Properties

Label 1.512.af
Base Field $\F_{2^{9}}$
Dimension $1$
Ordinary Yes
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{9}}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 512 x^{2}$
Frobenius angles:  $\pm0.464759447268$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-7}) \)
Galois group:  $C_2$

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 a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 508 263144 134225284 68719003024 35184365852108 18014398720839416 9223372041104766164 4722366482782680160800 2417851639226647529086108 1237940039285411746904576264

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 508 263144 134225284 68719003024 35184365852108 18014398720839416 9223372041104766164 4722366482782680160800 2417851639226647529086108 1237940039285411746904576264

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).
All geometric endomorphisms are defined over $\F_{2^{9}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.

SubfieldPrimitive Model
$\F_{2}$1.2.b

Twists

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