# Properties

 Label 1376.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 1376.b

Label Equation
1376.b.176128.1 $$y^2 + y = 4x^5 + 4x^4 + x^3 + 2x^2$$

## L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$2$$$$1 + 2 T^{2}$$
$$43$$$$( 1 + T )( 1 - 8 T + 43 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$1 - 2 T^{2} + 25 T^{4}$$
$$7$$$$1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}$$
$$11$$$$( 1 + 11 T^{2} )( 1 + 5 T + 11 T^{2} )$$
$$13$$$$1 + 3 T + 4 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 7 T + 28 T^{2} - 119 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 3 T + 22 T^{2} - 69 T^{3} + 529 T^{4}$$
$$29$$$$1 - 6 T + 34 T^{2} - 174 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

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.