# Properties

 Label 100312.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 100312.a

Label Equation
100312.a.200624.1 $$y^2 + (x^2 + 1)y = -x^6 - 2x^5 + x^4 + 3x^3 - x - 1$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 - T$$
$$12539$$$$( 1 + T )( 1 - 40 T + 12539 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4}$$
$$5$$$$1 - 2 T^{2} + 25 T^{4}$$
$$7$$$$1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 6 T + 11 T^{2} )( 1 + 11 T^{2} )$$
$$13$$$$1 + 4 T + 18 T^{2} + 52 T^{3} + 169 T^{4}$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 17 T^{2} )$$
$$19$$$$1 - 3 T + 22 T^{2} - 57 T^{3} + 361 T^{4}$$
$$23$$$$1 + 28 T^{2} + 529 T^{4}$$
$$29$$$$1 + 2 T + 6 T^{2} + 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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