Properties

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

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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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.