# Properties

 Label 644.b 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 644.b

Label Equation
644.b.14812.1 $$y^2 + (x^3 + 1)y = x^5 - x^4 - 4x^3 + 5x^2 - x - 1$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 + T^{2}$$
$$7$$$$( 1 + T )( 1 + 2 T + 7 T^{2} )$$
$$23$$$$( 1 + T )( 1 - 4 T + 23 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$1 + 4 T^{2} + 25 T^{4}$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 + 2 T - 6 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 - 12 T^{2} + 361 T^{4}$$
$$29$$$$( 1 - 8 T + 29 T^{2} )( 1 + 29 T^{2} )$$
$\cdots$$\cdots$

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