# Properties

 Label 1.241.a Base Field $\F_{241}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{241}$ Dimension: $1$ L-polynomial: $1 + 241 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-241})$$ Galois group: $C_2$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 242 58564 13997522 3373286400 812990017202 195930622140484 47219273189051282 11379844831814553600 2742542606093287451762 660952768070108255908804

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 242 58564 13997522 3373286400 812990017202 195930622140484 47219273189051282 11379844831814553600 2742542606093287451762 660952768070108255908804

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{241}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-241})$$.
Endomorphism algebra over $\overline{\F}_{241}$
 The base change of $A$ to $\F_{241^{2}}$ is the simple isogeny class 1.58081.so and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $241$ and $\infty$.
All geometric endomorphisms are defined over $\F_{241^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.