# Properties

 Label 1312.c 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 1312.c

Label Equation
1312.c.671744.1 $$y^2 + (x + 1)y = x^6 + 4x^5 + 7x^4 + 5x^3 + 2x^2$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 - T$$
$$41$$$$( 1 - T )( 1 + 2 T + 41 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$1 - 6 T^{2} + 49 T^{4}$$
$$11$$$$1 + 2 T + 8 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$1 + 6 T^{2} + 169 T^{4}$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )$$
$$19$$$$1 - 2 T + 8 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 2 T + 23 T^{2} )^{2}$$
$$29$$$$( 1 - 8 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$\cdots$$\cdots$

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