# Properties

 Label 686.a Conductor $686$ Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{CM} \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Learn more about

## Genus 2 curves in isogeny class 686.a

Label Equation
686.a.686.1 $$y^2 + (x^2 + x)y = x^5 + x^4 + 2x^3 + x^2 + x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$2$$$$( 1 + T )( 1 - T + 2 T^{2} )$$
$$7$$$$1 - T$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$( 1 + 5 T^{2} )^{2}$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 11 T^{2} )$$
$$13$$$$( 1 + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 17 T^{2} )$$
$$19$$$$( 1 - 2 T + 19 T^{2} )( 1 + 19 T^{2} )$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + 23 T^{2} )$$
$$29$$$$( 1 - 2 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $N(\mathrm{U}(1)\times\mathrm{SU}(2))$, $$\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 49.a

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism algebra over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-7})$$ with defining polynomial $$x^{2} - x + 2$$

Endomorphism algebra over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-7})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$

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