# Properties

 Label 5209.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 5209.a

Label Equation
5209.a.5209.1 $$y^2 + (x^3 + x + 1)y = x^4 + x^3 - x^2 - x$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$5209$$$$( 1 + T )( 1 - 142 T + 5209 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 3 T + 7 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 + 2 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 + T^{2} + 121 T^{4}$$
$$13$$$$1 + 2 T + 16 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 + 4 T + 27 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$1 - 2 T - 6 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 + 5 T + 23 T^{2} + 115 T^{3} + 529 T^{4}$$
$$29$$$$1 - 34 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

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