# Properties

 Label 997.b 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 997.b

Label Equation
997.b.997.1 $$y^2 + y = x^5 - 2x^4 + 2x^3 - x^2$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$997$$$$( 1 - T )( 1 + 28 T + 997 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}$$
$$5$$$$1 + 2 T + 4 T^{2} + 10 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 - 6 T + 11 T^{2} )( 1 + 2 T + 11 T^{2} )$$
$$13$$$$( 1 + 4 T + 13 T^{2} )^{2}$$
$$17$$$$1 - 14 T^{2} + 289 T^{4}$$
$$19$$$$1 - T + 9 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 2 T - 14 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 5 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

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