# Properties

 Label 1253.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 1253.a

Label Equation
1253.a.1253.1 $$y^2 + (x^3 + x^2 + 1)y = -x^6 + 2x^5 - 33x^3 + 43x^2 + 15x - 330$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$7$$$$( 1 - T )( 1 + 4 T + 7 T^{2} )$$
$$179$$$$( 1 - T )( 1 + 179 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + T^{2} + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 - T + 3 T^{2} - 3 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$$11$$$$1 - T + 7 T^{2} - 11 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 5 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$( 1 - 7 T + 17 T^{2} )( 1 + 17 T^{2} )$$
$$19$$$$1 + T + 23 T^{2} + 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 3 T + 19 T^{2} + 69 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 3 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$
Not of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.