# Properties

 Label 20736.a Conductor $20736$ Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

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## Genus 2 curves in isogeny class 20736.a

Label Equation
20736.a.20736.1 $$y^2 = x^5 + x^4 - 3x^3 - 4x^2 - x$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$2$$$$1$$
$$3$$$$1 + 3 T^{2}$$

Good L-factors:
Prime L-Factor
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$( 1 - 6 T + 17 T^{2} )^{2}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )^{2}$$
$$23$$$$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$$29$$$$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $E_3$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial:
$$x^{3} - 3 x - 1$$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -1638 b^{2} + 567 b + 4725$$
$$g_6 = -107892 b^{2} + \frac{74925}{2} b + 310689$$
Conductor norm: 4096

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{9})^+$$ with defining polynomial $$x^{3} - 3 x - 1$$

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

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

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