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

• Label: 5.2.ah_bb_act_fm_aiq
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $3$
• Ordinary: no
• 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:	$( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 7 x + 27 x^{2} - 71 x^{3} + 142 x^{4} - 224 x^{5} + 284 x^{6} - 284 x^{7} + 216 x^{8} - 112 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.546783656212$
Angle rank:	$3$ (numerical)
Jacobians:	$0$

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$5600$	$146692$	$2240000$	$65162284$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$10$	$20$	$30$	$56$	$82$	$108$	$222$	$488$	$930$

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 $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^2 $\times$ 1.2.ab $\times$ 2.2.ac_d 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.2.ab : \(\Q(\sqrt{-7}) \).
• 2.2.ac_d : 4.0.1088.2.

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i^2 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is:

• 1.16.ab : \(\Q(\sqrt{-7}) \).
• 1.16.i^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 2.16.ac_b : 4.0.1088.2.

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 $\times$ 1.4.d $\times$ 2.4.c_b. The endomorphism algebra for each factor is:

  • 1.4.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 2.4.c_b : 4.0.1088.2.

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.af_p_abh_ck_ads	$2$	(not in LMFDB)
5.2.ad_h_al_s_ay	$2$	(not in LMFDB)
5.2.ad_h_ah_g_a	$2$	(not in LMFDB)
5.2.ab_d_af_k_ai	$2$	(not in LMFDB)
5.2.ab_d_ab_g_a	$2$	(not in LMFDB)
5.2.b_d_b_g_a	$2$	(not in LMFDB)
5.2.b_d_f_k_i	$2$	(not in LMFDB)
5.2.d_h_h_g_a	$2$	(not in LMFDB)
5.2.d_h_l_s_y	$2$	(not in LMFDB)
5.2.f_p_bh_ck_ds	$2$	(not in LMFDB)
5.2.h_bb_ct_fm_iq	$2$	(not in LMFDB)
5.2.ab_d_b_ac_k	$3$	(not in LMFDB)