Properties

Label 1.8.af
Base field $\F_{2^{3}}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 8 x^{2}$
Frobenius angles:  $\pm0.154919815756$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-7}) \)
Galois group:  $C_2$
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $56$ $508$ $4144$ $33044$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $56$ $508$ $4144$ $33044$ $263144$ $2099948$ $16783200$ $134225284$ $1073731736$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{3}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \).
All geometric endomorphisms are defined over $\F_{2^{3}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.

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

Twists

Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
1.8.f$2$1.64.aj