# Properties

 Label 763.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 763.a

Label Equation
763.a.763.1 $$y^2 + (x^3 + x)y = -2x^4 + 2x^2 - x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$7$$$$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$109$$$$( 1 - T )( 1 + 10 T + 109 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 9 T^{4}$$
$$5$$$$( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$11$$$$1 + 2 T + 4 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 3 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$( 1 - 3 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$
$$19$$$$1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4}$$
$$23$$$$1 - 2 T + 8 T^{2} - 46 T^{3} + 529 T^{4}$$
$$29$$$$1 + 9 T + 68 T^{2} + 261 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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