# Properties

 Label 4.5.ao_dn_any_blk 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: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.384619558242$ 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 $A(\F_{q^r})$ 28 221200 230463856 148816281600 94313975316988 60307689652537600 37824215428924710268 23460952075775882035200 14573350564538393946716272 9090913800345378830015530000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -8 12 118 608 3092 15810 79316 393600 1955998 9761292

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 2.5.ag_r 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.ag_r : $$\Q(\sqrt{2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 $\times$ 1.15625.ja 2 . The endomorphism algebra for each factor is: 1.15625.afm 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$ 1.15625.ja 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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.ac_av. The endomorphism algebra for each factor is: 1.25.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.25.ac_av : $$\Q(\sqrt{2}, \sqrt{-3})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae 2 $\times$ 2.125.a_afm. The endomorphism algebra for each factor is: 1.125.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.125.a_afm : $$\Q(\sqrt{2}, \sqrt{-3})$$.

## 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.ag_l_g_aca $2$ (not in LMFDB) 4.5.ac_af_k_m $2$ (not in LMFDB) 4.5.c_af_ak_m $2$ (not in LMFDB) 4.5.g_l_ag_aca $2$ (not in LMFDB) 4.5.o_dn_ny_blk $2$ (not in LMFDB) 4.5.ai_bc_ace_dy $3$ (not in LMFDB) 4.5.ac_af_k_m $3$ (not in LMFDB) 4.5.ac_e_ai_ad $3$ (not in LMFDB) 4.5.e_n_bc_cu $3$ (not in LMFDB) 4.5.k_ca_hc_sj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ag_l_g_aca $2$ (not in LMFDB) 4.5.ac_af_k_m $2$ (not in LMFDB) 4.5.c_af_ak_m $2$ (not in LMFDB) 4.5.g_l_ag_aca $2$ (not in LMFDB) 4.5.o_dn_ny_blk $2$ (not in LMFDB) 4.5.ai_bc_ace_dy $3$ (not in LMFDB) 4.5.ac_af_k_m $3$ (not in LMFDB) 4.5.ac_e_ai_ad $3$ (not in LMFDB) 4.5.e_n_bc_cu $3$ (not in LMFDB) 4.5.k_ca_hc_sj $3$ (not in LMFDB) 4.5.am_ct_akk_bbo $4$ (not in LMFDB) 4.5.ak_cd_ahu_ui $4$ (not in LMFDB) 4.5.ai_bf_adi_hw $4$ (not in LMFDB) 4.5.ag_x_aco_fw $4$ (not in LMFDB) 4.5.ae_h_c_abk $4$ (not in LMFDB) 4.5.ac_h_aba_bw $4$ (not in LMFDB) 4.5.a_ab_ag_ae $4$ (not in LMFDB) 4.5.a_ab_g_ae $4$ (not in LMFDB) 4.5.c_h_ba_bw $4$ (not in LMFDB) 4.5.e_h_ac_abk $4$ (not in LMFDB) 4.5.g_x_co_fw $4$ (not in LMFDB) 4.5.i_bf_di_hw $4$ (not in LMFDB) 4.5.k_cd_hu_ui $4$ (not in LMFDB) 4.5.m_ct_kk_bbo $4$ (not in LMFDB) 4.5.ak_ca_ahc_sj $6$ (not in LMFDB) 4.5.ae_n_abc_cu $6$ (not in LMFDB) 4.5.a_ae_a_bm $6$ (not in LMFDB) 4.5.c_e_i_ad $6$ (not in LMFDB) 4.5.i_bc_ce_dy $6$ (not in LMFDB) 4.5.ag_j_s_adi $8$ (not in LMFDB) 4.5.ag_z_ada_he $8$ (not in LMFDB) 4.5.g_j_as_adi $8$ (not in LMFDB) 4.5.g_z_da_he $8$ (not in LMFDB) 4.5.ai_y_ay_ac $12$ (not in LMFDB) 4.5.ai_bc_acq_fx $12$ (not in LMFDB) 4.5.ag_q_as_o $12$ (not in LMFDB) 4.5.ag_u_abq_di $12$ (not in LMFDB) 4.5.ae_e_u_adj $12$ (not in LMFDB) 4.5.ae_j_am_bc $12$ (not in LMFDB) 4.5.ae_m_am_w $12$ (not in LMFDB) 4.5.ae_q_abc_da $12$ (not in LMFDB) 4.5.ac_ad_ag_ca $12$ (not in LMFDB) 4.5.ac_a_ag_bu $12$ (not in LMFDB) 4.5.ac_b_ao_bw $12$ (not in LMFDB) 4.5.ac_e_ao_cc $12$ (not in LMFDB) 4.5.a_ai_a_ck $12$ (not in LMFDB) 4.5.a_e_a_bm $12$ (not in LMFDB) 4.5.a_i_a_ck $12$ (not in LMFDB) 4.5.c_ad_g_ca $12$ (not in LMFDB) 4.5.c_a_g_bu $12$ (not in LMFDB) 4.5.c_b_o_bw $12$ (not in LMFDB) 4.5.c_e_o_cc $12$ (not in LMFDB) 4.5.e_e_au_adj $12$ (not in LMFDB) 4.5.e_j_m_bc $12$ (not in LMFDB) 4.5.e_m_m_w $12$ (not in LMFDB) 4.5.e_q_bc_da $12$ (not in LMFDB) 4.5.g_q_s_o $12$ (not in LMFDB) 4.5.g_u_bq_di $12$ (not in LMFDB) 4.5.i_y_y_ac $12$ (not in LMFDB) 4.5.i_bc_cq_fx $12$ (not in LMFDB) 4.5.am_co_aiu_wg $24$ (not in LMFDB) 4.5.ak_by_ago_qs $24$ (not in LMFDB) 4.5.ai_bj_aem_lm $24$ (not in LMFDB) 4.5.ai_bm_aey_mk $24$ (not in LMFDB) 4.5.ag_p_abq_es $24$ (not in LMFDB) 4.5.ag_s_acc_fq $24$ (not in LMFDB) 4.5.ae_a_m_ao $24$ (not in LMFDB) 4.5.ae_c_e_c $24$ (not in LMFDB) 4.5.ae_c_u_ack $24$ (not in LMFDB) 4.5.ae_o_abs_du $24$ (not in LMFDB) 4.5.ae_q_aca_ek $24$ (not in LMFDB) 4.5.ac_ab_k_abm $24$ (not in LMFDB) 4.5.ac_c_ac_ao $24$ (not in LMFDB) 4.5.ac_c_o_abu $24$ (not in LMFDB) 4.5.a_ak_a_co $24$ (not in LMFDB) 4.5.a_ag_a_bi $24$ (not in LMFDB) 4.5.a_d_am_aw $24$ (not in LMFDB) 4.5.a_d_m_aw $24$ (not in LMFDB) 4.5.a_g_ay_c $24$ (not in LMFDB) 4.5.a_g_a_bi $24$ (not in LMFDB) 4.5.a_g_y_c $24$ (not in LMFDB) 4.5.a_k_a_co $24$ (not in LMFDB) 4.5.c_ab_ak_abm $24$ (not in LMFDB) 4.5.c_c_ao_abu $24$ (not in LMFDB) 4.5.c_c_c_ao $24$ (not in LMFDB) 4.5.e_a_am_ao $24$ (not in LMFDB) 4.5.e_c_au_ack $24$ (not in LMFDB) 4.5.e_c_ae_c $24$ (not in LMFDB) 4.5.e_o_bs_du $24$ (not in LMFDB) 4.5.e_q_ca_ek $24$ (not in LMFDB) 4.5.g_p_bq_es $24$ (not in LMFDB) 4.5.g_s_cc_fq $24$ (not in LMFDB) 4.5.i_bj_em_lm $24$ (not in LMFDB) 4.5.i_bm_ey_mk $24$ (not in LMFDB) 4.5.k_by_go_qs $24$ (not in LMFDB) 4.5.m_co_iu_wg $24$ (not in LMFDB)