Properties

Label 3.8.ak_ce_ahl
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:  $( 1 - 3 x + 8 x^{2} )( 1 - 7 x + 27 x^{2} - 56 x^{3} + 64 x^{4} )$
  $1 - 10 x + 56 x^{2} - 193 x^{3} + 448 x^{4} - 640 x^{5} + 512 x^{6}$
Frobenius angles:  $\pm0.195988130520$, $\pm0.322067999368$, $\pm0.361652535788$
Angle rank:  $3$ (numerical)
Isomorphism classes:  9

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $174$ $323640$ $163179288$ $72174956400$ $35119392164274$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $77$ $614$ $4297$ $32709$ $261134$ $2096555$ $16785169$ $134238782$ $1073703757$

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.ad $\times$ 2.8.ah_bb 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.ae_o_abf$2$(not in LMFDB)
3.8.e_o_bf$2$(not in LMFDB)
3.8.k_ce_hl$2$(not in LMFDB)