Properties

Label 2.8.ah_y
Base field $\F_{2^{3}}$
Dimension $2$
$p$-rank $1$
Ordinary no
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:  $2$
L-polynomial:  $1 - 7 x + 24 x^{2} - 56 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.0585111942353$, $\pm0.418160225599$
Angle rank:  $2$ (numerical)
Number field:  4.0.2312.1
Galois group:  $D_{4}$
Jacobians:  $1$

This isogeny class is simple and geometrically simple, not primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $26$ $3952$ $258362$ $16132064$ $1055303626$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $64$ $506$ $3936$ $32202$ $261712$ $2099162$ $16780224$ $134202602$ $1073703984$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2^{3}}$
The endomorphism algebra of this simple isogeny class is 4.0.2312.1.

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}$2.2.ab_a

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.h_y$2$2.64.ab_adc