Properties

Label 2.8.ad_e
Base field $\F_{2^{3}}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $1 - 3 x + 4 x^{2} - 24 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.117638066933$, $\pm0.631737784799$
Angle rank:  $2$ (numerical)
Number field:  4.0.90972.1
Galois group:  $D_{4}$
Jacobians:  $6$

This isogeny class is simple and geometrically simple, 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $42$ $4032$ $231714$ $16781184$ $1087503522$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $6$ $64$ $450$ $4096$ $33186$ $261952$ $2098914$ $16793344$ $134227746$ $1073761984$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are 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 4.0.90972.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.8.d_e$2$2.64.ab_a