Genus 2 Curve 20736.l.373248.1

• Label: 20736.l.373248.1
• Conductor: 20736
• Discriminant: 373248
• Sato-Tate group: $J(E_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{QM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$

Invariants

\( N \)	=	\( 20736 \)	=	\( 2^{8} \cdot 3^{4} \)
\( \Delta \)	=	\(373248\)	=	\( 2^{9} \cdot 3^{6} \)

Igusa-Clebsch invariants

\( I_2 \)	=	\(7008\)	=	\( 2^{5} \cdot 3 \cdot 73 \)
\( I_4 \)	=	\(1700352\)	=	\( 2^{9} \cdot 3^{4} \cdot 41 \)
\( I_6 \)	=	\(3259367424\)	=	\( 2^{17} \cdot 3^{4} \cdot 307 \)
\( I_{10} \)	=	\(1528823808\)	=	\( 2^{21} \cdot 3^{6} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	\(C_2 \)	(GAP id : [2,1])
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	\(C_2 \)	(GAP id : [2,1])

Rational points

This curve is locally solvable everywhere.

All rational points: (-1 : -4 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)

Number of rational Weierstrass points: \(2\)

Invariants of the Jacobian:

Analytic rank: \(0\)

2-Selmer rank: \(2\)

Order of Ш*: square

Regulator: 1.0

Real period: 18.600159265208624917879119180

Tamagawa numbers: 2 (p = 2), 4 (p = 3)

Torsion: \(\Z/{2}\Z \times \Z/{6}\Z\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{3})\)

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.339738624.10 with defining polynomial \(x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(6\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{15} a^{6} - \frac{1}{6} a^{4} - \frac{2}{3} a^{2} - \frac{3}{5}\) with minimal polynomial \(x^{2} - x + 1\):

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

Sato Tate group: $E_4$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-\frac{1}{10} a^{6} - \frac{2}{5}\) with minimal polynomial \(x^{2} + 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R\)

Sato Tate group: $J(E_2)$
not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(-\frac{1}{6} a^{6} - \frac{2}{3} a^{4} - \frac{2}{3} a^{2} - 2\) with minimal polynomial \(x^{2} - 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R\)

Sato Tate group: $J(E_2)$
not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{-3})\) with generator \(-\frac{2}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{6}{5}\) with minimal polynomial \(x^{4} - 2 x^{2} + 4\):

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

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

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(\frac{2}{45} a^{7} - \frac{1}{18} a^{5} + \frac{1}{9} a^{3} + \frac{11}{15} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{7}{90} a^{7} + \frac{5}{18} a^{5} + \frac{4}{9} a^{3} + \frac{23}{15} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(-\frac{1}{15} a^{7} - \frac{1}{6} a^{5} - \frac{2}{3} a^{3} - \frac{3}{5} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple