# Properties

 Label 389.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 389.a

Label Equation
389.a.389.1 $$y^2 + (x^3 + x)y = x^5 - 2x^4 - 8x^3 + 16x + 7$$
389.a.389.2 $$y^2 + (x + 1)y = x^5 + 2x^4 + 2x^3 + x^2$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$389$$$$( 1 + T )( 1 + 10 T + 389 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$1 - T - 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 2 T + 11 T^{2} )^{2}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + T + 13 T^{2} )$$
$$17$$$$1 + 10 T^{2} + 289 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 5 T + 19 T^{2} )$$
$$23$$$$1 - 10 T^{2} + 529 T^{4}$$
$$29$$$$( 1 - 10 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )$$
$\cdots$$\cdots$

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