# Properties

 Label 5.2.ai_bi_adt_hw_amq Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - x + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.384973271919$, $\pm0.456881978294$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 3800 179816 1710000 35539702 1366601600 39982829098 1019710620000 31323168057272 1132739151995000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 9 22 25 35 78 149 241 454 1029

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 1.2.ab $\times$ 2.2.ad_f 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}($$$\Q(\sqrt{-1})$$$)$ 1.2.ab : $$\Q(\sqrt{-7})$$. 2.2.ad_f : $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.bv $\times$ 1.4096.ey 2 . The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.bv : $$\Q(\sqrt{-7})$$. 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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 $\times$ 1.4.d $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.4.d : $$\Q(\sqrt{-7})$$. 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 $\times$ 1.8.f $\times$ 2.8.a_l. The endomorphism algebra for each factor is: 1.8.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.8.f : $$\Q(\sqrt{-7})$$. 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.ab $\times$ 1.16.i 2 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 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 : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.a 2 $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: 1.64.aj : $$\Q(\sqrt{-7})$$. 1.64.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$

## 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_u_abv_dk_afg $2$ (not in LMFDB) 5.2.ae_k_ar_y_abg $2$ (not in LMFDB) 5.2.ac_e_ah_m_aq $2$ (not in LMFDB) 5.2.ac_e_ab_a_i $2$ (not in LMFDB) 5.2.a_c_ab_e_ai $2$ (not in LMFDB) 5.2.a_c_b_e_i $2$ (not in LMFDB) 5.2.c_e_b_a_ai $2$ (not in LMFDB) 5.2.c_e_h_m_q $2$ (not in LMFDB) 5.2.e_k_r_y_bg $2$ (not in LMFDB) 5.2.g_u_bv_dk_fg $2$ (not in LMFDB) 5.2.i_bi_dt_hw_mq $2$ (not in LMFDB) 5.2.af_n_at_s_aq $3$ (not in LMFDB) 5.2.ac_e_ab_ag_o $3$ (not in LMFDB) 5.2.ac_e_ab_a_i $3$ (not in LMFDB) 5.2.b_b_f_g_c $3$ (not in LMFDB) 5.2.e_k_x_bq_ck $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ag_u_abv_dk_afg $2$ (not in LMFDB) 5.2.ae_k_ar_y_abg $2$ (not in LMFDB) 5.2.ac_e_ah_m_aq $2$ (not in LMFDB) 5.2.ac_e_ab_a_i $2$ (not in LMFDB) 5.2.a_c_ab_e_ai $2$ (not in LMFDB) 5.2.a_c_b_e_i $2$ (not in LMFDB) 5.2.c_e_b_a_ai $2$ (not in LMFDB) 5.2.c_e_h_m_q $2$ (not in LMFDB) 5.2.e_k_r_y_bg $2$ (not in LMFDB) 5.2.g_u_bv_dk_fg $2$ (not in LMFDB) 5.2.i_bi_dt_hw_mq $2$ (not in LMFDB) 5.2.af_n_at_s_aq $3$ (not in LMFDB) 5.2.ac_e_ab_ag_o $3$ (not in LMFDB) 5.2.ac_e_ab_a_i $3$ (not in LMFDB) 5.2.b_b_f_g_c $3$ (not in LMFDB) 5.2.e_k_x_bq_ck $3$ (not in LMFDB) 5.2.ag_u_abx_dq_afq $6$ (not in LMFDB) 5.2.ae_k_ax_bq_ack $6$ (not in LMFDB) 5.2.ad_f_ah_k_ao $6$ (not in LMFDB) 5.2.ad_f_af_k_aq $6$ (not in LMFDB) 5.2.ab_b_af_g_ac $6$ (not in LMFDB) 5.2.ab_b_b_g_ai $6$ (not in LMFDB) 5.2.a_c_ab_ac_ac $6$ (not in LMFDB) 5.2.a_c_b_ac_c $6$ (not in LMFDB) 5.2.b_b_ab_g_i $6$ (not in LMFDB) 5.2.c_e_b_ag_ao $6$ (not in LMFDB) 5.2.d_f_f_k_q $6$ (not in LMFDB) 5.2.d_f_h_k_o $6$ (not in LMFDB) 5.2.f_n_t_s_q $6$ (not in LMFDB) 5.2.g_u_bx_dq_fq $6$ (not in LMFDB) 5.2.ag_w_acf_ek_agy $8$ (not in LMFDB) 5.2.ae_g_ab_aq_bk $8$ (not in LMFDB) 5.2.ae_m_abb_by_acy $8$ (not in LMFDB) 5.2.ae_o_abh_cm_adw $8$ (not in LMFDB) 5.2.ac_a_b_ae_m $8$ (not in LMFDB) 5.2.ac_g_aj_o_au $8$ (not in LMFDB) 5.2.ac_i_ap_bc_abs $8$ (not in LMFDB) 5.2.a_e_ad_g_am $8$ (not in LMFDB) 5.2.a_e_d_g_m $8$ (not in LMFDB) 5.2.c_a_ab_ae_am $8$ (not in LMFDB) 5.2.c_g_j_o_u $8$ (not in LMFDB) 5.2.c_i_p_bc_bs $8$ (not in LMFDB) 5.2.e_g_b_aq_abk $8$ (not in LMFDB) 5.2.e_m_bb_by_cy $8$ (not in LMFDB) 5.2.e_o_bh_cm_dw $8$ (not in LMFDB) 5.2.g_w_cf_ek_gy $8$ (not in LMFDB) 5.2.af_p_abd_bu_acm $12$ (not in LMFDB) 5.2.ad_h_an_w_abi $12$ (not in LMFDB) 5.2.ad_h_al_w_abg $12$ (not in LMFDB) 5.2.ab_d_ah_k_ao $12$ (not in LMFDB) 5.2.ab_d_ab_k_ai $12$ (not in LMFDB) 5.2.a_c_b_ac_c $12$ (not in LMFDB) 5.2.b_d_b_k_i $12$ (not in LMFDB) 5.2.b_d_h_k_o $12$ (not in LMFDB) 5.2.d_h_l_w_bg $12$ (not in LMFDB) 5.2.d_h_n_w_bi $12$ (not in LMFDB) 5.2.f_p_bd_bu_cm $12$ (not in LMFDB) 5.2.ae_i_aj_e_c $24$ (not in LMFDB) 5.2.ae_m_az_bs_aco $24$ (not in LMFDB) 5.2.ad_h_aj_m_am $24$ (not in LMFDB) 5.2.ad_j_ap_bc_abk $24$ (not in LMFDB) 5.2.ac_c_ad_e_ac $24$ (not in LMFDB) 5.2.ac_g_al_u_abe $24$ (not in LMFDB) 5.2.ab_ad_f_c_am $24$ (not in LMFDB) 5.2.ab_ab_d_ac_ae $24$ (not in LMFDB) 5.2.ab_ab_d_e_ak $24$ (not in LMFDB) 5.2.ab_b_b_e_ag $24$ (not in LMFDB) 5.2.ab_d_ad_i_ae $24$ (not in LMFDB) 5.2.ab_d_ab_i_ag $24$ (not in LMFDB) 5.2.ab_f_af_q_am $24$ (not in LMFDB) 5.2.ab_f_ad_k_ae $24$ (not in LMFDB) 5.2.ab_f_ad_q_ak $24$ (not in LMFDB) 5.2.ab_h_af_w_am $24$ (not in LMFDB) 5.2.b_ad_af_c_m $24$ (not in LMFDB) 5.2.b_ab_ad_ac_e $24$ (not in LMFDB) 5.2.b_ab_ad_e_k $24$ (not in LMFDB) 5.2.b_b_ab_e_g $24$ (not in LMFDB) 5.2.b_d_b_i_g $24$ (not in LMFDB) 5.2.b_d_d_i_e $24$ (not in LMFDB) 5.2.b_f_d_k_e $24$ (not in LMFDB) 5.2.b_f_d_q_k $24$ (not in LMFDB) 5.2.b_f_f_q_m $24$ (not in LMFDB) 5.2.b_h_f_w_m $24$ (not in LMFDB) 5.2.c_c_d_e_c $24$ (not in LMFDB) 5.2.c_g_l_u_be $24$ (not in LMFDB) 5.2.d_h_j_m_m $24$ (not in LMFDB) 5.2.d_j_p_bc_bk $24$ (not in LMFDB) 5.2.e_i_j_e_ac $24$ (not in LMFDB) 5.2.e_m_z_bs_co $24$ (not in LMFDB)