# Properties

 Label 3391.b 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 3391.b

Label Equation
3391.b.3391.1 $$y^2 + (x^3 + x + 1)y = -x^5 - x^4$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$3391$$$$( 1 + T )( 1 + 80 T + 3391 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 + 4 T + 11 T^{2} + 44 T^{3} + 121 T^{4}$$
$$13$$$$1 + 2 T - 3 T^{2} + 26 T^{3} + 169 T^{4}$$
$$17$$$$1 + 4 T + 14 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$1 - 2 T + 9 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - 32 T^{2} + 529 T^{4}$$
$$29$$$$1 + 2 T + 32 T^{2} + 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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