# Properties

 Label 464.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 464.a

Label Equation
464.a.464.1 $$y^2 + (x + 1)y = -x^6 - 2x^5 - 2x^4 - x^3$$
464.a.29696.1 $$y^2 + (x + 1)y = 8x^5 + 3x^4 - 4x^3 - 2x^2$$
464.a.29696.2 $$y^2 + xy = 4x^5 + 33x^4 + 72x^3 + 16x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 + T + 2 T^{2}$$
$$29$$$$( 1 + T )( 1 + 2 T + 29 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$1 + T + 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 2 T - 2 T^{2} + 14 T^{3} + 49 T^{4}$$
$$11$$$$1 - 5 T + 18 T^{2} - 55 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T + 12 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$\cdots$$\cdots$

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