# Properties

 Label 4.5.al_cj_aip_wm 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: $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.0878807261908$, $\pm0.147583617650$, $\pm0.352416382350$, $\pm0.450170915301$ 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 $A(\F_{q^r})$ 72 397440 272519208 144731750400 92214761542272 60251969345892480 37826217127905395688 23430229459787612160000 14576431850763169422818952 9100385940366051423171379200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 27 139 591 3020 15795 79319 393087 1956415 9771462

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 1.5.ac $\times$ 2.5.af_n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{4}}$ is 1.625.o 2 $\times$ 2.625.acl_dfd. The endomorphism algebra for each factor is: 1.625.o 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.625.acl_dfd : 4.0.4901.1.
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 $\times$ 1.25.g $\times$ 2.25.b_abf. The endomorphism algebra for each factor is:

## 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.ah_z_act_gu $2$ (not in LMFDB) 4.5.ad_f_b_ay $2$ (not in LMFDB) 4.5.ab_b_h_aq $2$ (not in LMFDB) 4.5.b_b_ah_aq $2$ (not in LMFDB) 4.5.d_f_ab_ay $2$ (not in LMFDB) 4.5.h_z_ct_gu $2$ (not in LMFDB) 4.5.l_cj_ip_wm $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ah_z_act_gu $2$ (not in LMFDB) 4.5.ad_f_b_ay $2$ (not in LMFDB) 4.5.ab_b_h_aq $2$ (not in LMFDB) 4.5.b_b_ah_aq $2$ (not in LMFDB) 4.5.d_f_ab_ay $2$ (not in LMFDB) 4.5.h_z_ct_gu $2$ (not in LMFDB) 4.5.l_cj_ip_wm $2$ (not in LMFDB) 4.5.an_db_aln_bei $4$ (not in LMFDB) 4.5.aj_bv_agl_qq $4$ (not in LMFDB) 4.5.af_h_f_abc $4$ (not in LMFDB) 4.5.af_t_acd_ey $4$ (not in LMFDB) 4.5.ad_ab_l_am $4$ (not in LMFDB) 4.5.ab_h_ax_bg $4$ (not in LMFDB) 4.5.b_h_x_bg $4$ (not in LMFDB) 4.5.d_ab_al_am $4$ (not in LMFDB) 4.5.f_h_af_abc $4$ (not in LMFDB) 4.5.f_t_cd_ey $4$ (not in LMFDB) 4.5.j_bv_gl_qq $4$ (not in LMFDB) 4.5.n_db_ln_bei $4$ (not in LMFDB) 4.5.af_f_p_acc $8$ (not in LMFDB) 4.5.af_v_acn_fy $8$ (not in LMFDB) 4.5.f_f_ap_acc $8$ (not in LMFDB) 4.5.f_v_cn_fy $8$ (not in LMFDB) 4.5.aj_bs_afw_pd $12$ (not in LMFDB) 4.5.ah_w_ace_fh $12$ (not in LMFDB) 4.5.ad_c_q_acl $12$ (not in LMFDB) 4.5.ab_e_ai_ah $12$ (not in LMFDB) 4.5.b_e_i_ah $12$ (not in LMFDB) 4.5.d_c_aq_acl $12$ (not in LMFDB) 4.5.h_w_ce_fh $12$ (not in LMFDB) 4.5.j_bs_fw_pd $12$ (not in LMFDB)