# Properties

 Label 6.2.aj_bs_afq_oa_abbk_brc Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{4}$ Frobenius angles: $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.5$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## Point counts

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 6 45000 3598686 56250000 2051502486 80970435000 2986701558798 166075312500000 14224186252708614 1163586566278125000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 12 30 40 54 72 78 128 390 1032

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 4 $\times$ 1.2.ab $\times$ 1.2.a 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 : $$\Q(\sqrt{-7})$$. 1.2.a : $$\Q(\sqrt{-2})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 5 $\times$ 1.256.bf. The endomorphism algebra for each factor is: 1.256.abg 5 : $\mathrm{M}_{5}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.256.bf : $$\Q(\sqrt{-7})$$.
All geometric endomorphisms are defined over $\F_{2^{8}}$.
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 $\times$ 1.4.e. The endomorphism algebra for each factor is: 1.4.a 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$. 1.4.e : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.ab $\times$ 1.16.i 4 . The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i 4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.

## 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.ah_bc_ada_gq_ama_sm $2$ (not in LMFDB) 6.2.af_q_abi_ci_adk_ey $2$ (not in LMFDB) 6.2.ad_i_ao_bc_abo_cm $2$ (not in LMFDB) 6.2.ab_e_ac_m_ai_bg $2$ (not in LMFDB) 6.2.b_e_c_m_i_bg $2$ (not in LMFDB) 6.2.d_i_o_bc_bo_cm $2$ (not in LMFDB) 6.2.f_q_bi_ci_dk_ey $2$ (not in LMFDB) 6.2.h_bc_da_gq_ma_sm $2$ (not in LMFDB) 6.2.j_bs_fq_oa_bbk_brc $2$ (not in LMFDB) 6.2.ad_i_ai_e_u_abg $3$ (not in LMFDB) 6.2.d_i_w_bo_cq_ei $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ah_bc_ada_gq_ama_sm $2$ (not in LMFDB) 6.2.af_q_abi_ci_adk_ey $2$ (not in LMFDB) 6.2.ad_i_ao_bc_abo_cm $2$ (not in LMFDB) 6.2.ab_e_ac_m_ai_bg $2$ (not in LMFDB) 6.2.b_e_c_m_i_bg $2$ (not in LMFDB) 6.2.d_i_o_bc_bo_cm $2$ (not in LMFDB) 6.2.f_q_bi_ci_dk_ey $2$ (not in LMFDB) 6.2.h_bc_da_gq_ma_sm $2$ (not in LMFDB) 6.2.j_bs_fq_oa_bbk_brc $2$ (not in LMFDB) 6.2.ad_i_ai_e_u_abg $3$ (not in LMFDB) 6.2.d_i_w_bo_cq_ei $3$ (not in LMFDB) 6.2.b_e_e_e_i_a $5$ (not in LMFDB) 6.2.ah_bc_adc_gy_amu_ts $6$ (not in LMFDB) 6.2.af_q_abq_dk_aga_jg $6$ (not in LMFDB) 6.2.af_q_abo_dg_afs_iq $6$ (not in LMFDB) 6.2.ad_i_aw_bo_acq_ei $6$ (not in LMFDB) 6.2.ad_i_aq_bc_abs_cm $6$ (not in LMFDB) 6.2.ab_e_ai_m_au_bg $6$ (not in LMFDB) 6.2.ab_e_ac_a_e_aq $6$ (not in LMFDB) 6.2.ab_e_a_e_m_a $6$ (not in LMFDB) 6.2.b_e_a_e_am_a $6$ (not in LMFDB) 6.2.b_e_c_a_ae_aq $6$ (not in LMFDB) 6.2.b_e_i_m_u_bg $6$ (not in LMFDB) 6.2.d_i_i_e_au_abg $6$ (not in LMFDB) 6.2.d_i_q_bc_bs_cm $6$ (not in LMFDB) 6.2.f_q_bo_dg_fs_iq $6$ (not in LMFDB) 6.2.f_q_bq_dk_ga_jg $6$ (not in LMFDB) 6.2.h_bc_dc_gy_mu_ts $6$ (not in LMFDB) 6.2.al_ck_aiw_xw_abwy_dae $8$ (not in LMFDB) 6.2.aj_bq_afa_lo_avg_bgq $8$ (not in LMFDB) 6.2.ah_w_abi_e_ds_aia $8$ (not in LMFDB) 6.2.ah_ba_ack_ee_afw_ia $8$ (not in LMFDB) 6.2.ah_be_adm_ie_apk_ya $8$ (not in LMFDB) 6.2.af_k_ag_am_bg_abw $8$ (not in LMFDB) 6.2.af_m_ao_ae_bw_ads $8$ (not in LMFDB) 6.2.af_o_aba_bs_acu_ei $8$ (not in LMFDB) 6.2.af_s_abu_dw_agu_km $8$ (not in LMFDB) 6.2.af_u_acc_eu_aiq_no $8$ (not in LMFDB) 6.2.ad_ac_s_au_ay_dc $8$ (not in LMFDB) 6.2.ad_c_g_am_a_q $8$ (not in LMFDB) 6.2.ad_e_ac_ae_q_abg $8$ (not in LMFDB) 6.2.ad_g_ag_ae_y_abw $8$ (not in LMFDB) 6.2.ad_g_ag_m_ay_bw $8$ (not in LMFDB) 6.2.ad_k_as_bk_abw_dc $8$ (not in LMFDB) 6.2.ad_m_aba_ci_ads_ge $8$ (not in LMFDB) 6.2.ad_o_abe_cy_aeq_ia $8$ (not in LMFDB) 6.2.ab_ag_g_m_ai_aq $8$ (not in LMFDB) 6.2.ab_ae_g_ae_ai_bg $8$ (not in LMFDB) 6.2.ab_ac_c_e_a_aq $8$ (not in LMFDB) 6.2.ab_a_c_ae_a_a $8$ (not in LMFDB) 6.2.ab_c_ac_ae_i_aq $8$ (not in LMFDB) 6.2.ab_c_ac_m_ai_q $8$ (not in LMFDB) 6.2.ab_e_ac_ae_i_abg $8$ (not in LMFDB) 6.2.ab_g_ag_u_aq_bw $8$ (not in LMFDB) 6.2.ab_i_ag_bc_aq_cm $8$ (not in LMFDB) 6.2.ab_k_ak_bs_abo_ei $8$ (not in LMFDB) 6.2.ab_m_ak_ci_abo_ge $8$ (not in LMFDB) 6.2.b_ag_ag_m_i_aq $8$ (not in LMFDB) 6.2.b_ae_ag_ae_i_bg $8$ (not in LMFDB) 6.2.b_ac_ac_e_a_aq $8$ (not in LMFDB) 6.2.b_a_ac_ae_a_a $8$ (not in LMFDB) 6.2.b_c_c_ae_ai_aq $8$ (not in LMFDB) 6.2.b_c_c_m_i_q $8$ (not in LMFDB) 6.2.b_e_c_ae_ai_abg $8$ (not in LMFDB) 6.2.b_g_g_u_q_bw $8$ (not in LMFDB) 6.2.b_i_g_bc_q_cm $8$ (not in LMFDB) 6.2.b_k_k_bs_bo_ei $8$ (not in LMFDB) 6.2.b_m_k_ci_bo_ge $8$ (not in LMFDB) 6.2.d_ac_as_au_y_dc $8$ (not in LMFDB) 6.2.d_c_ag_am_a_q $8$ (not in LMFDB) 6.2.d_e_c_ae_aq_abg $8$ (not in LMFDB) 6.2.d_g_g_ae_ay_abw $8$ (not in LMFDB) 6.2.d_g_g_m_y_bw $8$ (not in LMFDB) 6.2.d_k_s_bk_bw_dc $8$ (not in LMFDB) 6.2.d_m_ba_ci_ds_ge $8$ (not in LMFDB) 6.2.d_o_be_cy_eq_ia $8$ (not in LMFDB) 6.2.f_k_g_am_abg_abw $8$ (not in LMFDB) 6.2.f_m_o_ae_abw_ads $8$ (not in LMFDB) 6.2.f_o_ba_bs_cu_ei $8$ (not in LMFDB) 6.2.f_s_bu_dw_gu_km $8$ (not in LMFDB) 6.2.f_u_cc_eu_iq_no $8$ (not in LMFDB) 6.2.h_w_bi_e_ads_aia $8$ (not in LMFDB) 6.2.h_ba_ck_ee_fw_ia $8$ (not in LMFDB) 6.2.h_be_dm_ie_pk_ya $8$ (not in LMFDB) 6.2.j_bq_fa_lo_vg_bgq $8$ (not in LMFDB) 6.2.l_ck_iw_xw_bwy_dae $8$ (not in LMFDB) 6.2.ad_i_am_m_ai_a $10$ (not in LMFDB) 6.2.ab_e_ae_e_ai_a $10$ (not in LMFDB) 6.2.d_i_m_m_i_a $10$ (not in LMFDB) 6.2.ad_g_ag_e_a_a $16$ (not in LMFDB) 6.2.ab_c_ac_e_a_a $16$ (not in LMFDB) 6.2.ab_e_ac_e_a_a $16$ (not in LMFDB) 6.2.b_c_c_e_a_a $16$ (not in LMFDB) 6.2.b_e_c_e_a_a $16$ (not in LMFDB) 6.2.d_g_g_e_a_a $16$ (not in LMFDB) 6.2.aj_bq_afc_ma_awq_bjc $24$ (not in LMFDB) 6.2.ah_y_abw_ce_abc_a $24$ (not in LMFDB) 6.2.ah_ba_acs_fw_akq_qi $24$ (not in LMFDB) 6.2.ah_ba_acq_fo_aka_pk $24$ (not in LMFDB) 6.2.ah_bc_acy_ge_akq_qa $24$ (not in LMFDB) 6.2.af_k_ai_ai_bk_acm $24$ (not in LMFDB) 6.2.af_m_as_u_au_y $24$ (not in LMFDB) 6.2.af_m_aq_q_au_bg $24$ (not in LMFDB) 6.2.af_o_abi_cu_aeu_hc $24$ (not in LMFDB) 6.2.af_o_abc_bw_acy_ei $24$ (not in LMFDB) 6.2.af_o_ay_bc_au_q $24$ (not in LMFDB) 6.2.af_o_au_i_bk_adc $24$ (not in LMFDB) 6.2.af_q_abm_cy_afc_hs $24$ (not in LMFDB) 6.2.af_q_abk_cu_aeu_hk $24$ (not in LMFDB) 6.2.af_s_abw_ea_ahg_lc $24$ (not in LMFDB) 6.2.af_s_abs_do_aga_jg $24$ (not in LMFDB) 6.2.ad_a_m_aq_am_bw $24$ (not in LMFDB) 6.2.ad_c_a_a_m_abg $24$ (not in LMFDB) 6.2.ad_c_g_ai_am_bo $24$ (not in LMFDB) 6.2.ad_e_ag_m_am_i $24$ (not in LMFDB) 6.2.ad_e_ae_ae_u_abg $24$ (not in LMFDB) 6.2.ad_e_a_ai_m_aq $24$ (not in LMFDB) 6.2.ad_e_a_a_am_bg $24$ (not in LMFDB) 6.2.ad_g_ao_y_abk_ce $24$ (not in LMFDB) 6.2.ad_g_am_y_abk_bw $24$ (not in LMFDB) 6.2.ad_g_ak_m_am_q $24$ (not in LMFDB) 6.2.ad_g_ai_m_am_q $24$ (not in LMFDB) 6.2.ad_g_ag_a_m_ay $24$ (not in LMFDB) 6.2.ad_g_ag_i_am_y $24$ (not in LMFDB) 6.2.ad_g_ae_a_m_aq $24$ (not in LMFDB) 6.2.ad_i_as_bk_aci_dk $24$ (not in LMFDB) 6.2.ad_i_am_q_am_q $24$ (not in LMFDB) 6.2.ad_i_am_y_abk_cm $24$ (not in LMFDB) 6.2.ad_k_ay_bw_adg_ey $24$ (not in LMFDB) 6.2.ad_k_aw_bs_acy_ei $24$ (not in LMFDB) 6.2.ad_k_au_bs_acq_ei $24$ (not in LMFDB) 6.2.ad_k_as_bo_aci_ea $24$ (not in LMFDB) 6.2.ad_m_abc_ci_aee_ge $24$ (not in LMFDB) 6.2.ad_m_ay_ce_adg_fo $24$ (not in LMFDB) 6.2.ab_ae_e_i_ae_aq $24$ (not in LMFDB) 6.2.ab_ac_a_a_e_a $24$ (not in LMFDB) 6.2.ab_ac_c_i_ae_ai $24$ (not in LMFDB) 6.2.ab_ac_e_ae_ae_q $24$ (not in LMFDB) 6.2.ab_ac_i_ai_am_bg $24$ (not in LMFDB) 6.2.ab_a_ae_ae_m_a $24$ (not in LMFDB) 6.2.ab_a_ac_e_ae_i $24$ (not in LMFDB) 6.2.ab_a_a_a_e_aq $24$ (not in LMFDB) 6.2.ab_a_a_i_ae_a $24$ (not in LMFDB) 6.2.ab_a_c_a_ae_q $24$ (not in LMFDB) 6.2.ab_a_g_ae_ae_y $24$ (not in LMFDB) 6.2.ab_c_ak_i_am_bo $24$ (not in LMFDB) 6.2.ab_c_ag_e_ae_q $24$ (not in LMFDB) 6.2.ab_c_ae_i_am_q $24$ (not in LMFDB) 6.2.ab_c_ac_a_e_ai $24$ (not in LMFDB) 6.2.ab_c_ac_i_ae_i $24$ (not in LMFDB) 6.2.ab_c_a_ae_e_aq $24$ (not in LMFDB) 6.2.ab_c_a_e_ae_q $24$ (not in LMFDB) 6.2.ab_c_e_a_e_q $24$ (not in LMFDB) 6.2.ab_e_ag_m_au_y $24$ (not in LMFDB) 6.2.ab_e_ae_i_ae_q $24$ (not in LMFDB) 6.2.ab_e_ae_q_am_bg $24$ (not in LMFDB) 6.2.ab_e_ac_i_ae_q $24$ (not in LMFDB) 6.2.ab_e_c_e_m_i $24$ (not in LMFDB) 6.2.ab_g_ak_u_abk_bw $24$ (not in LMFDB) 6.2.ab_g_ai_q_abc_bg $24$ (not in LMFDB) 6.2.ab_g_ag_y_au_ce $24$ (not in LMFDB) 6.2.ab_g_ae_m_ae_q $24$ (not in LMFDB) 6.2.ab_g_ae_u_am_bw $24$ (not in LMFDB) 6.2.ab_g_a_i_u_a $24$ (not in LMFDB) 6.2.ab_i_am_bc_aca_cm $24$ (not in LMFDB) 6.2.ab_i_ai_bg_abc_dc $24$ (not in LMFDB) 6.2.ab_i_ag_bg_au_dc $24$ (not in LMFDB) 6.2.ab_k_ai_bs_abc_ei $24$ (not in LMFDB) 6.2.b_ae_ae_i_e_aq $24$ (not in LMFDB) 6.2.b_ac_ai_ai_m_bg $24$ (not in LMFDB) 6.2.b_ac_ae_ae_e_q $24$ (not in LMFDB) 6.2.b_ac_ac_i_e_ai $24$ (not in LMFDB) 6.2.b_ac_a_a_ae_a $24$ (not in LMFDB) 6.2.b_a_ag_ae_e_y $24$ (not in LMFDB) 6.2.b_a_ac_a_e_q $24$ (not in LMFDB) 6.2.b_a_a_a_ae_aq $24$ (not in LMFDB) 6.2.b_a_a_i_e_a $24$ (not in LMFDB) 6.2.b_a_c_e_e_i $24$ (not in LMFDB) 6.2.b_a_e_ae_am_a $24$ (not in LMFDB) 6.2.b_c_ae_a_ae_q $24$ (not in LMFDB) 6.2.b_c_a_ae_ae_aq $24$ (not in LMFDB) 6.2.b_c_a_e_e_q $24$ (not in LMFDB) 6.2.b_c_c_a_ae_ai $24$ (not in LMFDB) 6.2.b_c_c_i_e_i $24$ (not in LMFDB) 6.2.b_c_e_i_m_q $24$ (not in LMFDB) 6.2.b_c_g_e_e_q $24$ (not in LMFDB) 6.2.b_c_k_i_m_bo $24$ (not in LMFDB) 6.2.b_e_ac_e_am_i $24$ (not in LMFDB) 6.2.b_e_c_i_e_q $24$ (not in LMFDB) 6.2.b_e_e_i_e_q $24$ (not in LMFDB) 6.2.b_e_e_q_m_bg $24$ (not in LMFDB) 6.2.b_e_g_m_u_y $24$ (not in LMFDB) 6.2.b_g_a_i_au_a $24$ (not in LMFDB) 6.2.b_g_e_m_e_q $24$ (not in LMFDB) 6.2.b_g_e_u_m_bw $24$ (not in LMFDB) 6.2.b_g_g_y_u_ce $24$ (not in LMFDB) 6.2.b_g_i_q_bc_bg $24$ (not in LMFDB) 6.2.b_g_k_u_bk_bw $24$ (not in LMFDB) 6.2.b_i_g_bg_u_dc $24$ (not in LMFDB) 6.2.b_i_i_bg_bc_dc $24$ (not in LMFDB) 6.2.b_i_m_bc_ca_cm $24$ (not in LMFDB) 6.2.b_k_i_bs_bc_ei $24$ (not in LMFDB) 6.2.d_a_am_aq_m_bw $24$ (not in LMFDB) 6.2.d_c_ag_ai_m_bo $24$ (not in LMFDB) 6.2.d_c_a_a_am_abg $24$ (not in LMFDB) 6.2.d_e_a_ai_am_aq $24$ (not in LMFDB) 6.2.d_e_a_a_m_bg $24$ (not in LMFDB) 6.2.d_e_e_ae_au_abg $24$ (not in LMFDB) 6.2.d_e_g_m_m_i $24$ (not in LMFDB) 6.2.d_g_e_a_am_aq $24$ (not in LMFDB) 6.2.d_g_g_a_am_ay $24$ (not in LMFDB) 6.2.d_g_g_i_m_y $24$ (not in LMFDB) 6.2.d_g_i_m_m_q $24$ (not in LMFDB) 6.2.d_g_k_m_m_q $24$ (not in LMFDB) 6.2.d_g_m_y_bk_bw $24$ (not in LMFDB) 6.2.d_g_o_y_bk_ce $24$ (not in LMFDB) 6.2.d_i_m_q_m_q $24$ (not in LMFDB) 6.2.d_i_m_y_bk_cm $24$ (not in LMFDB) 6.2.d_i_s_bk_ci_dk $24$ (not in LMFDB) 6.2.d_k_s_bo_ci_ea $24$ (not in LMFDB) 6.2.d_k_u_bs_cq_ei $24$ (not in LMFDB) 6.2.d_k_w_bs_cy_ei $24$ (not in LMFDB) 6.2.d_k_y_bw_dg_ey $24$ (not in LMFDB) 6.2.d_m_y_ce_dg_fo $24$ (not in LMFDB) 6.2.d_m_bc_ci_ee_ge $24$ (not in LMFDB) 6.2.f_k_i_ai_abk_acm $24$ (not in LMFDB) 6.2.f_m_q_q_u_bg $24$ (not in LMFDB) 6.2.f_m_s_u_u_y $24$ (not in LMFDB) 6.2.f_o_u_i_abk_adc $24$ (not in LMFDB) 6.2.f_o_y_bc_u_q $24$ (not in LMFDB) 6.2.f_o_bc_bw_cy_ei $24$ (not in LMFDB) 6.2.f_o_bi_cu_eu_hc $24$ (not in LMFDB) 6.2.f_q_bk_cu_eu_hk $24$ (not in LMFDB) 6.2.f_q_bm_cy_fc_hs $24$ (not in LMFDB) 6.2.f_s_bs_do_ga_jg $24$ (not in LMFDB) 6.2.f_s_bw_ea_hg_lc $24$ (not in LMFDB) 6.2.h_y_bw_ce_bc_a $24$ (not in LMFDB) 6.2.h_ba_cq_fo_ka_pk $24$ (not in LMFDB) 6.2.h_ba_cs_fw_kq_qi $24$ (not in LMFDB) 6.2.h_bc_cy_ge_kq_qa $24$ (not in LMFDB) 6.2.j_bq_fc_ma_wq_bjc $24$ (not in LMFDB) 6.2.af_o_ay_y_ai_ai $40$ (not in LMFDB) 6.2.ad_e_a_ae_a_i $40$ (not in LMFDB) 6.2.ad_g_ai_i_ai_i $40$ (not in LMFDB) 6.2.ad_i_am_u_ay_bo $40$ (not in LMFDB) 6.2.ab_a_a_e_a_ai $40$ (not in LMFDB) 6.2.ab_c_a_a_ai_i $40$ (not in LMFDB) 6.2.ab_c_a_a_a_a $40$ (not in LMFDB) 6.2.ab_c_a_a_i_ai $40$ (not in LMFDB) 6.2.ab_e_ae_m_ai_y $40$ (not in LMFDB) 6.2.ab_g_ae_q_ai_bg $40$ (not in LMFDB) 6.2.b_a_a_e_a_ai $40$ (not in LMFDB) 6.2.b_c_a_a_ai_ai $40$ (not in LMFDB) 6.2.b_c_a_a_a_a $40$ (not in LMFDB) 6.2.b_c_a_a_i_i $40$ (not in LMFDB) 6.2.b_e_e_m_i_y $40$ (not in LMFDB) 6.2.b_g_e_q_i_bg $40$ (not in LMFDB) 6.2.d_e_a_ae_a_i $40$ (not in LMFDB) 6.2.d_g_i_i_i_i $40$ (not in LMFDB) 6.2.d_i_m_u_y_bo $40$ (not in LMFDB) 6.2.f_o_y_y_i_ai $40$ (not in LMFDB)