# Properties

 Label 1225.a Conductor $1225$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{RM}$$ $$\End(J) \otimes \Q$$ $$\mathsf{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

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## Genus 2 curves in isogeny class 1225.a

Label Equation
1225.a.6125.1 $$y^2 + (x^3 + x^2)y = 2x^3 + x^2 + x + 2$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$5$$$$( 1 - T )^{2}$$
$$7$$$$( 1 + T )^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$11$$$$1 - T + 18 T^{2} - 11 T^{3} + 121 T^{4}$$
$$13$$$$1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4}$$
$$17$$$$1 + 5 T + 36 T^{2} + 85 T^{3} + 289 T^{4}$$
$$19$$$$1 + 6 T + 30 T^{2} + 114 T^{3} + 361 T^{4}$$
$$23$$$$1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

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

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{17})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

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