# Properties

 Label 4.5.al_cg_ahp_te 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: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.206741677780$, $\pm0.540075011113$ 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 $A(\F_{q^r})$ 64 332800 235286272 159488409600 105937056318784 62857951053414400 37622750928200078656 23334330333320655667200 14573326513210221066708736 9091927125274661971438720000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 21 118 653 3455 16458 78899 391485 1955998 9762381

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 2.5.ad_i 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.ad_i : $$\Q(\sqrt{-3}, \sqrt{17})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ha 2 $\times$ 1.15625.ja 2 . The endomorphism algebra for each factor is: 1.15625.ha 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-51})$$$)$ 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.h_y. The endomorphism algebra for each factor is: 1.25.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.25.h_y : $$\Q(\sqrt{-3}, \sqrt{17})$$.
• Endomorphism algebra over $\F_{5^{3}}$  The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae 2 $\times$ 2.125.a_ha. The endomorphism algebra for each factor is: 1.125.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.125.a_ha : $$\Q(\sqrt{-3}, \sqrt{17})$$.

## 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.ad_c_d_c $2$ (not in LMFDB) 4.5.d_c_ad_c $2$ (not in LMFDB) 4.5.f_k_l_s $2$ (not in LMFDB) 4.5.l_cg_hp_te $2$ (not in LMFDB) 4.5.ai_t_q_afc $3$ (not in LMFDB) 4.5.af_k_al_s $3$ (not in LMFDB) 4.5.b_h_e_s $3$ (not in LMFDB) 4.5.e_e_ai_abb $3$ (not in LMFDB) 4.5.h_bf_dw_jy $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ad_c_d_c $2$ (not in LMFDB) 4.5.d_c_ad_c $2$ (not in LMFDB) 4.5.f_k_l_s $2$ (not in LMFDB) 4.5.l_cg_hp_te $2$ (not in LMFDB) 4.5.ai_t_q_afc $3$ (not in LMFDB) 4.5.af_k_al_s $3$ (not in LMFDB) 4.5.b_h_e_s $3$ (not in LMFDB) 4.5.e_e_ai_abb $3$ (not in LMFDB) 4.5.h_bf_dw_jy $3$ (not in LMFDB) 4.5.aj_bs_afr_ok $4$ (not in LMFDB) 4.5.ah_bi_aef_kw $4$ (not in LMFDB) 4.5.af_q_abv_ew $4$ (not in LMFDB) 4.5.ad_i_aj_o $4$ (not in LMFDB) 4.5.ad_o_abh_du $4$ (not in LMFDB) 4.5.ab_e_f_g $4$ (not in LMFDB) 4.5.ab_k_f_bq $4$ (not in LMFDB) 4.5.b_e_af_g $4$ (not in LMFDB) 4.5.b_k_af_bq $4$ (not in LMFDB) 4.5.d_i_j_o $4$ (not in LMFDB) 4.5.d_o_bh_du $4$ (not in LMFDB) 4.5.f_q_bv_ew $4$ (not in LMFDB) 4.5.h_bi_ef_kw $4$ (not in LMFDB) 4.5.j_bs_fr_ok $4$ (not in LMFDB) 4.5.ah_bf_adw_jy $6$ (not in LMFDB) 4.5.ae_e_i_abb $6$ (not in LMFDB) 4.5.ab_h_ae_s $6$ (not in LMFDB) 4.5.a_an_a_do $6$ (not in LMFDB) 4.5.i_t_aq_afc $6$ (not in LMFDB) 4.5.ad_a_j_ao $8$ (not in LMFDB) 4.5.ad_q_abn_ek $8$ (not in LMFDB) 4.5.d_a_aj_ao $8$ (not in LMFDB) 4.5.d_q_bn_ek $8$ (not in LMFDB) 4.5.ai_bh_ads_iy $12$ (not in LMFDB) 4.5.ag_l_m_acy $12$ (not in LMFDB) 4.5.ag_z_acu_gu $12$ (not in LMFDB) 4.5.af_n_abm_dy $12$ (not in LMFDB) 4.5.ae_h_i_abw $12$ (not in LMFDB) 4.5.ae_s_abw_ex $12$ (not in LMFDB) 4.5.ae_v_abw_fs $12$ (not in LMFDB) 4.5.ac_ai_e_cf $12$ (not in LMFDB) 4.5.ac_af_e_bk $12$ (not in LMFDB) 4.5.ac_g_ay_br $12$ (not in LMFDB) 4.5.ac_j_ay_cm $12$ (not in LMFDB) 4.5.ab_b_o_as $12$ (not in LMFDB) 4.5.a_ab_a_i $12$ (not in LMFDB) 4.5.a_b_a_i $12$ (not in LMFDB) 4.5.a_n_a_do $12$ (not in LMFDB) 4.5.b_b_ao_as $12$ (not in LMFDB) 4.5.c_ai_ae_cf $12$ (not in LMFDB) 4.5.c_af_ae_bk $12$ (not in LMFDB) 4.5.c_g_y_br $12$ (not in LMFDB) 4.5.c_j_y_cm $12$ (not in LMFDB) 4.5.e_h_ai_abw $12$ (not in LMFDB) 4.5.e_s_bw_ex $12$ (not in LMFDB) 4.5.e_v_bw_fs $12$ (not in LMFDB) 4.5.f_n_bm_dy $12$ (not in LMFDB) 4.5.g_l_am_acy $12$ (not in LMFDB) 4.5.g_z_cu_gu $12$ (not in LMFDB) 4.5.i_bh_ds_iy $12$ (not in LMFDB) 4.5.a_ap_a_ec $24$ (not in LMFDB) 4.5.a_ab_a_ag $24$ (not in LMFDB) 4.5.a_b_a_ag $24$ (not in LMFDB) 4.5.a_p_a_ec $24$ (not in LMFDB)