# Properties

 Label 3319.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 3319.a

Label Equation
3319.a.3319.1 $$y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + x^3$$

## L-function data

Analytic rank:$$2$$  (upper bound)
Mordell-Weil rank:$$2$$

Prime L-Factor
$$3319$$$$( 1 + T )( 1 - 74 T + 3319 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$1 + 5 T + 13 T^{2} + 25 T^{3} + 25 T^{4}$$
$$7$$$$1 + 3 T + 7 T^{2} + 21 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 3 T + 11 T^{2} )( 1 + 5 T + 11 T^{2} )$$
$$13$$$$( 1 + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$1 + 2 T + 7 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T^{2} + 361 T^{4}$$
$$23$$$$1 + T + 28 T^{2} + 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + 8 T + 37 T^{2} + 232 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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.