# Properties

 Label 1145.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 1145.a

Label Equation
1145.a.1145.1 $$y^2 + (x^3 + 1)y = -3x^4 + 3x^3 - x$$
1145.a.143125.1 $$y^2 + (x^3 + x^2 + x)y = 2x^4 + 4x^3 + 9x^2 + 10x + 9$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$5$$$$( 1 - T )( 1 + 3 T + 5 T^{2} )$$
$$229$$$$( 1 + T )( 1 + 2 T + 229 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$7$$$$1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 7 T + 17 T^{2} )$$
$$19$$$$1 - 3 T - 6 T^{2} - 57 T^{3} + 361 T^{4}$$
$$23$$$$1 - 4 T + 30 T^{2} - 92 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 6 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$\cdots$$\cdots$

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