# Properties

 Label 1137.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 1137.a

Label Equation
1137.a.1137.1 $$y^2 + (x^2 + x + 1)y = x^5 + x^4 + x^3$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$3$$$$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$379$$$$( 1 - T )( 1 - 20 T + 379 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T^{2} + 4 T^{4}$$
$$5$$$$( 1 - 3 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 + 2 T + 10 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$1 - 6 T + 22 T^{2} - 102 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 5 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$1 + T + 22 T^{2} + 23 T^{3} + 529 T^{4}$$
$$29$$$$1 - 6 T + 22 T^{2} - 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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