# Properties

 Label 3391.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 3391.a

Label Equation
3391.a.3391.1 $$y^2 + (x^3 + x + 1)y = -2x^4 + x^3 + 6x^2 + 2x$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Prime L-Factor
$$3391$$$$( 1 - T )( 1 - 14 T + 3391 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + T^{2} + 2 T^{3} + 4 T^{4}$$
$$3$$$$( 1 - T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + T - T^{2} + 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 4 T + 10 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$1 + 2 T + 5 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 3 T + 13 T^{2} )( 1 + T + 13 T^{2} )$$
$$17$$$$1 - 2 T + 6 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$( 1 + T + 19 T^{2} )( 1 + 5 T + 19 T^{2} )$$
$$23$$$$1 + 24 T^{2} + 529 T^{4}$$
$$29$$$$1 - 6 T + 20 T^{2} - 174 T^{3} + 841 T^{4}$$
$\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.