# Properties

 Label 731.a Conductor $731$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 731.a

Label Equation
731.a.12427.1 $$y^2 + (x^3 + x^2)y = x^5 + 2x^4 - x - 3$$

## L-function data

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

Prime L-Factor
$$17$$$$( 1 + T )( 1 - 3 T + 17 T^{2} )$$
$$43$$$$( 1 + T )( 1 - 4 T + 43 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 9 T^{4}$$
$$5$$$$1 + 4 T^{2} + 25 T^{4}$$
$$7$$$$1 + 2 T + 4 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$19$$$$( 1 - 8 T + 19 T^{2} )( 1 + 19 T^{2} )$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 3 T + 23 T^{2} )$$
$$29$$$$1 - 6 T + 28 T^{2} - 174 T^{3} + 841 T^{4}$$
$\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.