Abelian variety isogeny class 5.2.ah_ba_acq_fg_aih over $\F_{2}$

• Label: 5.2.ah_ba_acq_fg_aih
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $5$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$
	$1 - 7 x + 26 x^{2} - 68 x^{3} + 136 x^{4} - 215 x^{5} + 272 x^{6} - 272 x^{7} + 208 x^{8} - 112 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0435981566527$, $\pm0.123548644961$, $\pm0.329312442367$, $\pm0.456881978294$, $\pm0.527830414776$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	1
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$1349$	$31996$	$352089$	$24621781$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$8$	$8$	$0$	$21$	$74$	$129$	$248$	$575$	$1153$

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 $\F_{2^{42}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 2.2.ad_f $\times$ 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:

• 2.2.ad_f : \(\Q(\sqrt{-3}, \sqrt{5})\).
• 3.2.ae_j_ap : \(\Q(\zeta_{7})\).

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

The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd^3 $\times$ 1.4398046511104.hpued^2. The endomorphism algebra for each factor is:

• 1.4398046511104.ahxvrd^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
• 1.4398046511104.hpued^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 2.4.b_ad $\times$ 3.4.c_ad_an. The endomorphism algebra for each factor is:

  • 2.4.b_ad : \(\Q(\sqrt{-3}, \sqrt{5})\).
  • 3.4.c_ad_an : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_l $\times$ 3.8.ab_ag_bb. The endomorphism algebra for each factor is:

  • 2.8.a_l : \(\Q(\sqrt{-3}, \sqrt{5})\).
  • 3.8.ab_ag_bb : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.l^2 $\times$ 3.64.an_ag_bcp. The endomorphism algebra for each factor is:

  • 1.64.l^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
  • 3.64.an_ag_bcp : \(\Q(\zeta_{7})\).
• Endomorphism algebra over $\F_{2^{7}}$
  The base change of $A$ to $\F_{2^{7}}$ is 1.128.an^3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is:

  • 1.128.an^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 2.128.bn_yl : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{14}}$
  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.dj^3 $\times$ 2.16384.ajr_cqyz. The endomorphism algebra for each factor is:

  • 1.16384.dj^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 2.16384.ajr_cqyz : \(\Q(\sqrt{-3}, \sqrt{5})\).
• Endomorphism algebra over $\F_{2^{21}}$
  The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.edn^3 $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is:

  • 1.2097152.edn^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-7}) \)$)$
  • 2.2097152.a_hpued : \(\Q(\sqrt{-3}, \sqrt{5})\).

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
5.2.b_c_c_ac_ab	$2$	(not in LMFDB)
5.2.h_ba_cq_fg_ih	$2$	(not in LMFDB)
5.2.ae_i_al_n_ar	$3$	(not in LMFDB)
5.2.ab_c_ac_ac_b	$3$	(not in LMFDB)