# Properties

 Label 4428.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 4428.a

Label Equation
4428.a.239112.1 $$y^2 + (x^3 + x + 1)y = -2x^5 - x^4 - x$$

## L-function data

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

Prime L-Factor
$$2$$$$1 + T + T^{2}$$
$$3$$$$1 + T$$
$$41$$$$( 1 + T )( 1 + 3 T + 41 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 2 T + T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$1 + 2 T + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - T + 10 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 - 20 T^{2} + 361 T^{4}$$
$$23$$$$1 - T - 14 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + 2 T + 22 T^{2} + 58 T^{3} + 841 T^{4}$$
$\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.