# Properties

 Label 676.b Sato-Tate group $E_1$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 676.b

Label Equation
676.b.17576.1 $$y^2 + (x^2 + x)y = -x^6 + 3x^5 - 6x^4 + 6x^3 - 6x^2 + 3x - 1$$

## L-function data

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

Prime L-Factor
$$2$$$$( 1 + T )^{2}$$
$$13$$$$( 1 - T )^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 - T + 3 T^{2} )^{2}$$
$$5$$$$( 1 + 3 T + 5 T^{2} )^{2}$$
$$7$$$$( 1 + T + 7 T^{2} )^{2}$$
$$11$$$$( 1 - 6 T + 11 T^{2} )^{2}$$
$$17$$$$( 1 + 3 T + 17 T^{2} )^{2}$$
$$19$$$$( 1 - 2 T + 19 T^{2} )^{2}$$
$$23$$$$( 1 + 23 T^{2} )^{2}$$
$$29$$$$( 1 - 6 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_1$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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.