# Properties

 Label 5.2.ah_bc_acy_ga_ajo Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

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

This isogeny class is not simple.

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

## 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 9000 276822 2250000 50036646 1245699000 26430987246 738112500000 29572112791494 1135206406125000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 12 26 32 46 72 94 160 422 1032

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 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 4 $\times$ 1.256.bf. The endomorphism algebra for each factor is: 1.256.abg 4 : $\mathrm{M}_{4}(B)$, 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 3 $\times$ 1.4.d $\times$ 1.4.e. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 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 3 . 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 3 : $\mathrm{M}_{3}(B)$, 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 5.2.af_q_abk_cq_aea $2$ (not in LMFDB) 5.2.ad_i_am_u_ay $2$ (not in LMFDB) 5.2.ab_e_ae_m_ai $2$ (not in LMFDB) 5.2.b_e_e_m_i $2$ (not in LMFDB) 5.2.d_i_m_u_y $2$ (not in LMFDB) 5.2.f_q_bk_cq_ea $2$ (not in LMFDB) 5.2.h_bc_cy_ga_jo $2$ (not in LMFDB) 5.2.ab_e_c_a_q $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.af_q_abk_cq_aea $2$ (not in LMFDB) 5.2.ad_i_am_u_ay $2$ (not in LMFDB) 5.2.ab_e_ae_m_ai $2$ (not in LMFDB) 5.2.b_e_e_m_i $2$ (not in LMFDB) 5.2.d_i_m_u_y $2$ (not in LMFDB) 5.2.f_q_bk_cq_ea $2$ (not in LMFDB) 5.2.h_bc_cy_ga_jo $2$ (not in LMFDB) 5.2.ab_e_c_a_q $3$ (not in LMFDB) 5.2.af_q_abm_cu_aei $6$ (not in LMFDB) 5.2.ad_i_as_bg_abw $6$ (not in LMFDB) 5.2.ab_e_ag_i_aq $6$ (not in LMFDB) 5.2.b_e_ac_a_aq $6$ (not in LMFDB) 5.2.b_e_g_i_q $6$ (not in LMFDB) 5.2.d_i_s_bg_bw $6$ (not in LMFDB) 5.2.f_q_bm_cu_ei $6$ (not in LMFDB) 5.2.aj_bq_aey_ku_aro $8$ (not in LMFDB) 5.2.ah_ba_acm_eq_ahc $8$ (not in LMFDB) 5.2.af_k_ae_ay_ce $8$ (not in LMFDB) 5.2.af_o_ay_bg_abo $8$ (not in LMFDB) 5.2.af_s_abs_dk_afg $8$ (not in LMFDB) 5.2.ad_c_e_ai_i $8$ (not in LMFDB) 5.2.ad_e_a_am_y $8$ (not in LMFDB) 5.2.ad_g_ai_q_ay $8$ (not in LMFDB) 5.2.ad_k_au_bo_ace $8$ (not in LMFDB) 5.2.ad_m_ay_ca_acu $8$ (not in LMFDB) 5.2.ab_ag_i_i_ay $8$ (not in LMFDB) 5.2.ab_ac_e_a_ai $8$ (not in LMFDB) 5.2.ab_a_a_ae_i $8$ (not in LMFDB) 5.2.ab_c_a_ai_i $8$ (not in LMFDB) 5.2.ab_c_a_i_ai $8$ (not in LMFDB) 5.2.ab_g_ae_q_ai $8$ (not in LMFDB) 5.2.ab_i_ai_bc_ay $8$ (not in LMFDB) 5.2.ab_k_ai_bo_ay $8$ (not in LMFDB) 5.2.b_ag_ai_i_y $8$ (not in LMFDB) 5.2.b_ac_ae_a_i $8$ (not in LMFDB) 5.2.b_a_a_ae_ai $8$ (not in LMFDB) 5.2.b_c_a_ai_ai $8$ (not in LMFDB) 5.2.b_c_a_i_i $8$ (not in LMFDB) 5.2.b_g_e_q_i $8$ (not in LMFDB) 5.2.b_i_i_bc_y $8$ (not in LMFDB) 5.2.b_k_i_bo_y $8$ (not in LMFDB) 5.2.d_c_ae_ai_ai $8$ (not in LMFDB) 5.2.d_e_a_am_ay $8$ (not in LMFDB) 5.2.d_g_i_q_y $8$ (not in LMFDB) 5.2.d_k_u_bo_ce $8$ (not in LMFDB) 5.2.d_m_y_ca_cu $8$ (not in LMFDB) 5.2.f_k_e_ay_ace $8$ (not in LMFDB) 5.2.f_o_y_bg_bo $8$ (not in LMFDB) 5.2.f_s_bs_dk_fg $8$ (not in LMFDB) 5.2.h_ba_cm_eq_hc $8$ (not in LMFDB) 5.2.j_bq_ey_ku_ro $8$ (not in LMFDB) 5.2.ab_c_a_a_a $16$ (not in LMFDB) 5.2.b_c_a_a_a $16$ (not in LMFDB) 5.2.ah_ba_aco_ey_ahs $24$ (not in LMFDB) 5.2.af_m_ao_e_i $24$ (not in LMFDB) 5.2.af_o_abg_ci_ado $24$ (not in LMFDB) 5.2.af_o_abe_ce_adk $24$ (not in LMFDB) 5.2.af_q_abi_ci_adk $24$ (not in LMFDB) 5.2.ad_c_c_ai_q $24$ (not in LMFDB) 5.2.ad_e_ae_e_ae $24$ (not in LMFDB) 5.2.ad_e_ac_e_ai $24$ (not in LMFDB) 5.2.ad_g_aq_bc_abk $24$ (not in LMFDB) 5.2.ad_g_ak_q_ay $24$ (not in LMFDB) 5.2.ad_g_ag_e_a $24$ (not in LMFDB) 5.2.ad_g_ac_ai_y $24$ (not in LMFDB) 5.2.ad_i_aq_bc_abs $24$ (not in LMFDB) 5.2.ad_i_ao_bc_abo $24$ (not in LMFDB) 5.2.ad_k_aw_bo_acm $24$ (not in LMFDB) 5.2.ad_k_as_bk_abw $24$ (not in LMFDB) 5.2.ab_ae_g_e_aq $24$ (not in LMFDB) 5.2.ab_ac_ac_a_q $24$ (not in LMFDB) 5.2.ab_ac_e_e_am $24$ (not in LMFDB) 5.2.ab_a_ae_e_e $24$ (not in LMFDB) 5.2.ab_a_c_ae_a $24$ (not in LMFDB) 5.2.ab_a_c_e_ai $24$ (not in LMFDB) 5.2.ab_c_ag_i_ai $24$ (not in LMFDB) 5.2.ab_c_ac_e_a $24$ (not in LMFDB) 5.2.ab_c_a_ae_e $24$ (not in LMFDB) 5.2.ab_c_a_e_ae $24$ (not in LMFDB) 5.2.ab_c_c_a_i $24$ (not in LMFDB) 5.2.ab_e_ai_m_au $24$ (not in LMFDB) 5.2.ab_e_ac_e_a $24$ (not in LMFDB) 5.2.ab_e_ac_m_ai $24$ (not in LMFDB) 5.2.ab_g_ak_q_abg $24$ (not in LMFDB) 5.2.ab_g_ag_u_aq $24$ (not in LMFDB) 5.2.ab_g_ae_u_am $24$ (not in LMFDB) 5.2.ab_i_ag_bc_aq $24$ (not in LMFDB) 5.2.b_ae_ag_e_q $24$ (not in LMFDB) 5.2.b_ac_ae_e_m $24$ (not in LMFDB) 5.2.b_ac_c_a_aq $24$ (not in LMFDB) 5.2.b_a_ac_ae_a $24$ (not in LMFDB) 5.2.b_a_ac_e_i $24$ (not in LMFDB) 5.2.b_a_e_e_ae $24$ (not in LMFDB) 5.2.b_c_ac_a_ai $24$ (not in LMFDB) 5.2.b_c_a_ae_ae $24$ (not in LMFDB) 5.2.b_c_a_e_e $24$ (not in LMFDB) 5.2.b_c_c_e_a $24$ (not in LMFDB) 5.2.b_c_g_i_i $24$ (not in LMFDB) 5.2.b_e_c_e_a $24$ (not in LMFDB) 5.2.b_e_c_m_i $24$ (not in LMFDB) 5.2.b_e_i_m_u $24$ (not in LMFDB) 5.2.b_g_e_u_m $24$ (not in LMFDB) 5.2.b_g_g_u_q $24$ (not in LMFDB) 5.2.b_g_k_q_bg $24$ (not in LMFDB) 5.2.b_i_g_bc_q $24$ (not in LMFDB) 5.2.d_c_ac_ai_aq $24$ (not in LMFDB) 5.2.d_e_c_e_i $24$ (not in LMFDB) 5.2.d_e_e_e_e $24$ (not in LMFDB) 5.2.d_g_c_ai_ay $24$ (not in LMFDB) 5.2.d_g_g_e_a $24$ (not in LMFDB) 5.2.d_g_k_q_y $24$ (not in LMFDB) 5.2.d_g_q_bc_bk $24$ (not in LMFDB) 5.2.d_i_o_bc_bo $24$ (not in LMFDB) 5.2.d_i_q_bc_bs $24$ (not in LMFDB) 5.2.d_k_s_bk_bw $24$ (not in LMFDB) 5.2.d_k_w_bo_cm $24$ (not in LMFDB) 5.2.f_m_o_e_ai $24$ (not in LMFDB) 5.2.f_o_be_ce_dk $24$ (not in LMFDB) 5.2.f_o_bg_ci_do $24$ (not in LMFDB) 5.2.f_q_bi_ci_dk $24$ (not in LMFDB) 5.2.h_ba_co_ey_hs $24$ (not in LMFDB) 5.2.ad_g_ag_a_e $40$ (not in LMFDB) 5.2.ab_a_c_a_ae $40$ (not in LMFDB) 5.2.ab_c_ac_a_ae $40$ (not in LMFDB) 5.2.ab_e_ac_i_ae $40$ (not in LMFDB) 5.2.b_a_ac_a_e $40$ (not in LMFDB) 5.2.b_c_c_a_e $40$ (not in LMFDB) 5.2.b_e_c_i_e $40$ (not in LMFDB) 5.2.d_g_g_a_ae $40$ (not in LMFDB)