# Properties

 Label 1109.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 1109.a

Label Equation
1109.a.1109.1 $$y^2 + y = x^5 - 6x^4 - 36x^3 - 6x^2 + 63x - 36$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$1109$$$$( 1 + T )( 1 - 50 T + 1109 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 9 T^{4}$$
$$5$$$$1 - 3 T + 7 T^{2} - 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 3 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$1 + 5 T + 18 T^{2} + 55 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 5 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$1 - 3 T + 29 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$1 + 9 T + 53 T^{2} + 171 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 7 T + 23 T^{2} )$$
$$29$$$$1 - 5 T + 13 T^{2} - 145 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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