# Properties

 Label 6.2.ai_bg_adh_gq_ale_qq Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.139386741866$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.686170398078$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 2 7600 333944 44460000 2264848282 96443027200 6291945767774 262178930520000 15623004528210248 1325604375213190000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 5 10 33 55 86 177 241 442 1165

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 2.2.ad_f $\times$ 2.2.ab_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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.2.ad_f : $$\Q(\sqrt{-3}, \sqrt{5})$$. 2.2.ab_a : 4.0.2312.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.ey 2 $\times$ 2.4096.agf_vki. The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.4096.agf_vki : 4.0.2312.1.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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 $\times$ 2.4.ab_e $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.4.ab_e : 4.0.2312.1. 2.4.b_ad : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e 2 $\times$ 2.8.ah_y $\times$ 2.8.a_l. The endomorphism algebra for each factor is: 1.8.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.8.ah_y : 4.0.2312.1. 2.8.a_l : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 2.16.ah_bh $\times$ 2.16.h_bo. The endomorphism algebra for each factor is: 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$. 2.16.h_bo : 4.0.2312.1.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 $\times$ 1.64.l 2 $\times$ 2.64.ab_adc. The endomorphism algebra for each factor is: 1.64.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 2.64.ab_adc : 4.0.2312.1.

## 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.ag_s_abf_bc_c_abg $2$ (not in LMFDB) 6.2.ae_i_an_y_abq_cm $2$ (not in LMFDB) 6.2.ac_c_ab_e_ac_a $2$ (not in LMFDB) 6.2.ac_c_b_a_ag_q $2$ (not in LMFDB) 6.2.a_a_af_e_ac_q $2$ (not in LMFDB) 6.2.a_a_f_e_c_q $2$ (not in LMFDB) 6.2.c_c_ab_a_g_q $2$ (not in LMFDB) 6.2.c_c_b_e_c_a $2$ (not in LMFDB) 6.2.e_i_n_y_bq_cm $2$ (not in LMFDB) 6.2.g_s_bf_bc_ac_abg $2$ (not in LMFDB) 6.2.i_bg_dh_gq_le_qq $2$ (not in LMFDB) 6.2.af_l_an_q_abm_cu $3$ (not in LMFDB) 6.2.ac_c_ab_ac_e_a $3$ (not in LMFDB) 6.2.ac_c_ab_e_ac_a $3$ (not in LMFDB) 6.2.b_ab_ab_e_e_a $3$ (not in LMFDB) 6.2.e_i_l_k_e_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ag_s_abf_bc_c_abg $2$ (not in LMFDB) 6.2.ae_i_an_y_abq_cm $2$ (not in LMFDB) 6.2.ac_c_ab_e_ac_a $2$ (not in LMFDB) 6.2.ac_c_b_a_ag_q $2$ (not in LMFDB) 6.2.a_a_af_e_ac_q $2$ (not in LMFDB) 6.2.a_a_f_e_c_q $2$ (not in LMFDB) 6.2.c_c_ab_a_g_q $2$ (not in LMFDB) 6.2.c_c_b_e_c_a $2$ (not in LMFDB) 6.2.e_i_n_y_bq_cm $2$ (not in LMFDB) 6.2.g_s_bf_bc_ac_abg $2$ (not in LMFDB) 6.2.i_bg_dh_gq_le_qq $2$ (not in LMFDB) 6.2.af_l_an_q_abm_cu $3$ (not in LMFDB) 6.2.ac_c_ab_ac_e_a $3$ (not in LMFDB) 6.2.ac_c_ab_e_ac_a $3$ (not in LMFDB) 6.2.b_ab_ab_e_e_a $3$ (not in LMFDB) 6.2.e_i_l_k_e_a $3$ (not in LMFDB) 6.2.ag_s_abp_de_afk_ia $6$ (not in LMFDB) 6.2.ae_i_al_k_ae_a $6$ (not in LMFDB) 6.2.ad_d_af_q_au_q $6$ (not in LMFDB) 6.2.ad_d_f_ai_ak_bo $6$ (not in LMFDB) 6.2.ab_ab_ab_m_ag_ai $6$ (not in LMFDB) 6.2.ab_ab_b_e_ae_a $6$ (not in LMFDB) 6.2.a_a_af_ac_e_q $6$ (not in LMFDB) 6.2.a_a_f_ac_ae_q $6$ (not in LMFDB) 6.2.b_ab_b_m_g_ai $6$ (not in LMFDB) 6.2.c_c_b_ac_ae_a $6$ (not in LMFDB) 6.2.d_d_af_ai_k_bo $6$ (not in LMFDB) 6.2.d_d_f_q_u_q $6$ (not in LMFDB) 6.2.f_l_n_q_bm_cu $6$ (not in LMFDB) 6.2.g_s_bp_de_fk_ia $6$ (not in LMFDB) 6.2.ag_u_abx_du_agk_jo $8$ (not in LMFDB) 6.2.ae_e_d_ai_k_aq $8$ (not in LMFDB) 6.2.ae_k_ap_o_ac_ai $8$ (not in LMFDB) 6.2.ae_m_abd_ce_adq_fo $8$ (not in LMFDB) 6.2.ac_ac_j_ai_ak_bg $8$ (not in LMFDB) 6.2.ac_e_aj_o_aw_bo $8$ (not in LMFDB) 6.2.ac_g_ah_i_ac_a $8$ (not in LMFDB) 6.2.a_c_ab_c_c_i $8$ (not in LMFDB) 6.2.a_c_b_c_ac_i $8$ (not in LMFDB) 6.2.c_ac_aj_ai_k_bg $8$ (not in LMFDB) 6.2.c_e_j_o_w_bo $8$ (not in LMFDB) 6.2.c_g_h_i_c_a $8$ (not in LMFDB) 6.2.e_e_ad_ai_ak_aq $8$ (not in LMFDB) 6.2.e_k_p_o_c_ai $8$ (not in LMFDB) 6.2.e_m_bd_ce_dq_fo $8$ (not in LMFDB) 6.2.g_u_bx_du_gk_jo $8$ (not in LMFDB) 6.2.af_n_ax_bo_acw_eq $12$ (not in LMFDB) 6.2.ad_f_al_y_abk_bw $12$ (not in LMFDB) 6.2.ad_f_ab_a_ag_y $12$ (not in LMFDB) 6.2.ab_b_ad_m_ak_i $12$ (not in LMFDB) 6.2.ab_b_ab_e_ae_a $12$ (not in LMFDB) 6.2.b_b_b_e_e_a $12$ (not in LMFDB) 6.2.b_b_d_m_k_i $12$ (not in LMFDB) 6.2.d_f_b_a_g_y $12$ (not in LMFDB) 6.2.d_f_l_y_bk_bw $12$ (not in LMFDB) 6.2.f_n_x_bo_cw_eq $12$ (not in LMFDB) 6.2.ae_g_af_i_aq_y $24$ (not in LMFDB) 6.2.ae_k_av_bo_acq_ea $24$ (not in LMFDB) 6.2.ad_f_ah_o_aw_bg $24$ (not in LMFDB) 6.2.ad_h_an_ba_abq_cm $24$ (not in LMFDB) 6.2.ac_a_f_ae_ai_y $24$ (not in LMFDB) 6.2.ac_e_ad_e_ae_i $24$ (not in LMFDB) 6.2.ab_af_d_q_ac_abo $24$ (not in LMFDB) 6.2.ab_ad_b_i_c_ay $24$ (not in LMFDB) 6.2.ab_ad_b_o_ae_ay $24$ (not in LMFDB) 6.2.ab_ab_ab_k_ae_ai $24$ (not in LMFDB) 6.2.ab_b_ad_k_ai_i $24$ (not in LMFDB) 6.2.ab_b_d_c_ac_q $24$ (not in LMFDB) 6.2.ab_d_af_i_ak_y $24$ (not in LMFDB) 6.2.ab_d_af_o_aq_y $24$ (not in LMFDB) 6.2.ab_d_b_g_c_q $24$ (not in LMFDB) 6.2.ab_f_ah_q_aw_bo $24$ (not in LMFDB) 6.2.b_af_ad_q_c_abo $24$ (not in LMFDB) 6.2.b_ad_ab_i_ac_ay $24$ (not in LMFDB) 6.2.b_ad_ab_o_e_ay $24$ (not in LMFDB) 6.2.b_ab_b_k_e_ai $24$ (not in LMFDB) 6.2.b_b_ad_c_c_q $24$ (not in LMFDB) 6.2.b_b_d_k_i_i $24$ (not in LMFDB) 6.2.b_d_ab_g_ac_q $24$ (not in LMFDB) 6.2.b_d_f_i_k_y $24$ (not in LMFDB) 6.2.b_d_f_o_q_y $24$ (not in LMFDB) 6.2.b_f_h_q_w_bo $24$ (not in LMFDB) 6.2.c_a_af_ae_i_y $24$ (not in LMFDB) 6.2.c_e_d_e_e_i $24$ (not in LMFDB) 6.2.d_f_h_o_w_bg $24$ (not in LMFDB) 6.2.d_h_n_ba_bq_cm $24$ (not in LMFDB) 6.2.e_g_f_i_q_y $24$ (not in LMFDB) 6.2.e_k_v_bo_cq_ea $24$ (not in LMFDB)