Properties

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

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Invariants

Base field:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 8 x^{2} )( 1 + x + 8 x^{2} )$
  $1 - 3 x + 12 x^{2} - 24 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.556567041129$
Angle rank:  $1$ (numerical)
Jacobians:  $3$

This isogeny class is not simple, 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $50$ $5200$ $267050$ $16900000$ $1091476250$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $6$ $80$ $522$ $4128$ $33306$ $262640$ $2092362$ $16767808$ $134224506$ $1073728400$

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

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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.af_u$2$2.64.p_ey
2.8.d_m$2$2.64.p_ey
2.8.f_u$2$2.64.p_ey

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

TwistExtension degreeCommon base change
2.8.af_u$2$2.64.p_ey
2.8.d_m$2$2.64.p_ey
2.8.f_u$2$2.64.p_ey
2.8.ab_q$8$(not in LMFDB)
2.8.b_q$8$(not in LMFDB)