# Properties

 Label 65536.a Conductor $65536$ Sato-Tate group $D_{2,1}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathsf{CM})$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 65536.a

Label Equation
65536.a.65536.1 $$y^2 = x^5 + x$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Prime L-Factor
$$2$$$$1$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$5$$$$( 1 + 5 T^{2} )^{2}$$
$$7$$$$( 1 + 7 T^{2} )^{2}$$
$$11$$$$( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$( 1 + 13 T^{2} )^{2}$$
$$17$$$$( 1 + 6 T + 17 T^{2} )^{2}$$
$$19$$$$( 1 - 2 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$$
$$23$$$$( 1 + 23 T^{2} )^{2}$$
$$29$$$$( 1 + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## 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 256.d
Elliptic curve isogeny class 256.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 $$K = \Q (a) \simeq$$ $$\Q(\zeta_{8})$$ with defining polynomial $$x^{4} + 1$$

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

 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-2})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

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