Properties

Label 3.8.ak_bu_afp
Base field $\F_{2^{3}}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 8 x^{2} )( 1 - 5 x + 13 x^{2} - 40 x^{3} + 64 x^{4} )$
  $1 - 10 x + 46 x^{2} - 145 x^{3} + 368 x^{4} - 640 x^{5} + 512 x^{6}$
Frobenius angles:  $\pm0.0644257339289$, $\pm0.154919815756$, $\pm0.530510095142$
Angle rank:  $3$ (numerical)
Isomorphism classes:  15

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $132$ $227304$ $120901968$ $66087728784$ $35454761137092$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $57$ $458$ $3937$ $33019$ $263646$ $2097703$ $16777697$ $134248634$ $1073798097$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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.af $\times$ 2.8.af_n 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
3.8.a_ae_ap$2$(not in LMFDB)
3.8.a_ae_p$2$(not in LMFDB)
3.8.k_bu_fp$2$(not in LMFDB)