# Properties

 Label 169.a Conductor $169$ Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\End(J) \otimes \Q$$ $$\mathsf{CM}$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 169.a

Label Equation
169.a.169.1 $$y^2 + (x^3 + x + 1)y = x^5 + x^4$$

## L-function data

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

Prime L-Factor
$$13$$$$1 + 5 T + 13 T^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$1 - 7 T^{2} + 25 T^{4}$$
$$7$$$$1 + 7 T^{2} + 49 T^{4}$$
$$11$$$$1 + 11 T^{2} + 121 T^{4}$$
$$17$$$$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$$23$$$$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$$29$$$$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_6$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{13})^+$$ with defining polynomial:
$$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$$

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
$$g_4 = 3630 b^{5} + 1720 b^{4} - 15597 b^{3} - 8353 b^{2} + 9457 b + 2644$$
$$g_6 = 316316 b^{5} + 160160 b^{4} - 1342341 b^{3} - 759759 b^{2} + 759759 b + 207207$$
Conductor norm: 1

## 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 $$K = \Q (a) \simeq$$ $$\Q(\zeta_{13})^+$$ with defining polynomial $$x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1$$

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.