Properties

Label 3.2.ad_h_al
Base field $\F_{2}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - x + 2 x^{2} )( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
  $1 - 3 x + 7 x^{2} - 11 x^{3} + 14 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.174442860055$, $\pm0.384973271919$, $\pm0.546783656212$
Angle rank:  $3$ (numerical)
Jacobians:  $1$
Isomorphism classes:  4

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

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $224$ $868$ $3584$ $38764$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $10$ $12$ $14$ $40$ $82$ $140$ $286$ $552$ $930$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2}$.

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ab $\times$ 2.2.ac_d 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
3.2.ab_d_af$2$3.4.f_l_t
3.2.b_d_f$2$3.4.f_l_t
3.2.d_h_l$2$3.4.f_l_t