Genus 2 Isogeny Class 12700.a

• Label: 12700.a
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Genus 2 curves in isogeny class 12700.a

Label	Equation
12700.a.25400.1	\(y^2 + (x^3 + x + 1)y = x^5 - 3x^3 - x^2 + x\)

L-function data

Analytic rank:

\(2\)

Bad L-factors:

Prime	L-Factor
\(2\)	\( 1 + T + T^{2}\)
\(5\)	\( 1 + 4 T + 5 T^{2}\)
\(127\)	\( ( 1 + T )( 1 + 2 T + 127 T^{2} )\)

Good L-factors:

Prime	L-Factor
\(3\)	\( ( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )\)
\(7\)	\( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)	\( 1 + 5 T + 13 T^{2} + 55 T^{3} + 121 T^{4}\)
\(13\)	\( ( 1 - T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)	\( 1 + 7 T + 37 T^{2} + 119 T^{3} + 289 T^{4}\)
\(19\)	\( 1 - T + 15 T^{2} - 19 T^{3} + 361 T^{4}\)
\(23\)	\( 1 + 3 T + 37 T^{2} + 69 T^{3} + 529 T^{4}\)
\(29\)	\( 1 + 9 T + 40 T^{2} + 261 T^{3} + 841 T^{4}\)
$\cdots$	$\cdots$

See L-function page for more information

Sato-Tate group

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

Endomorphisms of the Jacobian

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

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.