Genus 2 Curve 169.a.169.1

• Label: 169.a.169.1
• Conductor: 169
• Discriminant: -169
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

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

$y^2 + (x^3 + x + 1)y = x^5 + x^4$

Invariants

\( N \)	=	\( 169 \)	=	\( 13^{2} \)
\( \Delta \)	=	\(-169\)	=	\( -1 \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	=	\(8\)	=	\( 2^{3} \)
\( I_4 \)	=	\(3172\)	=	\( 2^{2} \cdot 13 \cdot 61 \)
\( I_6 \)	=	\(30056\)	=	\( 2^{3} \cdot 13 \cdot 17^{2} \)
\( I_{10} \)	=	\(-692224\)	=	\( -1 \cdot 2^{12} \cdot 13^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	\(C_6 \)	(GAP id : [6,2])
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	\(D_6 \)	(GAP id : [12,4])

Rational points

This curve is locally solvable everywhere.

All rational points: (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0)

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

2-Selmer rank: \(0\)

Order of Ш*: square

Regulator: 1.0

Real period: 32.667031090507096110005902370

Tamagawa numbers: 1 (p = 13)

Torsion: \(\Z/{19}\Z\)

2-torsion field: 6.0.10816.1

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_6$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition

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:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) 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 \(\GL_2\)-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 \(x^{2} - x - 3\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

Sato Tate group: $E_3$
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 \(x^{3} - x^{2} - 4 x - 1\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

Sato Tate group: $E_2$
of \(\GL_2\)-type, simple

Additional information

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 $13$.