Abelian variety isogeny class 4.2.af_p_abg_bz over $\F_{2}$

• Label: 4.2.af_p_abg_bz
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$( 1 - x + 2 x^{2} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$
	$1 - 5 x + 15 x^{2} - 32 x^{3} + 51 x^{4} - 64 x^{5} + 60 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0435981566527$, $\pm0.329312442367$, $\pm0.384973271919$, $\pm0.527830414776$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$568$	$5894$	$32944$	$563662$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$10$	$13$	$6$	$13$	$43$	$103$	$262$	$580$	$1035$

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 $\F_{2^{14}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ab $\times$ 3.2.ae_j_ap 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 : \(\Q(\sqrt{-7}) \).
• 3.2.ae_j_ap : \(\Q(\zeta_{7})\).

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

The base change of $A$ to $\F_{2^{14}}$ is 1.16384.dj^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-7}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 3.4.c_ad_an. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 3.4.c_ad_an : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{7}}$
  The base change of $A$ to $\F_{2^{7}}$ is 1.128.an^3 $\times$ 1.128.n. The endomorphism algebra for each factor is:

  • 1.128.an^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 1.128.n : \(\Q(\sqrt{-7}) \).

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
4.2.ad_h_ao_v	$2$	4.4.f_h_ao_acl
4.2.d_h_o_v	$2$	4.4.f_h_ao_acl
4.2.f_p_bg_bz	$2$	4.4.f_h_ao_acl