Genus 2 isogeny class 13696.b

• Label: 13696.b
• Conductor: $13696$
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 13696.b

Label	Equation
13696.b.109568.1	\(y^2 + x^2y = x^5 + 12x^4 - 2x^3 - 253x^2 + 288x - 83\)

L-function data

Analytic rank:

\(0\)

Mordell-Weil rank:

\(0\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 - T\)
\(107\)	\( ( 1 - T )( 1 + 107 T^{2} )\)

Good L-factors:

Prime	L-Factor
\(3\)	\( ( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(5\)	\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}\)
\(7\)	\( ( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )\)
\(13\)	\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)	\( ( 1 - 6 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )\)
\(19\)	\( ( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )\)
\(23\)	\( 1 + 3 T + 22 T^{2} + 69 T^{3} + 529 T^{4}\)
\(29\)	\( 1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $\mathrm{USp}(4)$

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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