# Properties

 Label 1343.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 1343.b

Label Equation
1343.b.1343.1 $$y^2 + (x^3 + x + 1)y = -3x^4 + x^3 + 2x^2 + x$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$17$$$$( 1 + T )( 1 + 2 T + 17 T^{2} )$$
$$79$$$$( 1 - T )( 1 + 3 T + 79 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + T^{2} + 2 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + 4 T + 9 T^{2} + 20 T^{3} + 25 T^{4}$$
$$7$$$$1 - T - 7 T^{3} + 49 T^{4}$$
$$11$$$$1 - 3 T + 11 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$1 + T + 5 T^{2} + 13 T^{3} + 169 T^{4}$$
$$19$$$$1 - T - 12 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 10 T + 68 T^{2} + 230 T^{3} + 529 T^{4}$$
$$29$$$$1 - T - 27 T^{2} - 29 T^{3} + 841 T^{4}$$
$\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.