# Properties

 Label 53623.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 53623.a

Label Equation
53623.a.53623.1 $$y^2 + (x^2 + x + 1)y = x^6 - 2x^4 + x^3 - 2x$$

## L-function data

Analytic rank:$$3$$

Prime L-Factor
$$53623$$$$( 1 - T )( 1 - 48 T + 53623 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$$7$$$$1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4}$$
$$11$$$$1 + 2 T + 4 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 3 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$
$$17$$$$1 + 4 T + 10 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T + 25 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$( 1 + T + 23 T^{2} )( 1 + 9 T + 23 T^{2} )$$
$$29$$$$1 + 32 T^{2} + 841 T^{4}$$
$\cdots$$\cdots$

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