# Properties

 Label 18080.c 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 18080.c

Label Equation
18080.c.723200.1 $$y^2 + xy = x^6 - 2x^5 + x^3 + x^2 - 2x + 1$$

## L-function data

Analytic rank:$$2$$

Bad L-factors:
Prime L-Factor
$$2$$$$1 + T$$
$$5$$$$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$113$$$$( 1 + T )( 1 + 14 T + 113 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$7$$$$1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4}$$
$$13$$$$1 + 6 T + 22 T^{2} + 78 T^{3} + 169 T^{4}$$
$$17$$$$1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 18 T^{2} + 361 T^{4}$$
$$23$$$$1 + 2 T - 4 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$1 - 2 T - 6 T^{2} - 58 T^{3} + 841 T^{4}$$
$\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.