# Properties

 Label 1038.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 1038.a

Label Equation
1038.a.1038.2 $$y^2 + (x^3 + 1)y = x^4 + 2x^2 + x + 1$$
1038.a.1038.1 $$y^2 + (x^2 + x)y = x^5 - 12x^4 + 26x^3 + 46x^2 + 21x + 3$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$( 1 + T )( 1 - T + 2 T^{2} )$$
$$3$$$$( 1 + T )( 1 - T + 3 T^{2} )$$
$$173$$$$( 1 + T )( 1 - 6 T + 173 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 + 11 T^{2} )^{2}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$1 - 4 T + 10 T^{2} - 68 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$$23$$$$1 - T - 2 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )$$
$\cdots$$\cdots$

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