# Properties

 Label 574.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 574.a

Label Equation
574.a.293888.1 $$y^2 + (x^2 + x)y = x^5 - x^4 - 3x^2 + x + 1$$

## L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$2$$$$( 1 - T )( 1 + T + 2 T^{2} )$$
$$7$$$$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$41$$$$( 1 + T )( 1 - 2 T + 41 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$11$$$$1 - 2 T^{2} + 121 T^{4}$$
$$13$$$$1 - 4 T + 6 T^{2} - 52 T^{3} + 169 T^{4}$$
$$17$$$$( 1 + 2 T + 17 T^{2} )^{2}$$
$$19$$$$1 + 2 T + 18 T^{2} + 38 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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.