# Properties

 Label 1.241.abf Base Field $\F_{241}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{241}$ Dimension: $1$ L-polynomial: $1 - 31 x + 241 x^{2}$ Frobenius angles: $\pm0.0177663311058$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 1

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 211 57603 13990144 3373289283 812988283651 195930567705600 47219272787201971 11379844832476032003 2742542606001488007424 660952768067103056839203

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 211 57603 13990144 3373289283 812988283651 195930567705600 47219272787201971 11379844832476032003 2742542606001488007424 660952768067103056839203

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{241}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{241}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.241.bf $2$ (not in LMFDB) 1.241.o $3$ (not in LMFDB) 1.241.r $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.241.bf $2$ (not in LMFDB) 1.241.o $3$ (not in LMFDB) 1.241.r $3$ (not in LMFDB) 1.241.ar $6$ (not in LMFDB) 1.241.ao $6$ (not in LMFDB)