# Properties

 Label 1136.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 1136.a

Label Equation
1136.a.9088.1 $$y^2 + (x^3 + x)y = x^4 - x^3 + 2x^2 - x + 1$$
1136.a.290816.1 $$y^2 + (x^3 + x^2)y = -5x^4 - 9x^3 + 25x^2 + 40x - 24$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 - T$$
$$71$$$$( 1 + T )( 1 - 16 T + 71 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 2 T^{2} + 25 T^{4}$$
$$7$$$$1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + 2 T - 6 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T + 12 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 10 T^{2} + 289 T^{4}$$
$$19$$$$1 - 3 T + 6 T^{2} - 57 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 - 3 T + 23 T^{2} )$$
$$29$$$$1 - 4 T + 34 T^{2} - 116 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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