# Properties

 Label 4.5.an_df_ams_bie Base Field $\F_{5}$ Dimension $4$ Ordinary Yes $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

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## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )^{3}$ Frobenius angles: $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.265942140215$, $\pm0.265942140215$ 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 $A(\F_{q^r})$ 54 393660 364290048 196830000000 103259791417374 59657070843002880 36794213184728145438 23140362291178680000000 14542119714034827758444544 9103779873771429029456166300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 23 176 787 3373 15638 77161 388227 1951808 9775103

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 1.5.ad 3 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 : $$\Q(\sqrt{-1})$$. 1.5.ad 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-11})$$$)$
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.ah_x_abq_cu $2$ (not in LMFDB) 4.5.af_l_g_abw $2$ (not in LMFDB) 4.5.ab_ab_ag_bw $2$ (not in LMFDB) 4.5.b_ab_g_bw $2$ (not in LMFDB) 4.5.f_l_ag_abw $2$ (not in LMFDB) 4.5.h_x_bq_cu $2$ (not in LMFDB) 4.5.n_df_ms_bie $2$ (not in LMFDB) 4.5.ae_f_s_acu $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ah_x_abq_cu $2$ (not in LMFDB) 4.5.af_l_g_abw $2$ (not in LMFDB) 4.5.ab_ab_ag_bw $2$ (not in LMFDB) 4.5.b_ab_g_bw $2$ (not in LMFDB) 4.5.f_l_ag_abw $2$ (not in LMFDB) 4.5.h_x_bq_cu $2$ (not in LMFDB) 4.5.n_df_ms_bie $2$ (not in LMFDB) 4.5.ae_f_s_acu $3$ (not in LMFDB) 4.5.al_cn_ajm_ze $4$ (not in LMFDB) 4.5.ah_v_abc_bc $4$ (not in LMFDB) 4.5.ah_bd_ada_he $4$ (not in LMFDB) 4.5.af_p_au_bi $4$ (not in LMFDB) 4.5.af_r_abe_co $4$ (not in LMFDB) 4.5.ab_ad_ae_ca $4$ (not in LMFDB) 4.5.ab_d_ae_bu $4$ (not in LMFDB) 4.5.ab_f_ag_cc $4$ (not in LMFDB) 4.5.b_ad_e_ca $4$ (not in LMFDB) 4.5.b_d_e_bu $4$ (not in LMFDB) 4.5.b_f_g_cc $4$ (not in LMFDB) 4.5.f_p_u_bi $4$ (not in LMFDB) 4.5.f_r_be_co $4$ (not in LMFDB) 4.5.h_v_bc_bc $4$ (not in LMFDB) 4.5.h_bd_da_he $4$ (not in LMFDB) 4.5.l_cn_jm_ze $4$ (not in LMFDB) 4.5.ak_bv_afo_nk $6$ (not in LMFDB) 4.5.ae_f_as_cu $6$ (not in LMFDB) 4.5.ac_ab_a_m $6$ (not in LMFDB) 4.5.c_ab_a_m $6$ (not in LMFDB) 4.5.e_f_as_acu $6$ (not in LMFDB) 4.5.e_f_s_cu $6$ (not in LMFDB) 4.5.k_bv_fo_nk $6$ (not in LMFDB) 4.5.ai_bj_aee_ke $12$ (not in LMFDB) 4.5.ae_l_abk_ds $12$ (not in LMFDB) 4.5.ac_f_as_bk $12$ (not in LMFDB) 4.5.ac_f_s_abk $12$ (not in LMFDB) 4.5.c_f_as_abk $12$ (not in LMFDB) 4.5.c_f_s_bk $12$ (not in LMFDB) 4.5.e_l_bk_ds $12$ (not in LMFDB) 4.5.i_bj_ee_ke $12$ (not in LMFDB)