Properties

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

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 8 x^{2} )^{2}$
  $1 - 10 x + 41 x^{2} - 80 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.154919815756$, $\pm0.154919815756$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $16$ $3136$ $258064$ $17172736$ $1091905936$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $47$ $503$ $4191$ $33319$ $264143$ $2102743$ $16789183$ $134232839$ $1073721647$

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 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.ab_ab
$\F_{2}$2.2.c_f

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.a_aj$2$2.64.as_ib
2.8.k_bp$2$2.64.as_ib
2.8.f_r$3$2.512.ak_boj

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

TwistExtension degreeCommon base change
2.8.a_aj$2$2.64.as_ib
2.8.k_bp$2$2.64.as_ib
2.8.f_r$3$2.512.ak_boj
2.8.a_j$4$(not in LMFDB)
2.8.af_r$6$(not in LMFDB)