# Properties

 Label 4.5.am_cq_ajk_yg Base Field $\F_{5}$ Dimension $4$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{3}$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.5$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 48 288000 228796848 150994944000 102624949503408 63336001057056000 38055879314533355568 23397172002403909632000 14584065785848469246657328 9102096763908314775156000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 18 114 618 3354 16578 79794 392538 1957434 9773298

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 3 $\times$ 1.5.a 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.5.a : $$\Q(\sqrt{-5})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag 3 $\times$ 1.25.k. The endomorphism algebra for each factor is: 1.25.ag 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.25.k : the quaternion algebra over $$\Q$$ ramified at $5$ and $\infty$.
All geometric endomorphisms are defined over $\F_{5^{2}}$.

## 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.ae_e_e_ak $2$ (not in LMFDB) 4.5.e_e_ae_ak $2$ (not in LMFDB) 4.5.m_cq_jk_yg $2$ (not in LMFDB) 4.5.a_f_ae_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ae_e_e_ak $2$ (not in LMFDB) 4.5.e_e_ae_ak $2$ (not in LMFDB) 4.5.m_cq_jk_yg $2$ (not in LMFDB) 4.5.a_f_ae_a $3$ (not in LMFDB) 4.5.ak_ca_aha_sc $4$ (not in LMFDB) 4.5.ai_bo_afg_nm $4$ (not in LMFDB) 4.5.ag_u_acg_fu $4$ (not in LMFDB) 4.5.ag_bg_adu_kk $4$ (not in LMFDB) 4.5.ae_q_abs_eg $4$ (not in LMFDB) 4.5.ac_e_c_ak $4$ (not in LMFDB) 4.5.ac_q_aw_eg $4$ (not in LMFDB) 4.5.a_i_aq_be $4$ (not in LMFDB) 4.5.a_i_q_be $4$ (not in LMFDB) 4.5.c_e_ac_ak $4$ (not in LMFDB) 4.5.c_q_w_eg $4$ (not in LMFDB) 4.5.e_q_bs_eg $4$ (not in LMFDB) 4.5.g_u_cg_fu $4$ (not in LMFDB) 4.5.g_bg_du_kk $4$ (not in LMFDB) 4.5.i_bo_fg_nm $4$ (not in LMFDB) 4.5.k_ca_ha_sc $4$ (not in LMFDB) 4.5.ai_bl_aeu_mi $6$ (not in LMFDB) 4.5.a_f_e_a $6$ (not in LMFDB) 4.5.i_bl_eu_mi $6$ (not in LMFDB) 4.5.ae_c_m_abe $8$ (not in LMFDB) 4.5.ae_s_aca_fa $8$ (not in LMFDB) 4.5.ac_c_g_abe $8$ (not in LMFDB) 4.5.ac_s_aba_fa $8$ (not in LMFDB) 4.5.c_c_ag_abe $8$ (not in LMFDB) 4.5.c_s_ba_fa $8$ (not in LMFDB) 4.5.e_c_am_abe $8$ (not in LMFDB) 4.5.e_s_ca_fa $8$ (not in LMFDB) 4.5.ag_r_abu_eq $12$ (not in LMFDB) 4.5.ag_bd_ado_jg $12$ (not in LMFDB) 4.5.ae_n_abm_dc $12$ (not in LMFDB) 4.5.ac_b_o_abo $12$ (not in LMFDB) 4.5.ac_n_abc_dc $12$ (not in LMFDB) 4.5.a_f_aw_a $12$ (not in LMFDB) 4.5.a_f_w_a $12$ (not in LMFDB) 4.5.c_b_ao_abo $12$ (not in LMFDB) 4.5.c_n_bc_dc $12$ (not in LMFDB) 4.5.e_n_bm_dc $12$ (not in LMFDB) 4.5.g_r_bu_eq $12$ (not in LMFDB) 4.5.g_bd_do_jg $12$ (not in LMFDB)