Genus 2 isogeny class 13689.c

• Label: 13689.c
• Conductor: $13689$
• 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 13689.c

Label	Equation
13689.c.41067.1	\(y^2 + (x^3 + x)y = x^5 - 6x^3 - 13x^2 - 6x + 9\)

L-function data

Analytic rank:

\(1\)

Mordell-Weil rank:

\(1\)

Bad L-factors:

Prime	L-Factor
\(3\)	\( 1 + T\)
\(13\)	\( 1 + 3 T + 13 T^{2}\)

Good L-factors:

Prime	L-Factor
\(2\)	\( 1 + T + 2 T^{3} + 4 T^{4}\)
\(5\)	\( ( 1 + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)	\( ( 1 - 3 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )\)
\(11\)	\( 1 - 2 T + 10 T^{2} - 22 T^{3} + 121 T^{4}\)
\(17\)	\( 1 - T + 18 T^{2} - 17 T^{3} + 289 T^{4}\)
\(19\)	\( 1 + T - 12 T^{2} + 19 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)	\( ( 1 + 29 T^{2} )( 1 + 9 T + 29 T^{2} )\)
$\cdots$	$\cdots$

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.