# Properties

 Label 65814.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 65814.a

Label Equation
65814.a.394884.1 $$y^2 + (x^3 + 1)y = x^5 + x^4 - x^2 - 3x + 2$$

## L-function data

Analytic rank:$$3$$

Prime L-Factor
$$2$$$$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3$$$$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$7$$$$( 1 + T )( 1 + 4 T + 7 T^{2} )$$
$$1567$$$$( 1 - T )( 1 + 59 T + 1567 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$( 1 + 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$11$$$$1 + 10 T + 45 T^{2} + 110 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T + 7 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 8 T^{2} + 289 T^{4}$$
$$19$$$$1 + 5 T + 13 T^{2} + 95 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 3 T + 23 T^{2} )( 1 + 9 T + 23 T^{2} )$$
$$29$$$$1 + 2 T - 9 T^{2} + 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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