# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.718306605252$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 1 875 35152 4046875 51759671 861224000 26120764391 663788671875 28718559595024 1194516157296875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 2 11 42 46 47 94 130 407 1082

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 2.2.ab_ab 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}($$$\Q(\sqrt{-1})$$$)$ 2.2.ab_ab : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv 2 $\times$ 1.4096.ey 3 . The endomorphism algebra for each factor is: 1.4096.bv 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.4096.ey 3 : $\mathrm{M}_{3}(B)$, 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 3 $\times$ 2.4.ad_f. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 2.4.ad_f : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 1.8.e 3 . The endomorphism algebra for each factor is: 1.8.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.8.e 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 2.16.b_ap. The endomorphism algebra for each factor is: 1.16.i 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.b_ap : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 $\times$ 1.64.a 3 . The endomorphism algebra for each factor is: 1.64.aj 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.64.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$

## 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 5.2.af_l_ag_aw_ce $2$ (not in LMFDB) 5.2.ad_d_ac_k_ay $2$ (not in LMFDB) 5.2.ab_ab_g_c_ai $2$ (not in LMFDB) 5.2.b_ab_ag_c_i $2$ (not in LMFDB) 5.2.d_d_c_k_y $2$ (not in LMFDB) 5.2.f_l_g_aw_ace $2$ (not in LMFDB) 5.2.h_x_bu_co_dk $2$ (not in LMFDB) 5.2.ae_l_aw_bq_acm $3$ (not in LMFDB) 5.2.ab_ab_c_a_ae $3$ (not in LMFDB) 5.2.c_f_i_m_u $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.af_l_ag_aw_ce $2$ (not in LMFDB) 5.2.ad_d_ac_k_ay $2$ (not in LMFDB) 5.2.ab_ab_g_c_ai $2$ (not in LMFDB) 5.2.b_ab_ag_c_i $2$ (not in LMFDB) 5.2.d_d_c_k_y $2$ (not in LMFDB) 5.2.f_l_g_aw_ace $2$ (not in LMFDB) 5.2.h_x_bu_co_dk $2$ (not in LMFDB) 5.2.ae_l_aw_bq_acm $3$ (not in LMFDB) 5.2.ab_ab_c_a_ae $3$ (not in LMFDB) 5.2.c_f_i_m_u $3$ (not in LMFDB) 5.2.ai_bj_ady_ik_ano $6$ (not in LMFDB) 5.2.ag_v_aca_dw_aga $6$ (not in LMFDB) 5.2.ag_v_aby_dq_afo $6$ (not in LMFDB) 5.2.af_l_as_bg_aca $6$ (not in LMFDB) 5.2.ae_l_ay_bs_acq $6$ (not in LMFDB) 5.2.ae_l_as_ba_abg $6$ (not in LMFDB) 5.2.ad_d_c_ai_m $6$ (not in LMFDB) 5.2.ac_f_am_u_abc $6$ (not in LMFDB) 5.2.ac_f_ai_m_au $6$ (not in LMFDB) 5.2.ac_f_ag_o_aq $6$ (not in LMFDB) 5.2.ac_f_a_ae_u $6$ (not in LMFDB) 5.2.ab_ab_ag_i_e $6$ (not in LMFDB) 5.2.a_d_ae_e_am $6$ (not in LMFDB) 5.2.a_d_ac_k_a $6$ (not in LMFDB) 5.2.a_d_c_k_a $6$ (not in LMFDB) 5.2.a_d_e_e_m $6$ (not in LMFDB) 5.2.b_ab_ac_a_e $6$ (not in LMFDB) 5.2.b_ab_g_i_ae $6$ (not in LMFDB) 5.2.c_f_a_ae_au $6$ (not in LMFDB) 5.2.c_f_g_o_q $6$ (not in LMFDB) 5.2.c_f_m_u_bc $6$ (not in LMFDB) 5.2.d_d_ac_ai_am $6$ (not in LMFDB) 5.2.e_l_s_ba_bg $6$ (not in LMFDB) 5.2.e_l_w_bq_cm $6$ (not in LMFDB) 5.2.e_l_y_bs_cq $6$ (not in LMFDB) 5.2.f_l_s_bg_ca $6$ (not in LMFDB) 5.2.g_v_by_dq_fo $6$ (not in LMFDB) 5.2.g_v_ca_dw_ga $6$ (not in LMFDB) 5.2.i_bj_dy_ik_no $6$ (not in LMFDB) 5.2.af_n_ay_bm_ace $8$ (not in LMFDB) 5.2.ad_ab_k_ac_aq $8$ (not in LMFDB) 5.2.ad_f_a_ak_y $8$ (not in LMFDB) 5.2.ad_h_ao_w_abg $8$ (not in LMFDB) 5.2.ab_af_k_g_abg $8$ (not in LMFDB) 5.2.ab_ad_a_c_i $8$ (not in LMFDB) 5.2.ab_b_ae_g_ai $8$ (not in LMFDB) 5.2.ab_d_c_ac_q $8$ (not in LMFDB) 5.2.ab_f_ai_k_ay $8$ (not in LMFDB) 5.2.b_af_ak_g_bg $8$ (not in LMFDB) 5.2.b_ad_a_c_ai $8$ (not in LMFDB) 5.2.b_b_e_g_i $8$ (not in LMFDB) 5.2.b_d_ac_ac_aq $8$ (not in LMFDB) 5.2.b_f_i_k_y $8$ (not in LMFDB) 5.2.d_ab_ak_ac_q $8$ (not in LMFDB) 5.2.d_f_a_ak_ay $8$ (not in LMFDB) 5.2.d_h_o_w_bg $8$ (not in LMFDB) 5.2.f_n_y_bm_ce $8$ (not in LMFDB) 5.2.ag_p_ao_ao_bw $12$ (not in LMFDB) 5.2.ae_f_a_ae_e $12$ (not in LMFDB) 5.2.ac_ab_g_c_aq $12$ (not in LMFDB) 5.2.a_ad_ae_e_m $12$ (not in LMFDB) 5.2.a_ad_e_e_am $12$ (not in LMFDB) 5.2.c_ab_ag_c_q $12$ (not in LMFDB) 5.2.e_f_a_ae_ae $12$ (not in LMFDB) 5.2.g_p_o_ao_abw $12$ (not in LMFDB) 5.2.ag_x_aci_es_ahk $24$ (not in LMFDB) 5.2.ae_h_ae_ag_q $24$ (not in LMFDB) 5.2.ae_h_ac_as_bo $24$ (not in LMFDB) 5.2.ae_j_ak_e_e $24$ (not in LMFDB) 5.2.ae_n_abe_ce_adk $24$ (not in LMFDB) 5.2.ae_n_abc_cc_adc $24$ (not in LMFDB) 5.2.ae_n_aba_bw_acq $24$ (not in LMFDB) 5.2.ae_p_abi_cs_aea $24$ (not in LMFDB) 5.2.ad_b_e_e_au $24$ (not in LMFDB) 5.2.ad_f_am_u_ay $24$ (not in LMFDB) 5.2.ad_f_ai_q_abc $24$ (not in LMFDB) 5.2.ac_af_o_g_abo $24$ (not in LMFDB) 5.2.ac_ad_k_e_abc $24$ (not in LMFDB) 5.2.ac_b_ac_a_i $24$ (not in LMFDB) 5.2.ac_b_c_ag_i $24$ (not in LMFDB) 5.2.ac_b_c_a_ae $24$ (not in LMFDB) 5.2.ac_d_ac_ac_i $24$ (not in LMFDB) 5.2.ac_d_ac_e_ae $24$ (not in LMFDB) 5.2.ac_d_a_ak_q $24$ (not in LMFDB) 5.2.ac_f_ae_e_a $24$ (not in LMFDB) 5.2.ac_h_ao_y_abo $24$ (not in LMFDB) 5.2.ac_h_am_ba_abg $24$ (not in LMFDB) 5.2.ac_h_ak_y_abc $24$ (not in LMFDB) 5.2.ac_h_ai_s_aq $24$ (not in LMFDB) 5.2.ac_j_ao_bi_abo $24$ (not in LMFDB) 5.2.ac_j_am_bg_abg $24$ (not in LMFDB) 5.2.ac_l_aq_bu_abw $24$ (not in LMFDB) 5.2.ab_ad_i_e_au $24$ (not in LMFDB) 5.2.ab_ab_ac_e_a $24$ (not in LMFDB) 5.2.ab_b_a_ae_i $24$ (not in LMFDB) 5.2.ab_b_e_a_e $24$ (not in LMFDB) 5.2.ab_d_ag_i_aq $24$ (not in LMFDB) 5.2.a_af_a_g_a $24$ (not in LMFDB) 5.2.a_ad_a_e_a $24$ (not in LMFDB) 5.2.a_ab_ac_ac_i $24$ (not in LMFDB) 5.2.a_ab_a_c_a $24$ (not in LMFDB) 5.2.a_ab_c_ac_ai $24$ (not in LMFDB) 5.2.a_b_ac_e_e $24$ (not in LMFDB) 5.2.a_b_a_ag_a $24$ (not in LMFDB) 5.2.a_b_a_a_a $24$ (not in LMFDB) 5.2.a_b_c_e_ae $24$ (not in LMFDB) 5.2.a_d_a_ac_a $24$ (not in LMFDB) 5.2.a_d_a_e_a $24$ (not in LMFDB) 5.2.a_f_ag_i_ay $24$ (not in LMFDB) 5.2.a_f_ac_q_ae $24$ (not in LMFDB) 5.2.a_f_a_o_a $24$ (not in LMFDB) 5.2.a_f_c_q_e $24$ (not in LMFDB) 5.2.a_f_g_i_y $24$ (not in LMFDB) 5.2.a_h_ac_w_ai $24$ (not in LMFDB) 5.2.a_h_a_y_a $24$ (not in LMFDB) 5.2.a_h_c_w_i $24$ (not in LMFDB) 5.2.a_j_a_bi_a $24$ (not in LMFDB) 5.2.b_ad_ai_e_u $24$ (not in LMFDB) 5.2.b_ab_c_e_a $24$ (not in LMFDB) 5.2.b_b_ae_a_ae $24$ (not in LMFDB) 5.2.b_b_a_ae_ai $24$ (not in LMFDB) 5.2.b_d_g_i_q $24$ (not in LMFDB) 5.2.c_af_ao_g_bo $24$ (not in LMFDB) 5.2.c_ad_ak_e_bc $24$ (not in LMFDB) 5.2.c_b_ac_ag_ai $24$ (not in LMFDB) 5.2.c_b_ac_a_e $24$ (not in LMFDB) 5.2.c_b_c_a_ai $24$ (not in LMFDB) 5.2.c_d_a_ak_aq $24$ (not in LMFDB) 5.2.c_d_c_ac_ai $24$ (not in LMFDB) 5.2.c_d_c_e_e $24$ (not in LMFDB) 5.2.c_f_e_e_a $24$ (not in LMFDB) 5.2.c_h_i_s_q $24$ (not in LMFDB) 5.2.c_h_k_y_bc $24$ (not in LMFDB) 5.2.c_h_m_ba_bg $24$ (not in LMFDB) 5.2.c_h_o_y_bo $24$ (not in LMFDB) 5.2.c_j_m_bg_bg $24$ (not in LMFDB) 5.2.c_j_o_bi_bo $24$ (not in LMFDB) 5.2.c_l_q_bu_bw $24$ (not in LMFDB) 5.2.d_b_ae_e_u $24$ (not in LMFDB) 5.2.d_f_i_q_bc $24$ (not in LMFDB) 5.2.d_f_m_u_y $24$ (not in LMFDB) 5.2.e_h_c_as_abo $24$ (not in LMFDB) 5.2.e_h_e_ag_aq $24$ (not in LMFDB) 5.2.e_j_k_e_ae $24$ (not in LMFDB) 5.2.e_n_ba_bw_cq $24$ (not in LMFDB) 5.2.e_n_bc_cc_dc $24$ (not in LMFDB) 5.2.e_n_be_ce_dk $24$ (not in LMFDB) 5.2.e_p_bi_cs_ea $24$ (not in LMFDB) 5.2.g_x_ci_es_hk $24$ (not in LMFDB)