# Properties

 Label 2304.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 2304.a

Label Equation
2304.a.13824.2 $$y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + 3x^2 + x + 2$$
2304.a.13824.1 $$y^2 + y = 2x^5 + 3x^4 - x^3 - 2x^2$$

## L-function data

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

Prime L-Factor
$$2$$$$1 + 2 T + 2 T^{2}$$
$$3$$$$1 + 2 T + 3 T^{2}$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$1 + 2 T - 6 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 2 T - 2 T^{2} + 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - 2 T^{2} + 529 T^{4}$$
$$29$$$$1 - 2 T + 30 T^{2} - 58 T^{3} + 841 T^{4}$$
$\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.