# Properties

 Label 3.13.ar_fd_axm Base Field $\F_{13}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{13}$ Dimension: $3$ L-polynomial: $( 1 - 7 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.187167041811$, $\pm0.312832958189$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

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

## 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 $A(\F_{q^r})$ 560 4233600 10798833920 23532042240000 51258148307970800 112443608063027404800 247055866870251819395120 542815143389447140638720000 1192553296286215309816730420480 2620006291306630482093779865840000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 147 2238 28847 371817 4826304 62746317 815752319 10604677254 137859052107

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 1.13.ag $\times$ 1.13.ae 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 $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{4}}$ is 1.28561.ahj $\times$ 1.28561.je 2 . The endomorphism algebra for each factor is: 1.28561.ahj : $$\Q(\sqrt{-3})$$. 1.28561.je 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{13^{2}}$  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.ak $\times$ 1.169.k. 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 3.13.aj_bd_aco $2$ (not in LMFDB) 3.13.af_b_bm $2$ (not in LMFDB) 3.13.ad_ah_dm $2$ (not in LMFDB) 3.13.d_ah_adm $2$ (not in LMFDB) 3.13.f_b_abm $2$ (not in LMFDB) 3.13.j_bd_co $2$ (not in LMFDB) 3.13.r_fd_xm $2$ (not in LMFDB) 3.13.ai_br_age $3$ (not in LMFDB) 3.13.af_n_ak $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.aj_bd_aco $2$ (not in LMFDB) 3.13.af_b_bm $2$ (not in LMFDB) 3.13.ad_ah_dm $2$ (not in LMFDB) 3.13.d_ah_adm $2$ (not in LMFDB) 3.13.f_b_abm $2$ (not in LMFDB) 3.13.j_bd_co $2$ (not in LMFDB) 3.13.r_fd_xm $2$ (not in LMFDB) 3.13.ai_br_age $3$ (not in LMFDB) 3.13.af_n_ak $3$ (not in LMFDB) 3.13.at_gd_abcs $4$ (not in LMFDB) 3.13.ap_eh_ati $4$ (not in LMFDB) 3.13.ah_d_cs $4$ (not in LMFDB) 3.13.ah_x_acs $4$ (not in LMFDB) 3.13.af_aj_es $4$ (not in LMFDB) 3.13.ab_ab_di $4$ (not in LMFDB) 3.13.b_ab_adi $4$ (not in LMFDB) 3.13.f_aj_aes $4$ (not in LMFDB) 3.13.h_d_acs $4$ (not in LMFDB) 3.13.h_x_cs $4$ (not in LMFDB) 3.13.p_eh_ti $4$ (not in LMFDB) 3.13.t_gd_bcs $4$ (not in LMFDB) 3.13.ap_ej_atq $6$ (not in LMFDB) 3.13.am_df_anw $6$ (not in LMFDB) 3.13.ah_z_ack $6$ (not in LMFDB) 3.13.ae_t_ace $6$ (not in LMFDB) 3.13.ad_f_bq $6$ (not in LMFDB) 3.13.a_l_abw $6$ (not in LMFDB) 3.13.a_l_bw $6$ (not in LMFDB) 3.13.d_f_abq $6$ (not in LMFDB) 3.13.e_t_ce $6$ (not in LMFDB) 3.13.f_n_k $6$ (not in LMFDB) 3.13.h_z_ck $6$ (not in LMFDB) 3.13.i_br_ge $6$ (not in LMFDB) 3.13.m_df_nw $6$ (not in LMFDB) 3.13.p_ej_tq $6$ (not in LMFDB) 3.13.ah_al_gm $8$ (not in LMFDB) 3.13.ah_bl_agm $8$ (not in LMFDB) 3.13.h_al_agm $8$ (not in LMFDB) 3.13.h_bl_gm $8$ (not in LMFDB) 3.13.ar_ff_axy $12$ (not in LMFDB) 3.13.ao_dv_aqu $12$ (not in LMFDB) 3.13.an_da_amf $12$ (not in LMFDB) 3.13.an_dr_aqc $12$ (not in LMFDB) 3.13.al_bs_aev $12$ (not in LMFDB) 3.13.al_co_akl $12$ (not in LMFDB) 3.13.ak_bz_ahg $12$ (not in LMFDB) 3.13.ak_ct_alg $12$ (not in LMFDB) 3.13.aj_bk_aep $12$ (not in LMFDB) 3.13.ai_bw_ahu $12$ (not in LMFDB) 3.13.ah_p_ac $12$ (not in LMFDB) 3.13.ag_y_aeg $12$ (not in LMFDB) 3.13.ag_bn_aeu $12$ (not in LMFDB) 3.13.af_d_by $12$ (not in LMFDB) 3.13.af_x_aby $12$ (not in LMFDB) 3.13.ae_y_aeg $12$ (not in LMFDB) 3.13.ad_am_df $12$ (not in LMFDB) 3.13.ad_p_c $12$ (not in LMFDB) 3.13.ac_d_u $12$ (not in LMFDB) 3.13.ac_i_adu $12$ (not in LMFDB) 3.13.ac_x_au $12$ (not in LMFDB) 3.13.ab_ag_af $12$ (not in LMFDB) 3.13.ab_ae_dl $12$ (not in LMFDB) 3.13.ab_g_abp $12$ (not in LMFDB) 3.13.b_ag_f $12$ (not in LMFDB) 3.13.b_ae_adl $12$ (not in LMFDB) 3.13.b_g_bp $12$ (not in LMFDB) 3.13.c_d_au $12$ (not in LMFDB) 3.13.c_i_du $12$ (not in LMFDB) 3.13.c_x_u $12$ (not in LMFDB) 3.13.d_am_adf $12$ (not in LMFDB) 3.13.d_f_abq $12$ (not in LMFDB) 3.13.d_p_ac $12$ (not in LMFDB) 3.13.e_y_eg $12$ (not in LMFDB) 3.13.f_d_aby $12$ (not in LMFDB) 3.13.f_x_by $12$ (not in LMFDB) 3.13.g_y_eg $12$ (not in LMFDB) 3.13.g_bn_eu $12$ (not in LMFDB) 3.13.h_p_c $12$ (not in LMFDB) 3.13.i_bw_hu $12$ (not in LMFDB) 3.13.j_bk_ep $12$ (not in LMFDB) 3.13.k_bz_hg $12$ (not in LMFDB) 3.13.k_ct_lg $12$ (not in LMFDB) 3.13.l_bs_ev $12$ (not in LMFDB) 3.13.l_co_kl $12$ (not in LMFDB) 3.13.n_da_mf $12$ (not in LMFDB) 3.13.n_dr_qc $12$ (not in LMFDB) 3.13.o_dv_qu $12$ (not in LMFDB) 3.13.r_ff_xy $12$ (not in LMFDB) 3.13.af_al_eq $24$ (not in LMFDB) 3.13.af_bl_aeq $24$ (not in LMFDB) 3.13.ac_al_bw $24$ (not in LMFDB) 3.13.ac_bl_abw $24$ (not in LMFDB) 3.13.c_al_abw $24$ (not in LMFDB) 3.13.c_bl_bw $24$ (not in LMFDB) 3.13.f_al_aeq $24$ (not in LMFDB) 3.13.f_bl_eq $24$ (not in LMFDB)