Properties

Label 1.269.az
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:  $1 - 25 x + 269 x^{2}$
Frobenius angles:  $\pm0.224149536542$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-451}) \)
Galois group:  $C_2$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 245 72275 19469660 5236251475 1408516957225 378890486609600 101921535857307565 27416893174242574275 7375144265942864927180 1983913807582720356756875

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 245 72275 19469660 5236251475 1408516957225 378890486609600 101921535857307565 27416893174242574275 7375144265942864927180 1983913807582720356756875

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{269}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-451}) \).
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.z$2$(not in LMFDB)