# Properties

 Label 4.5.am_cu_akp_bcc Base Field $\F_{5}$ Dimension $4$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{5}$ Dimension: $4$ L-polynomial: $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.200000000000$, $\pm0.265942140215$, $\pm0.400000000000$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 66 421740 336444768 175781232000 99467228373216 60704070058494720 37575140666632160658 23314520028921029568000 14532517594099089483772896 9086545273779316874774707200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 26 165 714 3259 15911 78800 391154 1950519 9756601

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae $\times$ 1.5.ad $\times$ 2.5.af_p 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 $\F_{5^{10}}$ is 1.9765625.ajgk 2 $\times$ 1.9765625.sg $\times$ 1.9765625.ell. The endomorphism algebra for each factor is: 1.9765625.ajgk 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $5$ and $\infty$. 1.9765625.sg : $$\Q(\sqrt{-1})$$. 1.9765625.ell : $$\Q(\sqrt{-11})$$.
All geometric endomorphisms are defined over $\F_{5^{10}}$.
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.b $\times$ 2.25.f_z. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{5^{5}}$  The base change of $A$ to $\F_{5^{5}}$ is 1.3125.cf $\times$ 1.3125.cy $\times$ 2.3125.a_ajgk. The endomorphism algebra for each factor is: 1.3125.cf : $$\Q(\sqrt{-11})$$. 1.3125.cy : $$\Q(\sqrt{-1})$$. 2.3125.a_ajgk : the quaternion algebra over $$\Q(\sqrt{5})$$ ramified at both real infinite places.

## 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.ag_s_abj_cs $2$ (not in LMFDB) 4.5.ae_i_f_abe $2$ (not in LMFDB) 4.5.ac_c_af_be $2$ (not in LMFDB) 4.5.c_c_f_be $2$ (not in LMFDB) 4.5.e_i_af_abe $2$ (not in LMFDB) 4.5.g_s_bj_cs $2$ (not in LMFDB) 4.5.m_cu_kp_bcc $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.ag_s_abj_cs $2$ (not in LMFDB) 4.5.ae_i_f_abe $2$ (not in LMFDB) 4.5.ac_c_af_be $2$ (not in LMFDB) 4.5.c_c_f_be $2$ (not in LMFDB) 4.5.e_i_af_abe $2$ (not in LMFDB) 4.5.g_s_bj_cs $2$ (not in LMFDB) 4.5.m_cu_kp_bcc $2$ (not in LMFDB) 4.5.ak_ce_ahx_uu $4$ (not in LMFDB) 4.5.ag_y_acn_ge $4$ (not in LMFDB) 4.5.ae_o_az_ci $4$ (not in LMFDB) 4.5.a_g_af_bo $4$ (not in LMFDB) 4.5.a_g_f_bo $4$ (not in LMFDB) 4.5.e_o_z_ci $4$ (not in LMFDB) 4.5.g_y_cn_ge $4$ (not in LMFDB) 4.5.k_ce_hx_uu $4$ (not in LMFDB) 4.5.ah_m_bj_ago $5$ (not in LMFDB) 4.5.ac_c_af_be $5$ (not in LMFDB) 4.5.ab_am_f_cs $10$ (not in LMFDB) 4.5.b_am_af_cs $10$ (not in LMFDB) 4.5.h_m_abj_ago $10$ (not in LMFDB) 4.5.ah_bb_acs_ge $15$ (not in LMFDB) 4.5.ah_bg_aeb_kk $20$ (not in LMFDB) 4.5.af_g_z_aeg $20$ (not in LMFDB) 4.5.af_ba_acx_ic $20$ (not in LMFDB) 4.5.ab_ag_f_k $20$ (not in LMFDB) 4.5.ab_i_ap_be $20$ (not in LMFDB) 4.5.ab_o_ap_dm $20$ (not in LMFDB) 4.5.b_ag_af_k $20$ (not in LMFDB) 4.5.b_i_p_be $20$ (not in LMFDB) 4.5.b_o_p_dm $20$ (not in LMFDB) 4.5.f_g_az_aeg $20$ (not in LMFDB) 4.5.f_ba_cx_ic $20$ (not in LMFDB) 4.5.h_bg_eb_kk $20$ (not in LMFDB) 4.5.ab_d_ak_bo $30$ (not in LMFDB) 4.5.b_d_k_bo $30$ (not in LMFDB) 4.5.h_bb_cs_ge $30$ (not in LMFDB) 4.5.ah_w_abj_by $40$ (not in LMFDB) 4.5.af_q_az_by $40$ (not in LMFDB) 4.5.ab_ac_af_by $40$ (not in LMFDB) 4.5.ab_e_af_by $40$ (not in LMFDB) 4.5.b_ac_f_by $40$ (not in LMFDB) 4.5.b_e_f_by $40$ (not in LMFDB) 4.5.f_q_z_by $40$ (not in LMFDB) 4.5.h_w_bj_by $40$ (not in LMFDB) 4.5.ah_r_a_aci $60$ (not in LMFDB) 4.5.af_l_a_abe $60$ (not in LMFDB) 4.5.af_v_aby_fa $60$ (not in LMFDB) 4.5.ab_ah_a_ci $60$ (not in LMFDB) 4.5.ab_ab_a_be $60$ (not in LMFDB) 4.5.ab_j_ak_cs $60$ (not in LMFDB) 4.5.b_ah_a_ci $60$ (not in LMFDB) 4.5.b_ab_a_be $60$ (not in LMFDB) 4.5.b_j_k_cs $60$ (not in LMFDB) 4.5.f_l_a_abe $60$ (not in LMFDB) 4.5.f_v_by_fa $60$ (not in LMFDB) 4.5.h_r_a_aci $60$ (not in LMFDB)