# Properties

 Label 100234.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 100234.a

Label Equation
100234.a.801872.1 $$y^2 + (x^3 + x)y = x^5 - x^4 - 5x^3 - x^2 + 5x - 3$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$2$$$$( 1 + T )( 1 + 2 T^{2} )$$
$$23$$$$( 1 + T )( 1 + 6 T + 23 T^{2} )$$
$$2179$$$$( 1 + T )( 1 - 88 T + 2179 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$1 - 4 T^{2} + 25 T^{4}$$
$$7$$$$1 + 5 T + 15 T^{2} + 35 T^{3} + 49 T^{4}$$
$$11$$$$1 - T + 13 T^{2} - 11 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T - 2 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$1 + 8 T + 42 T^{2} + 152 T^{3} + 361 T^{4}$$
$$29$$$$1 + 6 T + 38 T^{2} + 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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