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

• Label: 5.2.ai_bh_adm_ha_alc
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-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:	$5$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x + 3 x^{2} - 4 x^{3} + 4 x^{4} )$
	$1 - 8 x + 33 x^{2} - 90 x^{3} + 182 x^{4} - 288 x^{5} + 364 x^{6} - 360 x^{7} + 264 x^{8} - 128 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.174442860055$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.546783656212$
Angle rank:	$2$ (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:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2$	$3500$	$136214$	$3500000$	$121438802$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$7$	$19$	$39$	$75$	$91$	$79$	$159$	$451$	$987$

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^3 $\times$ 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^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
• 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.i^3 $\times$ 2.16.ac_b. The endomorphism algebra for each factor is:

• 1.16.i^3 : $\mathrm{M}_{3}(B)$, 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^3 $\times$ 2.4.c_b. The endomorphism algebra for each factor is:

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 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.ae_j_ao_w_abg	$2$	(not in LMFDB)
5.2.ae_j_ak_g_a	$2$	(not in LMFDB)
5.2.a_b_ac_g_a	$2$	(not in LMFDB)
5.2.a_b_c_g_a	$2$	(not in LMFDB)
5.2.e_j_k_g_a	$2$	(not in LMFDB)
5.2.e_j_o_w_bg	$2$	(not in LMFDB)
5.2.i_bh_dm_ha_lc	$2$	(not in LMFDB)
5.2.ac_d_a_ae_m	$3$	(not in LMFDB)