Properties

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

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 8 x^{2} )( 1 + 8 x^{2} )$
  $1 - 5 x + 16 x^{2} - 40 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.154919815756$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  $3$

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $36$ $4536$ $260604$ $16447536$ $1082818836$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $72$ $508$ $4016$ $33044$ $264168$ $2099948$ $16775008$ $134225284$ $1073797272$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{3}}$
The isogeny class factors as 1.8.af $\times$ 1.8.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2^{3}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.q. The endomorphism algebra for each factor is:

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.b_e

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.f_q$2$2.64.h_aq
2.8.aj_bk$8$(not in LMFDB)
2.8.ab_ae$8$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.8.f_q$2$2.64.h_aq
2.8.aj_bk$8$(not in LMFDB)
2.8.ab_ae$8$(not in LMFDB)
2.8.b_ae$8$(not in LMFDB)
2.8.j_bk$8$(not in LMFDB)