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

• Label: 3.2.ad_j_an
• 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:	$( 1 - x + 2 x^{2} )^{3}$
	$1 - 3 x + 9 x^{2} - 13 x^{3} + 18 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.384973271919$, $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$512$	$2744$	$4096$	$10648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$14$	$24$	$14$	$0$	$38$	$168$	$350$	$528$	$854$

Jacobians and polarizations

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

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

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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$

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.ab_f_ad	$2$	3.4.j_bn_dv
3.2.b_f_d	$2$	3.4.j_bn_dv
3.2.d_j_n	$2$	3.4.j_bn_dv
3.2.a_a_f	$3$	3.8.p_dv_ob