# Properties

 Label 816.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

# Related objects

## Genus 2 curves in isogeny class 816.a

Label Equation
816.a.13872.1 $$y^2 + (x^3 + x^2)y = -2x^4 + 6x^2 - 8x + 3$$
816.a.39168.1 $$y^2 + (x^2 + 1)y = 3x^5 - 4x^3 - x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 + T$$
$$3$$$$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$17$$$$( 1 - T )( 1 + 6 T + 17 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 - 2 T^{2} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$( 1 - 13 T^{2} )^{2}$$
$$19$$$$( 1 - 8 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$( 1 - 6 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$1 - 8 T + 46 T^{2} - 232 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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