# Properties

 Label 709.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 709.a

Label Equation
709.a.709.1 $$y^2 + xy = x^5 - 2x^2 + x$$

## L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$709$$$$( 1 + T )( 1 - 6 T + 709 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$1 + T + 5 T^{3} + 25 T^{4}$$
$$7$$$$( 1 + 7 T^{2} )^{2}$$
$$11$$$$1 + T - 6 T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$1 - 10 T^{2} + 361 T^{4}$$
$$23$$$$1 + 2 T + 14 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 9 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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.