# Properties

 Label 741.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

# Learn more about

## Genus 2 curves in isogeny class 741.a

Label Equation
741.a.28899.1 $$y^2 + (x + 1)y = -3x^5 - x^4 + 2x^2 + x$$

## L-function data

Analytic rank:$$0$$

Bad L-factors:
Prime L-Factor
$$3$$$$( 1 + T )( 1 + 3 T^{2} )$$
$$13$$$$( 1 + T )( 1 + 2 T + 13 T^{2} )$$
$$19$$$$( 1 + T )( 1 - 4 T + 19 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$5$$$$1 + T + 5 T^{3} + 25 T^{4}$$
$$7$$$$1 - T + 6 T^{2} - 7 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 5 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$17$$$$1 - 3 T + 20 T^{2} - 51 T^{3} + 289 T^{4}$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$( 1 - 8 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$\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.