# Properties

 Label 4.5.ao_dp_aoo_bnk Base Field $\F_{5}$ Dimension $4$ Ordinary Yes $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 5 x^{2} )^{2}$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.265942140215$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary. $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 36 291600 308634624 186624000000 103876170945156 60838550898278400 37298113629404912964 23271370696986624000000 14559654807532466321826816 9101424859935705708138090000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -8 16 154 752 3392 15946 78224 390432 1954162 9772576

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 1.5.ad 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.5.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.5.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$
All geometric endomorphisms are defined over $\F_{5}$.

## 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 4.5.ai_bb_abw_cy $2$ (not in LMFDB) 4.5.ag_n_g_acm $2$ (not in LMFDB) 4.5.ac_ad_ag_cm $2$ (not in LMFDB) 4.5.a_af_a_bs $2$ (not in LMFDB) 4.5.c_ad_g_cm $2$ (not in LMFDB) 4.5.g_n_ag_acm $2$ (not in LMFDB) 4.5.i_bb_bw_cy $2$ (not in LMFDB) 4.5.o_dp_oo_bnk $2$ (not in LMFDB) 4.5.af_g_v_adi $3$ (not in LMFDB) 4.5.ac_g_a_t $3$ (not in LMFDB) 4.5.h_bb_dg_ig $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ai_bb_abw_cy $2$ (not in LMFDB) 4.5.ag_n_g_acm $2$ (not in LMFDB) 4.5.ac_ad_ag_cm $2$ (not in LMFDB) 4.5.a_af_a_bs $2$ (not in LMFDB) 4.5.c_ad_g_cm $2$ (not in LMFDB) 4.5.g_n_ag_acm $2$ (not in LMFDB) 4.5.i_bb_bw_cy $2$ (not in LMFDB) 4.5.o_dp_oo_bnk $2$ (not in LMFDB) 4.5.af_g_v_adi $3$ (not in LMFDB) 4.5.ac_g_a_t $3$ (not in LMFDB) 4.5.h_bb_dg_ig $3$ (not in LMFDB) 4.5.am_cv_akw_bcy $4$ (not in LMFDB) 4.5.ak_cf_aic_vk $4$ (not in LMFDB) 4.5.ai_z_abg_y $4$ (not in LMFDB) 4.5.ai_bh_adm_ia $4$ (not in LMFDB) 4.5.ag_r_ay_bg $4$ (not in LMFDB) 4.5.ag_t_abk_cq $4$ (not in LMFDB) 4.5.ag_z_aco_gi $4$ (not in LMFDB) 4.5.ae_j_g_abg $4$ (not in LMFDB) 4.5.ae_n_aq_bk $4$ (not in LMFDB) 4.5.ae_p_ay_cm $4$ (not in LMFDB) 4.5.ac_b_ai_bw $4$ (not in LMFDB) 4.5.ac_d_am_ca $4$ (not in LMFDB) 4.5.ac_j_as_cy $4$ (not in LMFDB) 4.5.a_ah_a_ce $4$ (not in LMFDB) 4.5.a_b_ag_bg $4$ (not in LMFDB) 4.5.a_b_g_bg $4$ (not in LMFDB) 4.5.a_f_a_bs $4$ (not in LMFDB) 4.5.a_h_a_ce $4$ (not in LMFDB) 4.5.c_b_i_bw $4$ (not in LMFDB) 4.5.c_d_m_ca $4$ (not in LMFDB) 4.5.c_j_s_cy $4$ (not in LMFDB) 4.5.e_j_ag_abg $4$ (not in LMFDB) 4.5.e_n_q_bk $4$ (not in LMFDB) 4.5.e_p_y_cm $4$ (not in LMFDB) 4.5.g_r_y_bg $4$ (not in LMFDB) 4.5.g_t_bk_cq $4$ (not in LMFDB) 4.5.g_z_co_gi $4$ (not in LMFDB) 4.5.i_z_bg_y $4$ (not in LMFDB) 4.5.i_bh_dm_ia $4$ (not in LMFDB) 4.5.k_cf_ic_vk $4$ (not in LMFDB) 4.5.m_cv_kw_bcy $4$ (not in LMFDB) 4.5.al_cc_agj_pe $6$ (not in LMFDB) 4.5.ak_cc_ahk_tf $6$ (not in LMFDB) 4.5.ah_bb_adg_ig $6$ (not in LMFDB) 4.5.ae_m_ay_cj $6$ (not in LMFDB) 4.5.ad_ac_d_ba $6$ (not in LMFDB) 4.5.ab_d_m_aba $6$ (not in LMFDB) 4.5.b_d_am_aba $6$ (not in LMFDB) 4.5.c_g_a_t $6$ (not in LMFDB) 4.5.d_ac_ad_ba $6$ (not in LMFDB) 4.5.e_m_y_cj $6$ (not in LMFDB) 4.5.f_g_av_adi $6$ (not in LMFDB) 4.5.k_cc_hk_tf $6$ (not in LMFDB) 4.5.l_cc_gj_pe $6$ (not in LMFDB) 4.5.ag_l_s_ady $8$ (not in LMFDB) 4.5.ag_bb_ada_hu $8$ (not in LMFDB) 4.5.a_aj_a_cg $8$ (not in LMFDB) 4.5.a_ah_a_bq $8$ (not in LMFDB) 4.5.a_h_a_bq $8$ (not in LMFDB) 4.5.a_j_a_cg $8$ (not in LMFDB) 4.5.g_l_as_ady $8$ (not in LMFDB) 4.5.g_bb_da_hu $8$ (not in LMFDB) 4.5.aj_bo_aet_lq $12$ (not in LMFDB) 4.5.ai_be_acu_fv $12$ (not in LMFDB) 4.5.ah_be_adp_is $12$ (not in LMFDB) 4.5.af_j_abe_ec $12$ (not in LMFDB) 4.5.af_m_abn_eo $12$ (not in LMFDB) 4.5.ae_g_y_adl $12$ (not in LMFDB) 4.5.ae_k_aq_bn $12$ (not in LMFDB) 4.5.ad_e_p_acg $12$ (not in LMFDB) 4.5.ad_k_abh_cw $12$ (not in LMFDB) 4.5.ac_ac_ai_bz $12$ (not in LMFDB) 4.5.ac_a_am_bx $12$ (not in LMFDB) 4.5.ab_ad_g_ao $12$ (not in LMFDB) 4.5.ab_a_ad_ac $12$ (not in LMFDB) 4.5.ab_g_v_ao $12$ (not in LMFDB) 4.5.b_ad_ag_ao $12$ (not in LMFDB) 4.5.b_a_d_ac $12$ (not in LMFDB) 4.5.b_g_av_ao $12$ (not in LMFDB) 4.5.c_ac_i_bz $12$ (not in LMFDB) 4.5.c_a_m_bx $12$ (not in LMFDB) 4.5.d_e_ap_acg $12$ (not in LMFDB) 4.5.d_k_bh_cw $12$ (not in LMFDB) 4.5.e_g_ay_adl $12$ (not in LMFDB) 4.5.e_k_q_bn $12$ (not in LMFDB) 4.5.f_j_be_ec $12$ (not in LMFDB) 4.5.f_m_bn_eo $12$ (not in LMFDB) 4.5.h_be_dp_is $12$ (not in LMFDB) 4.5.i_be_cu_fv $12$ (not in LMFDB) 4.5.j_bo_et_lq $12$ (not in LMFDB) 4.5.ad_ae_j_s $24$ (not in LMFDB) 4.5.ad_m_abn_de $24$ (not in LMFDB) 4.5.d_ae_aj_s $24$ (not in LMFDB) 4.5.d_m_bn_de $24$ (not in LMFDB)