# Properties

 Label 997.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 997.a

Label Equation
997.a.997.1 $$y^2 + xy = x^5 - 8x^4 + 16x^3 - x$$
997.a.997.2 $$y^2 + (x + 1)y = x^5 + x^4$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$997$$$$( 1 + T )( 1 + 26 T + 997 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$1 - T + 4 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 4 T + 19 T^{2} )$$
$$23$$$$1 + 3 T + 38 T^{2} + 69 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 10 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$\cdots$$\cdots$

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