# Properties

 Label 4.5.an_dd_amd_bgi 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: $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 5 x + 15 x^{2} - 25 x^{3} + 25 x^{4} )$ Frobenius angles: $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.200000000000$, $\pm0.400000000000$ Angle rank: $1$ (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 $A(\F_{q^r})$ 44 312400 285043484 166666649600 100060969290304 61906285437739600 38089735991599372124 23446514423076677222400 14550041109224787799036844 9084194718280899304552960000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -7 19 143 679 3278 16219 79863 393359 1952873 9754074

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{5}$
 The isogeny class factors as 1.5.ae 2 $\times$ 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: 1.5.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.5.af_p : $$\Q(\zeta_{5})$$.
Endomorphism algebra over $\overline{\F}_{5}$
 The base change of $A$ to $\F_{5^{10}}$ is 1.9765625.ajgk 2 $\times$ 1.9765625.sg 2 . 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
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 2 $\times$ 2.25.f_z. The endomorphism algebra for each factor is: 1.25.ag 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.25.f_z : $$\Q(\zeta_{5})$$.
• Endomorphism algebra over $\F_{5^{5}}$  The base change of $A$ to $\F_{5^{5}}$ is 1.3125.cy 2 $\times$ 2.3125.a_ajgk. The endomorphism algebra for each factor is: 1.3125.cy 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 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.af_j_f_abo $2$ (not in LMFDB) 4.5.ad_b_af_bo $2$ (not in LMFDB) 4.5.d_b_f_bo $2$ (not in LMFDB) 4.5.f_j_af_abo $2$ (not in LMFDB) 4.5.n_dd_md_bgi $2$ (not in LMFDB) 4.5.ab_g_a_p $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.5.af_j_f_abo $2$ (not in LMFDB) 4.5.ad_b_af_bo $2$ (not in LMFDB) 4.5.d_b_f_bo $2$ (not in LMFDB) 4.5.f_j_af_abo $2$ (not in LMFDB) 4.5.n_dd_md_bgi $2$ (not in LMFDB) 4.5.ab_g_a_p $3$ (not in LMFDB) 4.5.al_cl_ajb_xw $4$ (not in LMFDB) 4.5.aj_bx_agt_rs $4$ (not in LMFDB) 4.5.ah_bb_acx_gy $4$ (not in LMFDB) 4.5.af_v_acd_fk $4$ (not in LMFDB) 4.5.ad_h_f_au $4$ (not in LMFDB) 4.5.ab_d_af_u $4$ (not in LMFDB) 4.5.ab_j_ap_ci $4$ (not in LMFDB) 4.5.b_d_f_u $4$ (not in LMFDB) 4.5.b_j_p_ci $4$ (not in LMFDB) 4.5.d_h_af_au $4$ (not in LMFDB) 4.5.f_v_cd_fk $4$ (not in LMFDB) 4.5.h_bb_cx_gy $4$ (not in LMFDB) 4.5.j_bx_gt_rs $4$ (not in LMFDB) 4.5.l_cl_jb_xw $4$ (not in LMFDB) 4.5.ai_q_bo_aic $5$ (not in LMFDB) 4.5.aj_bu_age_pz $6$ (not in LMFDB) 4.5.b_g_a_p $6$ (not in LMFDB) 4.5.j_bu_ge_pz $6$ (not in LMFDB) 4.5.af_h_p_acs $8$ (not in LMFDB) 4.5.af_x_acn_go $8$ (not in LMFDB) 4.5.f_h_ap_acs $8$ (not in LMFDB) 4.5.f_x_cn_go $8$ (not in LMFDB) 4.5.a_aq_a_eg $10$ (not in LMFDB) 4.5.i_q_abo_aic $10$ (not in LMFDB) 4.5.ah_y_aci_ff $12$ (not in LMFDB) 4.5.ad_e_u_acn $12$ (not in LMFDB) 4.5.d_e_au_acn $12$ (not in LMFDB) 4.5.h_y_ci_ff $12$ (not in LMFDB) 4.5.ai_bf_adc_gy $15$ (not in LMFDB) 4.5.e_b_au_aci $15$ (not in LMFDB) 4.5.e_q_bo_eb $15$ (not in LMFDB) 4.5.ai_bk_aeq_ly $20$ (not in LMFDB) 4.5.ag_i_be_afa $20$ (not in LMFDB) 4.5.ag_bc_adm_iw $20$ (not in LMFDB) 4.5.ae_e_u_adm $20$ (not in LMFDB) 4.5.ae_y_aci_hi $20$ (not in LMFDB) 4.5.ac_ai_k_be $20$ (not in LMFDB) 4.5.ac_m_abe_cs $20$ (not in LMFDB) 4.5.a_ae_a_ak $20$ (not in LMFDB) 4.5.a_e_a_ak $20$ (not in LMFDB) 4.5.a_q_a_eg $20$ (not in LMFDB) 4.5.c_ai_ak_be $20$ (not in LMFDB) 4.5.c_m_be_cs $20$ (not in LMFDB) 4.5.e_e_au_adm $20$ (not in LMFDB) 4.5.e_y_ci_hi $20$ (not in LMFDB) 4.5.g_i_abe_afa $20$ (not in LMFDB) 4.5.g_bc_dm_iw $20$ (not in LMFDB) 4.5.i_bk_eq_ly $20$ (not in LMFDB) 4.5.ae_b_u_aci $30$ (not in LMFDB) 4.5.ae_q_abo_eb $30$ (not in LMFDB) 4.5.a_ab_a_u $30$ (not in LMFDB) 4.5.i_bf_dc_gy $30$ (not in LMFDB) 4.5.ai_ba_abo_by $40$ (not in LMFDB) 4.5.ag_s_abe_by $40$ (not in LMFDB) 4.5.ae_o_au_by $40$ (not in LMFDB) 4.5.ac_c_ak_by $40$ (not in LMFDB) 4.5.a_as_a_fa $40$ (not in LMFDB) 4.5.a_ai_a_by $40$ (not in LMFDB) 4.5.a_ag_a_by $40$ (not in LMFDB) 4.5.a_ac_a_abe $40$ (not in LMFDB) 4.5.a_c_a_abe $40$ (not in LMFDB) 4.5.a_g_a_by $40$ (not in LMFDB) 4.5.a_i_a_by $40$ (not in LMFDB) 4.5.a_s_a_fa $40$ (not in LMFDB) 4.5.c_c_k_by $40$ (not in LMFDB) 4.5.e_o_u_by $40$ (not in LMFDB) 4.5.g_s_be_by $40$ (not in LMFDB) 4.5.i_ba_bo_by $40$ (not in LMFDB) 4.5.ai_v_a_adc $60$ (not in LMFDB) 4.5.ag_n_a_abo $60$ (not in LMFDB) 4.5.ag_x_aci_fk $60$ (not in LMFDB) 4.5.ae_g_a_af $60$ (not in LMFDB) 4.5.ae_j_a_au $60$ (not in LMFDB) 4.5.ae_t_abo_eq $60$ (not in LMFDB) 4.5.ae_v_aci_ge $60$ (not in LMFDB) 4.5.ac_al_k_ci $60$ (not in LMFDB) 4.5.ac_ag_a_cd $60$ (not in LMFDB) 4.5.ac_ad_a_bo $60$ (not in LMFDB) 4.5.ac_e_au_bt $60$ (not in LMFDB) 4.5.ac_h_au_ci $60$ (not in LMFDB) 4.5.ac_j_abe_bo $60$ (not in LMFDB) 4.5.a_al_a_dc $60$ (not in LMFDB) 4.5.a_b_a_u $60$ (not in LMFDB) 4.5.a_l_a_dc $60$ (not in LMFDB) 4.5.c_al_ak_ci $60$ (not in LMFDB) 4.5.c_ag_a_cd $60$ (not in LMFDB) 4.5.c_ad_a_bo $60$ (not in LMFDB) 4.5.c_e_u_bt $60$ (not in LMFDB) 4.5.c_h_u_ci $60$ (not in LMFDB) 4.5.c_j_be_bo $60$ (not in LMFDB) 4.5.e_g_a_af $60$ (not in LMFDB) 4.5.e_j_a_au $60$ (not in LMFDB) 4.5.e_t_bo_eq $60$ (not in LMFDB) 4.5.e_v_ci_ge $60$ (not in LMFDB) 4.5.g_n_a_abo $60$ (not in LMFDB) 4.5.g_x_ci_fk $60$ (not in LMFDB) 4.5.i_v_a_adc $60$ (not in LMFDB) 4.5.ae_l_au_by $120$ (not in LMFDB) 4.5.ac_ab_ak_by $120$ (not in LMFDB) 4.5.a_an_a_dm $120$ (not in LMFDB) 4.5.a_ad_a_k $120$ (not in LMFDB) 4.5.a_d_a_k $120$ (not in LMFDB) 4.5.a_n_a_dm $120$ (not in LMFDB) 4.5.c_ab_k_by $120$ (not in LMFDB) 4.5.e_l_u_by $120$ (not in LMFDB)