Properties

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

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Invariants

Base field:  $\F_{241}$
Dimension:  $1$
L-polynomial:  $1 - 17 x + 241 x^{2}$
Frobenius angles:  $\pm0.315567002228$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  10

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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 225 58275 14004900 3373481475 812989580625 195930567705600 47219272844789025 11379844839080902275 2742542606185086896100 660952768069917656431875

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 225 58275 14004900 3373481475 812989580625 195930567705600 47219272844789025 11379844839080902275 2742542606185086896100 660952768069917656431875

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.
TwistExtension DegreeCommon base change
1.241.r$2$(not in LMFDB)
1.241.ao$3$(not in LMFDB)
1.241.bf$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.241.r$2$(not in LMFDB)
1.241.ao$3$(not in LMFDB)
1.241.bf$3$(not in LMFDB)
1.241.abf$6$(not in LMFDB)
1.241.o$6$(not in LMFDB)