# Properties

 Label 1.269.a Base Field $\F_{269}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{269}$ Dimension: $1$ L-polynomial: $1 + 269 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-269})$$ Galois group: $C_2$ Jacobians: 22

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular.

 $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 22 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 270 72900 19465110 5235969600 1408514752350 378890507312100 101921535994725990 27416893172109062400 7375144266114367290030 1983913807587581830522500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 270 72900 19465110 5235969600 1408514752350 378890507312100 101921535994725990 27416893172109062400 7375144266114367290030 1983913807587581830522500

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{269}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-269})$$.
Endomorphism algebra over $\overline{\F}_{269}$
 The base change of $A$ to $\F_{269^{2}}$ is the simple isogeny class 1.72361.us and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $269$ and $\infty$.
All geometric endomorphisms are defined over $\F_{269^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.