# Properties

 Label 1088.a Conductor $1088$ Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 1088.a

Label Equation
1088.a.1088.1 $$y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + 2x^2 + x + 1$$

## L-function data

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

Prime L-Factor
$$2$$$$1 + 2 T^{2}$$
$$17$$$$( 1 + T )( 1 - 6 T + 17 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$1 - 2 T^{2} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 + 10 T^{2} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$19$$$$1 - 2 T^{2} + 361 T^{4}$$
$$23$$$$( 1 + 23 T^{2} )( 1 + 6 T + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\sqrt{2})$$ with defining polynomial:
$$x^{2} - 2$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 2.2.8.1-17.2-a
Elliptic curve isogeny class 2.2.8.1-17.1-a

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

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

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

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

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