# Properties

 Label 1311.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 1311.a

Label Equation
1311.a.814131.1 $$y^2 + xy = x^5 + 5x^4 + 5x^3 + 4x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$3$$$$( 1 - T )( 1 + 3 T^{2} )$$
$$19$$$$( 1 - T )( 1 - 4 T + 19 T^{2} )$$
$$23$$$$( 1 + T )( 1 - 8 T + 23 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$5$$$$1 + T + 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}$$
$$11$$$$1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 + 3 T + 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$$29$$$$( 1 + 2 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

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