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

• Label: 4.2.af_l_ao_q
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 - 2 x + 2 x^{2} )^{2}( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$
	$1 - 5 x + 11 x^{2} - 14 x^{3} + 16 x^{4} - 28 x^{5} + 44 x^{6} - 40 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.718306605252$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$175$	$2704$	$161875$	$1262431$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$2$	$7$	$34$	$38$	$47$	$110$	$162$	$439$	$1082$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^2 $\times$ 2.2.ab_ab 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.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.2.ab_ab : \(\Q(\sqrt{-3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv^2 $\times$ 1.4096.ey^2. The endomorphism algebra for each factor is:

• 1.4096.bv^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.4096.ey^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a^2 $\times$ 2.4.ad_f. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 2.4.ad_f : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af^2 $\times$ 1.8.e^2. The endomorphism algebra for each factor is:

  • 1.8.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.8.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i^2 $\times$ 2.16.b_ap. The endomorphism algebra for each factor is:

  • 1.16.i^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.b_ap : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj^2 $\times$ 1.64.a^2. The endomorphism algebra for each factor is:

  • 1.64.aj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.64.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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_d_g_aq	$2$	4.4.ad_n_abk_cu
4.2.ab_ab_ac_i	$2$	4.4.ad_n_abk_cu
4.2.b_ab_c_i	$2$	4.4.ad_n_abk_cu
4.2.d_d_ag_aq	$2$	4.4.ad_n_abk_cu
4.2.f_l_o_q	$2$	4.4.ad_n_abk_cu
4.2.ac_f_ai_q	$3$	(not in LMFDB)
4.2.b_ab_ac_ac	$3$	(not in LMFDB)
4.2.e_l_w_bi	$3$	(not in LMFDB)