# Properties

 Label 1309.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 1309.a

Label Equation
1309.a.9163.1 $$y^2 + (x^2 + 1)y = 7x^5 - x^4 - 5x^3 - x^2 + x$$
1309.a.9163.2 $$y^2 + x^2y = -x^6 + 20x^5 - 128x^4 + 248x^3 + 32x^2 + x$$

## L-function data

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

Prime L-Factor
$$7$$$$( 1 - T )( 1 + 7 T^{2} )$$
$$11$$$$( 1 + T )( 1 - 4 T + 11 T^{2} )$$
$$17$$$$( 1 - T )( 1 + 6 T + 17 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$13$$$$1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$$
$$23$$$$( 1 - 7 T + 23 T^{2} )( 1 + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$\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.