# 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

## 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:$$0$$

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$

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$
Not of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.