Properties

Label 713.b
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

Related objects

Learn more about

Genus 2 curves in isogeny class 713.b

Label Equation
713.b.713.1 \(y^2 + (x^3 + x + 1)y = -x^4\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(23\)\( ( 1 + T )( 1 - 6 T + 23 T^{2} )\)
\(31\)\( ( 1 + T )( 1 - 5 T + 31 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2} + 2 T^{3} + 4 T^{4}\)
\(3\)\( ( 1 - T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )\)
\(5\)\( ( 1 - 4 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)\( 1 + T - 4 T^{2} + 7 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 2 T + 10 T^{2} - 22 T^{3} + 121 T^{4}\)
\(13\)\( 1 + 3 T + 4 T^{2} + 39 T^{3} + 169 T^{4}\)
\(17\)\( ( 1 - 8 T + 17 T^{2} )( 1 + 17 T^{2} )\)
\(19\)\( ( 1 - 2 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )\)
\(29\)\( 1 + 3 T + 22 T^{2} + 87 T^{3} + 841 T^{4}\)
$\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.