# Properties

 Label 66601.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 66601.a

Label Equation
66601.a.66601.1 $$y^2 + (x^3 + 1)y = -x^4 + 5x^3 + 16x^2 + 11x + 2$$

## L-function data

Analytic rank:$$3$$

Prime L-Factor
$$66601$$$$( 1 - T )( 1 + 178 T + 66601 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}$$
$$5$$$$( 1 + 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$( 1 + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$$17$$$$1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 2 T - 14 T^{2} + 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - T - 21 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + 6 T + 35 T^{2} + 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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