Genus 2 isogeny class 20736.l

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

Genus 2 curves in isogeny class 20736.l

Label	Equation
20736.l.373248.1	\(y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + 2 T^{2}\)
\(3\)	\( 1\)

Good L-factors:

Prime	L-Factor
\(5\)	\( 1 - 2 T^{2} + 25 T^{4}\)
\(7\)	\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}\)
\(11\)	\( 1 - 2 T^{2} + 121 T^{4}\)
\(13\)	\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)	\( 1 + 10 T^{2} + 289 T^{4}\)
\(19\)	\( 1 - 2 T^{2} + 361 T^{4}\)
\(23\)	\( 1 - 2 T^{2} + 529 T^{4}\)
\(29\)	\( 1 + 46 T^{2} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

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

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.