# Properties

 Label 45413.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 45413.a

Label Equation
45413.a.45413.1 $$y^2 + (x^3 + x + 1)y = -x^5 + 2x^2 - 3x$$

## L-function data

Analytic rank:$$3$$

Prime L-Factor
$$45413$$$$( 1 - T )( 1 - 186 T + 45413 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$1 + 4 T + 8 T^{2} + 20 T^{3} + 25 T^{4}$$
$$7$$$$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$$11$$$$( 1 + 3 T + 11 T^{2} )( 1 + 5 T + 11 T^{2} )$$
$$13$$$$( 1 + 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$1 - T - 12 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 - T + 21 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 8 T + 38 T^{2} + 184 T^{3} + 529 T^{4}$$
$$29$$$$1 + 9 T + 67 T^{2} + 261 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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