( 0.090490 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(13\) | \(2\) | \(2\) | \(1\) | \(1 + 5 T + 13 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.40.3 | no |
\(3\) | 3.480.12 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial:
\(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 3630 b^{5} + 1720 b^{4} - 15597 b^{3} - 8353 b^{2} + 9457 b + 2644\)
\(g_6 = 316316 b^{5} + 160160 b^{4} - 1342341 b^{3} - 759759 b^{2} + 759759 b + 207207\)
Conductor norm: 1
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\) with minimal polynomial \(x^{2} - x - 3\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.169.1 with generator \(a^{3} - a^{2} - 3 a + 2\) with minimal polynomial \(x^{3} - x^{2} - 4 x - 1\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Additional information
The conductor $169$ of the Jacobian of $X_1(13)$ is the smallest known to arise for a simple abelian surface (and for any rational $L$-function of motivic weight $1$ and degree $4$ that is not the product of two rational $L$-functions of lower degree).
Mazur and Tate (Invent. Math. 22 (1973), 41-49) attribute the discovery of $19$-torsion point on the Jacobian of $X_1(13)$ to Ogg, and use it to prove that this point generates the full Mordell-Weil group using a 19-descent(!). It follows that there are no rational points other than the three pairs above $x=0,-1,\infty$; since these points are cusps of $X_1(13)$, Mazur and Tate thus prove that there is no elliptic curve $E/\Q$ with a rational torsion point of order $13$.