# Properties

 Label 1109.b 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

# Learn more about

## Genus 2 curves in isogeny class 1109.b

Label Equation
1109.b.1109.1 $$y^2 + y = x^5 - x^4 - x^3 + x^2 + x$$

## L-function data

Analytic rank:$$0$$

Bad L-factors:
Prime L-Factor
$$1109$$$$( 1 + T )( 1 + 24 T + 1109 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T^{2} + 4 T^{4}$$
$$3$$$$1 + 4 T^{2} + 9 T^{4}$$
$$5$$$$1 + 2 T^{2} + 25 T^{4}$$
$$7$$$$1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 + T + 13 T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 - 5 T + 19 T^{2} - 65 T^{3} + 169 T^{4}$$
$$17$$$$1 - 5 T + 17 T^{2} - 85 T^{3} + 289 T^{4}$$
$$19$$$$1 + 3 T + 26 T^{2} + 57 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 9 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$$29$$$$1 + 5 T + 37 T^{2} + 145 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.