Properties

Label 2.8.aj_bk
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 no

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 5 x + 8 x^{2} )( 1 - 4 x + 8 x^{2} )$
  $1 - 9 x + 36 x^{2} - 72 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.154919815756$, $\pm0.250000000000$
Angle rank:  $1$ (numerical)
Jacobians:  $0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $20$ $3640$ $276860$ $17508400$ $1091278100$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $56$ $540$ $4272$ $33300$ $263144$ $2097900$ $16775008$ $134208900$ $1073731736$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2^{3}}$
The isogeny class factors as 1.8.af $\times$ 1.8.ae 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 $\F_{2^{12}}$ is 1.4096.bv $\times$ 1.4096.ey. The endomorphism algebra for each factor is:
Remainder of endomorphism lattice by field

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.d_g

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.ab_ae$2$2.64.aj_ey
2.8.b_ae$2$2.64.aj_ey
2.8.j_bk$2$2.64.aj_ey

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

TwistExtension degreeCommon base change
2.8.ab_ae$2$2.64.aj_ey
2.8.b_ae$2$2.64.aj_ey
2.8.j_bk$2$2.64.aj_ey
2.8.ab_ae$4$(not in LMFDB)
2.8.af_q$8$(not in LMFDB)
2.8.f_q$8$(not in LMFDB)