# Properties

 Label 12544.g Conductor $12544$ Sato-Tate group $J(E_4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 12544.g

Label Equation
12544.g.175616.1 $$y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$7$$$$1 + 7 T^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$$5$$$$1 + 8 T^{2} + 25 T^{4}$$
$$11$$$$1 - 6 T^{2} + 121 T^{4}$$
$$13$$$$1 + 8 T^{2} + 169 T^{4}$$
$$17$$$$1 + 2 T + 2 T^{2} + 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T + 8 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$$29$$$$( 1 - 8 T + 29 T^{2} )( 1 + 8 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $J(E_4)$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ 4.0.2744.1 with defining polynomial:
$$x^{4} - x^{3} + 3 x^{2} + 3 x + 2$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112$$
$$g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124$$
Conductor norm: 1024
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112$$
$$g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764$$
Conductor norm: 1024

## 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 $$K = \Q (a) \simeq$$ 8.0.481890304.3 with defining polynomial $$x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2$$

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.