Abelian variety isogeny class 3.2.ab_ab_d over $\F_{2}$

• Label: 3.2.ab_ab_d
• 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

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 - 3 x^{2} + 4 x^{4} )$
	$1 - x - x^{2} + 3 x^{3} - 2 x^{4} - 4 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.115026728081$, $\pm0.384973271919$, $\pm0.884973271919$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$32$	$1036$	$4096$	$23804$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$2$	$14$	$14$	$22$	$74$	$142$	$350$	$518$	$1082$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:

• $x^4+x^3z+x^2y^2+x^2yz+xy^3+xz^3+y^2z^2=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 2.2.a_ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.a_ad : \(\Q(i, \sqrt{7})\).

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.ad^2 $\times$ 1.4.d. The endomorphism algebra for each factor is:

  • 1.4.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.4.d : \(\Q(\sqrt{-7}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.2.b_ab_ad	$2$	3.4.ad_d_d
3.2.ad_j_an	$4$	3.16.ad_bz_adt
3.2.ab_f_ad	$4$	3.16.ad_bz_adt