# Properties

 Label 4.5.al_cj_ain_we 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: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 11 x^{2} - 15 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.300933478836$, $\pm0.472779926746$ Angle rank: $3$ (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 $A(\F_{q^r})$ 76 418000 288734716 157943808000 98651495818816 61458772341538000 37682909808445408156 23329649713854578688000 14571525193312758813944236 9105474366436704604480000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 27 145 647 3230 16107 79025 391407 1955755 9776922

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 2.5.ad_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.ad_l : 4.0.6525.1.
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.af_n_abj_ds $2$ (not in LMFDB) 4.5.ad_f_d_aq $2$ (not in LMFDB) 4.5.d_f_ad_aq $2$ (not in LMFDB) 4.5.f_n_bj_ds $2$ (not in LMFDB) 4.5.l_cj_in_we $2$ (not in LMFDB) 4.5.b_k_q_bz $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.af_n_abj_ds $2$ (not in LMFDB) 4.5.ad_f_d_aq $2$ (not in LMFDB) 4.5.d_f_ad_aq $2$ (not in LMFDB) 4.5.f_n_bj_ds $2$ (not in LMFDB) 4.5.l_cj_in_we $2$ (not in LMFDB) 4.5.b_k_q_bz $3$ (not in LMFDB) 4.5.aj_bv_agj_qm $4$ (not in LMFDB) 4.5.ah_bl_aer_mm $4$ (not in LMFDB) 4.5.af_t_acb_fc $4$ (not in LMFDB) 4.5.ad_l_abb_cq $4$ (not in LMFDB) 4.5.ad_r_abh_em $4$ (not in LMFDB) 4.5.ab_h_l_m $4$ (not in LMFDB) 4.5.ab_n_ah_dg $4$ (not in LMFDB) 4.5.b_h_al_m $4$ (not in LMFDB) 4.5.b_n_h_dg $4$ (not in LMFDB) 4.5.d_l_bb_cq $4$ (not in LMFDB) 4.5.d_r_bh_em $4$ (not in LMFDB) 4.5.f_t_cb_fc $4$ (not in LMFDB) 4.5.h_bl_er_mm $4$ (not in LMFDB) 4.5.j_bv_gj_qm $4$ (not in LMFDB) 4.5.ah_bi_aei_lf $6$ (not in LMFDB) 4.5.ab_k_aq_bz $6$ (not in LMFDB) 4.5.h_bi_ei_lf $6$ (not in LMFDB) 4.5.ad_d_j_abm $8$ (not in LMFDB) 4.5.ad_t_abn_fi $8$ (not in LMFDB) 4.5.d_d_aj_abm $8$ (not in LMFDB) 4.5.d_t_bn_fi $8$ (not in LMFDB) 4.5.af_q_abs_dv $12$ (not in LMFDB) 4.5.ab_e_u_av $12$ (not in LMFDB) 4.5.b_e_au_av $12$ (not in LMFDB) 4.5.f_q_bs_dv $12$ (not in LMFDB)