# Properties

 Label 1544.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 1544.a

Label Equation
1544.a.3088.1 $$y^2 + (x^3 + x^2 + x + 1)y = x^2$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$2$$$$1 + 2 T + 2 T^{2}$$
$$193$$$$( 1 + T )( 1 + 19 T + 193 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$( 1 + T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$1 - 3 T^{2} + 49 T^{4}$$
$$11$$$$( 1 - 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$( 1 - 5 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$1 - T - 20 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 - 2 T + 6 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - 4 T + 11 T^{2} - 92 T^{3} + 529 T^{4}$$
$$29$$$$1 + 5 T + 36 T^{2} + 145 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

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.