Properties

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

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 8 x^{2} )( 1 - x + 8 x^{2} )$
  $1 - 4 x + 19 x^{2} - 32 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.322067999368$, $\pm0.443432958871$
Angle rank:  $2$ (numerical)
Jacobians:  $9$
Isomorphism classes:  36

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $48$ $5760$ $299088$ $16704000$ $1060603248$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $87$ $581$ $4079$ $32365$ $261639$ $2097877$ $16777951$ $134216573$ $1073778807$

Jacobians and polarizations

This isogeny class contains the Jacobians of 9 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 isogeny class factors as 1.8.ad $\times$ 1.8.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

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.ac_n$2$2.64.w_iz
2.8.c_n$2$2.64.w_iz
2.8.e_t$2$2.64.w_iz