Properties

Label 1.512.bn
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:  $1 + 39 x + 512 x^{2}$
Frobenius angles:  $\pm0.830654189980$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-527}) \)
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
$A(\F_{q^r})$ 552 261648 134217144 68719754016 35184361574472 18014398777575216 9223372031782492632 4722366482930200523328 2417851639229493701323368 1237940039285340091856063568

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 552 261648 134217144 68719754016 35184361574472 18014398777575216 9223372031782492632 4722366482930200523328 2417851639229493701323368 1237940039285340091856063568

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{9}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-527}) \).
All geometric endomorphisms are defined over $\F_{2^{9}}$.

Base change

This is a primitive isogeny class.

Twists

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