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

• Label: 4.2.a_ad_a_f
• 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:	$( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )( 1 + x - x^{2} + 2 x^{3} + 4 x^{4} )$
	$1 - 3 x^{2} + 5 x^{4} - 12 x^{6} + 16 x^{8}$
Frobenius angles:	$\pm0.0516399385854$, $\pm0.281693394748$, $\pm0.718306605252$, $\pm0.948360061415$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$49$	$3136$	$67081$	$1109227$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$-1$	$9$	$19$	$33$	$29$	$129$	$195$	$513$	$1139$

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 $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2}$

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

• 2.2.ab_ab : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• 2.2.b_ab : \(\Q(\sqrt{-3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj^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 2.4.ad_f^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-7})\)$)$

• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af^2 $\times$ 1.8.f^2. The endomorphism algebra for each factor is:

  • 1.8.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.8.f^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.ac_ab_ac_n	$2$	4.4.ag_t_acc_ez
4.2.c_ab_c_n	$2$	4.4.ag_t_acc_ez
4.2.ad_g_aj_l	$3$	(not in LMFDB)
4.2.a_g_a_r	$3$	(not in LMFDB)
4.2.d_g_j_l	$3$	(not in LMFDB)