Properties

Label 1.269.abe
Base Field $\F_{269}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{269}$
Dimension:  $1$
L-polynomial:  $1 - 30 x + 269 x^{2}$
Frobenius angles:  $\pm0.132532287154$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-11}) \)
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
$A(\F_{q^r})$ 240 72000 19462320 5236128000 1408515913200 378890499528000 101921536616840880 27416893192866432000 7375144266255572613360 1983913807586234257800000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 240 72000 19462320 5236128000 1408515913200 378890499528000 101921536616840880 27416893192866432000 7375144266255572613360 1983913807586234257800000

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{269}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \).
All geometric endomorphisms are defined over $\F_{269}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.269.be$2$(not in LMFDB)