# Properties

 Label 4.5.an_de_aml_bhi 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: $( 1 - 3 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.352416382350$ 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 $A(\F_{q^r})$ 48 345600 317207808 176947200000 99339951101808 60212639880806400 37513870763864295408 23403120802607923200000 14584385684232990885973248 9098187204184464865659840000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 21 158 717 3253 15786 78673 392637 1957478 9769101

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 1.5.ad $\times$ 1.5.ac 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 : $$\Q(\sqrt{-11})$$. 1.5.ac : $$\Q(\sqrt{-1})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{4}}$ is 1.625.o 3 $\times$ 1.625.bx. The endomorphism algebra for each factor is: 1.625.o 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.625.bx : $$\Q(\sqrt{-11})$$.
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{5^{2}}$  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag 2 $\times$ 1.25.b $\times$ 1.25.g. The endomorphism algebra for each factor is: 1.25.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.25.b : $$\Q(\sqrt{-11})$$. 1.25.g : $$\Q(\sqrt{-1})$$.

## 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 4.5.aj_bm_adz_ja $2$ (not in LMFDB) 4.5.ah_w_abp_cw $2$ (not in LMFDB) 4.5.af_k_f_abu $2$ (not in LMFDB) 4.5.ad_c_an_co $2$ (not in LMFDB) 4.5.ab_ac_b_ba $2$ (not in LMFDB) 4.5.b_ac_ab_ba $2$ (not in LMFDB) 4.5.d_c_n_co $2$ (not in LMFDB) 4.5.f_k_af_abu $2$ (not in LMFDB) 4.5.h_w_bp_cw $2$ (not in LMFDB) 4.5.j_bm_dz_ja $2$ (not in LMFDB) 4.5.n_de_ml_bhi $2$ (not in LMFDB) 4.5.ab_h_e_ba $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.aj_bm_adz_ja $2$ (not in LMFDB) 4.5.ah_w_abp_cw $2$ (not in LMFDB) 4.5.af_k_f_abu $2$ (not in LMFDB) 4.5.ad_c_an_co $2$ (not in LMFDB) 4.5.ab_ac_b_ba $2$ (not in LMFDB) 4.5.b_ac_ab_ba $2$ (not in LMFDB) 4.5.d_c_n_co $2$ (not in LMFDB) 4.5.f_k_af_abu $2$ (not in LMFDB) 4.5.h_w_bp_cw $2$ (not in LMFDB) 4.5.j_bm_dz_ja $2$ (not in LMFDB) 4.5.n_de_ml_bhi $2$ (not in LMFDB) 4.5.ab_h_e_ba $3$ (not in LMFDB) 4.5.ap_ea_aqr_btm $4$ (not in LMFDB) 4.5.al_cm_ajh_yo $4$ (not in LMFDB) 4.5.aj_bg_acd_da $4$ (not in LMFDB) 4.5.aj_by_agx_sg $4$ (not in LMFDB) 4.5.ah_q_h_ade $4$ (not in LMFDB) 4.5.ah_bc_acz_ha $4$ (not in LMFDB) 4.5.af_q_abf_ck $4$ (not in LMFDB) 4.5.af_w_acd_fq $4$ (not in LMFDB) 4.5.ad_i_az_da $4$ (not in LMFDB) 4.5.ad_i_h_as $4$ (not in LMFDB) 4.5.ad_o_ar_co $4$ (not in LMFDB) 4.5.ab_ai_b_ck $4$ (not in LMFDB) 4.5.ab_e_al_bm $4$ (not in LMFDB) 4.5.ab_k_al_cw $4$ (not in LMFDB) 4.5.b_ai_ab_ck $4$ (not in LMFDB) 4.5.b_e_l_bm $4$ (not in LMFDB) 4.5.b_k_l_cw $4$ (not in LMFDB) 4.5.d_i_ah_as $4$ (not in LMFDB) 4.5.d_i_z_da $4$ (not in LMFDB) 4.5.d_o_r_co $4$ (not in LMFDB) 4.5.f_q_bf_ck $4$ (not in LMFDB) 4.5.f_w_cd_fq $4$ (not in LMFDB) 4.5.h_q_ah_ade $4$ (not in LMFDB) 4.5.h_bc_cz_ha $4$ (not in LMFDB) 4.5.j_bg_cd_da $4$ (not in LMFDB) 4.5.j_by_gx_sg $4$ (not in LMFDB) 4.5.l_cm_jh_yo $4$ (not in LMFDB) 4.5.p_ea_qr_btm $4$ (not in LMFDB) 4.5.aj_bv_agi_qk $6$ (not in LMFDB) 4.5.af_t_aca_fe $6$ (not in LMFDB) 4.5.ad_l_au_cc $6$ (not in LMFDB) 4.5.b_h_ae_ba $6$ (not in LMFDB) 4.5.d_l_u_cc $6$ (not in LMFDB) 4.5.f_t_ca_fe $6$ (not in LMFDB) 4.5.j_bv_gi_qk $6$ (not in LMFDB) 4.5.ah_o_v_aew $8$ (not in LMFDB) 4.5.ah_be_adn_is $8$ (not in LMFDB) 4.5.af_i_p_ada $8$ (not in LMFDB) 4.5.af_y_acn_gw $8$ (not in LMFDB) 4.5.ab_ak_d_co $8$ (not in LMFDB) 4.5.ab_ae_d_s $8$ (not in LMFDB) 4.5.ab_g_an_bi $8$ (not in LMFDB) 4.5.ab_m_an_de $8$ (not in LMFDB) 4.5.b_ak_ad_co $8$ (not in LMFDB) 4.5.b_ae_ad_s $8$ (not in LMFDB) 4.5.b_g_n_bi $8$ (not in LMFDB) 4.5.b_m_n_de $8$ (not in LMFDB) 4.5.f_i_ap_ada $8$ (not in LMFDB) 4.5.f_y_cn_gw $8$ (not in LMFDB) 4.5.h_o_av_aew $8$ (not in LMFDB) 4.5.h_be_dn_is $8$ (not in LMFDB) 4.5.al_cj_aim_wa $12$ (not in LMFDB) 4.5.aj_bj_ade_gm $12$ (not in LMFDB) 4.5.ah_z_ack_fe $12$ (not in LMFDB) 4.5.af_h_ba_aei $12$ (not in LMFDB) 4.5.af_n_abc_cq $12$ (not in LMFDB) 4.5.ad_ab_ak_cu $12$ (not in LMFDB) 4.5.ad_f_aw_co $12$ (not in LMFDB) 4.5.ad_f_ae_m $12$ (not in LMFDB) 4.5.ad_f_e_am $12$ (not in LMFDB) 4.5.ad_f_w_aco $12$ (not in LMFDB) 4.5.ab_af_ac_bg $12$ (not in LMFDB) 4.5.ab_b_ao_ba $12$ (not in LMFDB) 4.5.b_af_c_bg $12$ (not in LMFDB) 4.5.b_b_o_ba $12$ (not in LMFDB) 4.5.d_ab_k_cu $12$ (not in LMFDB) 4.5.d_f_aw_aco $12$ (not in LMFDB) 4.5.d_f_ae_am $12$ (not in LMFDB) 4.5.d_f_e_m $12$ (not in LMFDB) 4.5.d_f_w_co $12$ (not in LMFDB) 4.5.f_h_aba_aei $12$ (not in LMFDB) 4.5.f_n_bc_cq $12$ (not in LMFDB) 4.5.h_z_ck_fe $12$ (not in LMFDB) 4.5.j_bj_de_gm $12$ (not in LMFDB) 4.5.l_cj_im_wa $12$ (not in LMFDB)