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

• Label: 5.2.ai_bj_ady_ik_ano
• 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:	$( 1 - x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} )^{3}$
	$1 - 8 x + 35 x^{2} - 102 x^{3} + 218 x^{4} - 352 x^{5} + 436 x^{6} - 408 x^{7} + 280 x^{8} - 128 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.384973271919$
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:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$8000$	$430612$	$4000000$	$33357764$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$11$	$31$	$39$	$35$	$47$	$107$	$223$	$427$	$911$

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 $\times$ 1.2.ab^2 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 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
• 1.2.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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

The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab^2 $\times$ 1.16.i^3. The endomorphism algebra for each factor is:

• 1.16.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i^3 : $\mathrm{M}_{3}(B)$, 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^3 $\times$ 1.4.d^2. The endomorphism algebra for each factor is:

  • 1.4.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$
  • 1.4.d^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
5.2.ag_v_aby_dq_afo	$2$	(not in LMFDB)
5.2.ae_l_aw_bq_acm	$2$	(not in LMFDB)
5.2.ae_l_as_ba_abg	$2$	(not in LMFDB)
5.2.ac_f_ag_o_aq	$2$	(not in LMFDB)
5.2.a_d_ac_k_a	$2$	(not in LMFDB)
5.2.a_d_c_k_a	$2$	(not in LMFDB)
5.2.c_f_g_o_q	$2$	(not in LMFDB)
5.2.e_l_s_ba_bg	$2$	(not in LMFDB)
5.2.e_l_w_bq_cm	$2$	(not in LMFDB)
5.2.g_v_by_dq_fo	$2$	(not in LMFDB)
5.2.i_bj_dy_ik_no	$2$	(not in LMFDB)
5.2.af_l_ag_aw_ce	$3$	(not in LMFDB)
5.2.ac_f_a_ae_u	$3$	(not in LMFDB)
5.2.b_ab_g_i_ae	$3$	(not in LMFDB)