# Properties

 Label 8212.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 8212.a

Label Equation
8212.a.32848.1 $$y^2 + (x^3 + x)y = 2x^3 - x^2 - 2x + 1$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$2$$$$1 + T + 2 T^{2}$$
$$2053$$$$( 1 - T )( 1 + 16 T + 2053 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$( 1 + T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$1 + 2 T + 5 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 + 6 T + 28 T^{2} + 66 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 + 5 T + 12 T^{2} + 85 T^{3} + 289 T^{4}$$
$$19$$$$1 - 4 T + 5 T^{2} - 76 T^{3} + 361 T^{4}$$
$$23$$$$1 - 2 T^{2} + 529 T^{4}$$
$$29$$$$( 1 - 3 T + 29 T^{2} )( 1 + 9 T + 29 T^{2} )$$
$\cdots$$\cdots$

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