# Properties

 Label 4.5.am_cr_ajs_zg 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: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.185749715683$, $\pm0.480916950984$ Angle rank: $1$ (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 $A(\F_{q^r})$ 52 317200 251539600 156234956800 102588049661812 63272170368160000 38056342149202802932 23345947240459252531200 14551929298156324491648400 9095389395817645702197130000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 20 126 640 3354 16562 79794 391680 1953126 9766100

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 2.5.ae_l 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}($$$\Q(\sqrt{-1})$$$)$ 2.5.ae_l : $$\Q(\zeta_{12})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ja 4 and its endomorphism algebra is $\mathrm{M}_{4}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{5^{6}}$.
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 $\times$ 2.25.g_l. The endomorphism algebra for each factor is: 1.25.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.25.g_l : $$\Q(\zeta_{12})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae 2 $\times$ 1.125.e 2 . The endomorphism algebra for each factor is: 1.125.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.125.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$

## 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.ae_f_ae_q $2$ (not in LMFDB) 4.5.ae_f_e_aq $2$ (not in LMFDB) 4.5.e_f_ae_aq $2$ (not in LMFDB) 4.5.e_f_e_q $2$ (not in LMFDB) 4.5.m_cr_js_zg $2$ (not in LMFDB) 4.5.a_am_a_di $3$ (not in LMFDB) 4.5.a_g_a_l $3$ (not in LMFDB) 4.5.m_cr_js_zg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ae_f_ae_q $2$ (not in LMFDB) 4.5.ae_f_e_aq $2$ (not in LMFDB) 4.5.e_f_ae_aq $2$ (not in LMFDB) 4.5.e_f_e_q $2$ (not in LMFDB) 4.5.m_cr_js_zg $2$ (not in LMFDB) 4.5.a_am_a_di $3$ (not in LMFDB) 4.5.a_g_a_l $3$ (not in LMFDB) 4.5.m_cr_js_zg $3$ (not in LMFDB) 4.5.ak_bp_adq_hc $4$ (not in LMFDB) 4.5.ak_cb_ahg_su $4$ (not in LMFDB) 4.5.ai_bd_acs_fw $4$ (not in LMFDB) 4.5.ai_bp_afk_oa $4$ (not in LMFDB) 4.5.ag_j_be_afg $4$ (not in LMFDB) 4.5.ag_v_aci_fw $4$ (not in LMFDB) 4.5.ag_v_acc_em $4$ (not in LMFDB) 4.5.ae_f_aw_dk $4$ (not in LMFDB) 4.5.ae_f_w_adk $4$ (not in LMFDB) 4.5.ae_r_abs_em $4$ (not in LMFDB) 4.5.ac_ah_c_ce $4$ (not in LMFDB) 4.5.ac_f_aw_bs $4$ (not in LMFDB) 4.5.ac_f_ae_i $4$ (not in LMFDB) 4.5.ac_f_e_ai $4$ (not in LMFDB) 4.5.ac_f_w_abs $4$ (not in LMFDB) 4.5.a_ad_ag_i $4$ (not in LMFDB) 4.5.a_ad_g_i $4$ (not in LMFDB) 4.5.a_j_am_bs $4$ (not in LMFDB) 4.5.a_j_m_bs $4$ (not in LMFDB) 4.5.c_ah_ac_ce $4$ (not in LMFDB) 4.5.c_f_aw_abs $4$ (not in LMFDB) 4.5.c_f_ae_ai $4$ (not in LMFDB) 4.5.c_f_e_i $4$ (not in LMFDB) 4.5.c_f_w_bs $4$ (not in LMFDB) 4.5.e_f_aw_adk $4$ (not in LMFDB) 4.5.e_f_w_dk $4$ (not in LMFDB) 4.5.e_r_bs_em $4$ (not in LMFDB) 4.5.g_j_abe_afg $4$ (not in LMFDB) 4.5.g_v_cc_em $4$ (not in LMFDB) 4.5.g_v_ci_fw $4$ (not in LMFDB) 4.5.i_bd_cs_fw $4$ (not in LMFDB) 4.5.i_bp_fk_oa $4$ (not in LMFDB) 4.5.k_bp_dq_hc $4$ (not in LMFDB) 4.5.k_cb_hg_su $4$ (not in LMFDB) 4.5.aq_em_atc_cao $6$ (not in LMFDB) 4.5.ai_u_i_aec $6$ (not in LMFDB) 4.5.ai_bm_aey_mt $6$ (not in LMFDB) 4.5.i_u_ai_aec $6$ (not in LMFDB) 4.5.i_bm_ey_mt $6$ (not in LMFDB) 4.5.q_em_tc_cao $6$ (not in LMFDB) 4.5.ae_d_m_abm $8$ (not in LMFDB) 4.5.ae_t_aca_fi $8$ (not in LMFDB) 4.5.ac_aj_g_cg $8$ (not in LMFDB) 4.5.ac_h_aba_bq $8$ (not in LMFDB) 4.5.c_aj_ag_cg $8$ (not in LMFDB) 4.5.c_h_ba_bq $8$ (not in LMFDB) 4.5.e_d_am_abm $8$ (not in LMFDB) 4.5.e_t_ca_fi $8$ (not in LMFDB) 4.5.ao_do_aog_bmk $12$ (not in LMFDB) 4.5.am_cu_akq_bcg $12$ (not in LMFDB) 4.5.ak_bs_aeo_kc $12$ (not in LMFDB) 4.5.ak_ce_ahy_uw $12$ (not in LMFDB) 4.5.ai_bg_adk_hy $12$ (not in LMFDB) 4.5.ai_bs_afw_pq $12$ (not in LMFDB) 4.5.ag_m_g_acg $12$ (not in LMFDB) 4.5.ag_s_abw_ep $12$ (not in LMFDB) 4.5.ag_y_aco_gc $12$ (not in LMFDB) 4.5.ae_c_aq_dn $12$ (not in LMFDB) 4.5.ae_i_abc_dq $12$ (not in LMFDB) 4.5.ae_i_e_abi $12$ (not in LMFDB) 4.5.ae_u_abs_fe $12$ (not in LMFDB) 4.5.ac_ae_c_bm $12$ (not in LMFDB) 4.5.ac_c_q_abp $12$ (not in LMFDB) 4.5.ac_i_aw_ck $12$ (not in LMFDB) 4.5.ac_i_k_ac $12$ (not in LMFDB) 4.5.a_ag_a_l $12$ (not in LMFDB) 4.5.a_a_a_o $12$ (not in LMFDB) 4.5.a_m_a_di $12$ (not in LMFDB) 4.5.c_ae_ac_bm $12$ (not in LMFDB) 4.5.c_c_aq_abp $12$ (not in LMFDB) 4.5.c_i_ak_ac $12$ (not in LMFDB) 4.5.c_i_w_ck $12$ (not in LMFDB) 4.5.e_c_q_dn $12$ (not in LMFDB) 4.5.e_i_ae_abi $12$ (not in LMFDB) 4.5.e_i_bc_dq $12$ (not in LMFDB) 4.5.e_u_bs_fe $12$ (not in LMFDB) 4.5.g_m_ag_acg $12$ (not in LMFDB) 4.5.g_s_bw_ep $12$ (not in LMFDB) 4.5.g_y_co_gc $12$ (not in LMFDB) 4.5.i_bg_dk_hy $12$ (not in LMFDB) 4.5.i_bs_fw_pq $12$ (not in LMFDB) 4.5.k_bs_eo_kc $12$ (not in LMFDB) 4.5.k_ce_hy_uw $12$ (not in LMFDB) 4.5.m_cu_kq_bcg $12$ (not in LMFDB) 4.5.o_do_og_bmk $12$ (not in LMFDB) 4.5.ai_s_y_agc $24$ (not in LMFDB) 4.5.ai_bi_aea_jy $24$ (not in LMFDB) 4.5.ag_k_s_adq $24$ (not in LMFDB) 4.5.ag_ba_ada_hm $24$ (not in LMFDB) 4.5.ae_g_m_ack $24$ (not in LMFDB) 4.5.ae_w_aca_gg $24$ (not in LMFDB) 4.5.ac_ag_g_bi $24$ (not in LMFDB) 4.5.ac_k_aba_co $24$ (not in LMFDB) 4.5.a_aq_a_ek $24$ (not in LMFDB) 4.5.a_ao_a_du $24$ (not in LMFDB) 4.5.a_ai_a_bn $24$ (not in LMFDB) 4.5.a_ac_a_c $24$ (not in LMFDB) 4.5.a_a_a_ao $24$ (not in LMFDB) 4.5.a_c_a_c $24$ (not in LMFDB) 4.5.a_i_a_bn $24$ (not in LMFDB) 4.5.a_o_a_du $24$ (not in LMFDB) 4.5.a_q_a_ek $24$ (not in LMFDB) 4.5.c_ag_ag_bi $24$ (not in LMFDB) 4.5.c_k_ba_co $24$ (not in LMFDB) 4.5.e_g_am_ack $24$ (not in LMFDB) 4.5.e_w_ca_gg $24$ (not in LMFDB) 4.5.g_k_as_adq $24$ (not in LMFDB) 4.5.g_ba_da_hm $24$ (not in LMFDB) 4.5.i_s_ay_agc $24$ (not in LMFDB) 4.5.i_bi_ea_jy $24$ (not in LMFDB) 4.5.ae_l_ay_bp $30$ (not in LMFDB) 4.5.e_l_y_bp $30$ (not in LMFDB) 4.5.a_a_a_abw $48$ (not in LMFDB) 4.5.a_a_a_bw $48$ (not in LMFDB) 4.5.ac_ab_m_at $60$ (not in LMFDB) 4.5.c_ab_am_at $60$ (not in LMFDB)