# Properties

 Label 587.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 587.a

Label Equation
587.a.587.1 $$y^2 + (x^3 + x + 1)y = -x^2 - x$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$587$$$$( 1 - T )( 1 + 36 T + 587 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 2 T + T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$1 - 5 T^{2} + 49 T^{4}$$
$$11$$$$1 + T - T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T + 21 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - 8 T^{2} + 289 T^{4}$$
$$19$$$$1 + 7 T + 31 T^{2} + 133 T^{3} + 361 T^{4}$$
$$23$$$$1 + 5 T + 35 T^{2} + 115 T^{3} + 529 T^{4}$$
$$29$$$$1 - 5 T + 33 T^{2} - 145 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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