# Properties

 Label 70351.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 70351.a

Label Equation
70351.a.70351.1 $$y^2 + (x^3 + x^2 + x)y = -x^4 + x^3 + 3x^2 - 4x + 1$$

## L-function data

Analytic rank:$$3$$

Prime L-Factor
$$70351$$$$( 1 - T )( 1 + 408 T + 70351 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$3$$$$1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}$$
$$5$$$$1 + 5 T + 13 T^{2} + 25 T^{3} + 25 T^{4}$$
$$7$$$$1 + 6 T + 20 T^{2} + 42 T^{3} + 49 T^{4}$$
$$11$$$$1 + 4 T + 13 T^{2} + 44 T^{3} + 121 T^{4}$$
$$13$$$$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$( 1 - 3 T + 17 T^{2} )( 1 + 7 T + 17 T^{2} )$$
$$19$$$$1 + 10 T + 52 T^{2} + 190 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )$$
$$29$$$$1 + 6 T + 11 T^{2} + 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.