Abelian Variety Isogeny Class 6.2.ak_cb_ahg_sy_abme_ciq over $\F_{2}$

• Label: 6.2.ak_cb_ahg_sy_abme_ciq
• Base Field: $\F_{2}$
• Dimension: $6$
• Ordinary: No
• $p$-rank: $2$
• Principally polarizable: Yes
• Contains a Jacobian: No

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$( 1 - x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} )^{4}$
Frobenius angles:	$\pm0.25$, $\pm0.25$, $\pm0.25$, $\pm0.25$, $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:	$1$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$	1	2	3	4	5	6	7	8	9	10
$A(\F_{q^r})$	4	40000	5597956	100000000	1367668324	55979560000	3287686987204	212576400000000	14362823545620004	1034299170025000000

Point counts of the (virtual) curve

$r$	1	2	3	4	5	6	7	8	9	10
$C(\F_{q^r})$	-7	11	35	47	43	47	91	191	395	911

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.ac^4 $\times$ 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^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
• 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^4. The endomorphism algebra for each factor is:

• 1.16.ab^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.16.i^4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

All geometric endomorphisms are defined over $\F_{2^{4}}$.
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^4 $\times$ 1.4.d^2. The endomorphism algebra for each factor is:

  • 1.4.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$
  • 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
6.2.ai_bj_aea_jc_aqq_zo	$2$	(not in LMFDB)
6.2.ag_v_aca_ee_ahk_lk	$2$	(not in LMFDB)
6.2.ag_v_abw_dg_aeq_gm	$2$	(not in LMFDB)
6.2.ae_l_au_bk_ace_dk	$2$	(not in LMFDB)
6.2.ac_f_ai_u_ay_bo	$2$	(not in LMFDB)
6.2.ac_f_ae_m_aq_bo	$2$	(not in LMFDB)
6.2.a_d_a_m_a_y	$2$	(not in LMFDB)
6.2.c_f_e_m_q_bo	$2$	(not in LMFDB)
6.2.c_f_i_u_y_bo	$2$	(not in LMFDB)
6.2.e_l_u_bk_ce_dk	$2$	(not in LMFDB)
6.2.g_v_bw_dg_eq_gm	$2$	(not in LMFDB)
6.2.g_v_ca_ee_hk_lk	$2$	(not in LMFDB)
6.2.i_bj_ea_jc_qq_zo	$2$	(not in LMFDB)
6.2.k_cb_hg_sy_bme_ciq	$2$	(not in LMFDB)
6.2.ah_x_abm_m_dk_ahs	$3$	(not in LMFDB)
6.2.ae_l_ao_g_bc_ace	$3$	(not in LMFDB)
6.2.ab_ab_k_ag_ai_bo	$3$	(not in LMFDB)
6.2.c_f_q_y_bo_cy	$3$	(not in LMFDB)
6.2.f_l_w_bw_cy_dw	$3$	(not in LMFDB)