# Properties

 Label 3.13.aq_es_avk 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: $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$ Frobenius angles: $\pm0.0772104791556$, $\pm0.256122854178$, $\pm0.312832958189$ 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 $A(\F_{q^r})$ 630 4524660 11052236160 23612933635200 51200489634024150 112337097044228259840 246971216439371889206910 542777263568205236101440000 1192552670567179325263778029440 2620021970752207141908792554997300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 158 2290 28946 371398 4821728 62724814 815695394 10604671690 137859877118

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 1.13.af $\times$ 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 $\F_{13^{6}}$ is 1.4826809.agao $\times$ 1.4826809.atm 2 . The endomorphism algebra for each factor is: 1.4826809.agao : $$\Q(\sqrt{-1})$$. 1.4826809.atm 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{13^{6}}$.
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.b $\times$ 1.169.k. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{13^{3}}$  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs $\times$ 1.2197.do. 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.ai_ba_acq $2$ (not in LMFDB) 3.13.ag_m_aq $2$ (not in LMFDB) 3.13.ac_ae_dk $2$ (not in LMFDB) 3.13.c_ae_adk $2$ (not in LMFDB) 3.13.g_m_q $2$ (not in LMFDB) 3.13.i_ba_cq $2$ (not in LMFDB) 3.13.q_es_vk $2$ (not in LMFDB) 3.13.an_dl_ape $3$ (not in LMFDB) 3.13.ah_bp_afm $3$ (not in LMFDB) 3.13.ae_ak_do $3$ (not in LMFDB) 3.13.ae_o_ae $3$ (not in LMFDB) 3.13.ae_bj_adk $3$ (not in LMFDB) 3.13.ab_r_o $3$ (not in LMFDB) 3.13.f_r_cw $3$ (not in LMFDB) 3.13.i_ba_cq $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ai_ba_acq $2$ (not in LMFDB) 3.13.ag_m_aq $2$ (not in LMFDB) 3.13.ac_ae_dk $2$ (not in LMFDB) 3.13.c_ae_adk $2$ (not in LMFDB) 3.13.g_m_q $2$ (not in LMFDB) 3.13.i_ba_cq $2$ (not in LMFDB) 3.13.q_es_vk $2$ (not in LMFDB) 3.13.an_dl_ape $3$ (not in LMFDB) 3.13.ah_bp_afm $3$ (not in LMFDB) 3.13.ae_ak_do $3$ (not in LMFDB) 3.13.ae_o_ae $3$ (not in LMFDB) 3.13.ae_bj_adk $3$ (not in LMFDB) 3.13.ab_r_o $3$ (not in LMFDB) 3.13.f_r_cw $3$ (not in LMFDB) 3.13.i_ba_cq $3$ (not in LMFDB) 3.13.as_fq_abac $4$ (not in LMFDB) 3.13.ai_q_c $4$ (not in LMFDB) 3.13.ag_c_cc $4$ (not in LMFDB) 3.13.ae_ai_ec $4$ (not in LMFDB) 3.13.e_ai_aec $4$ (not in LMFDB) 3.13.g_c_acc $4$ (not in LMFDB) 3.13.i_q_ac $4$ (not in LMFDB) 3.13.s_fq_bac $4$ (not in LMFDB) 3.13.as_fo_azo $6$ (not in LMFDB) 3.13.ao_ea_arw $6$ (not in LMFDB) 3.13.al_cz_amo $6$ (not in LMFDB) 3.13.ak_bg_acm $6$ (not in LMFDB) 3.13.aj_bt_agw $6$ (not in LMFDB) 3.13.ai_ch_aiq $6$ (not in LMFDB) 3.13.ag_y_ace $6$ (not in LMFDB) 3.13.af_r_acw $6$ (not in LMFDB) 3.13.ad_v_abm $6$ (not in LMFDB) 3.13.ab_f_ade $6$ (not in LMFDB) 3.13.a_bb_aq $6$ (not in LMFDB) 3.13.a_bb_q $6$ (not in LMFDB) 3.13.b_f_de $6$ (not in LMFDB) 3.13.b_r_ao $6$ (not in LMFDB) 3.13.d_v_bm $6$ (not in LMFDB) 3.13.e_ak_ado $6$ (not in LMFDB) 3.13.e_o_e $6$ (not in LMFDB) 3.13.e_bj_dk $6$ (not in LMFDB) 3.13.g_y_ce $6$ (not in LMFDB) 3.13.h_bp_fm $6$ (not in LMFDB) 3.13.i_ch_iq $6$ (not in LMFDB) 3.13.j_bt_gw $6$ (not in LMFDB) 3.13.k_bg_cm $6$ (not in LMFDB) 3.13.l_cz_mo $6$ (not in LMFDB) 3.13.n_dl_pe $6$ (not in LMFDB) 3.13.o_ea_rw $6$ (not in LMFDB) 3.13.s_fo_zo $6$ (not in LMFDB) 3.13.au_gq_abfi $12$ (not in LMFDB) 3.13.aq_eu_avu $12$ (not in LMFDB) 3.13.ap_ed_asg $12$ (not in LMFDB) 3.13.an_dn_api $12$ (not in LMFDB) 3.13.al_cd_ahu $12$ (not in LMFDB) 3.13.ak_cp_aky $12$ (not in LMFDB) 3.13.aj_bv_ags $12$ (not in LMFDB) 3.13.ai_e_di $12$ (not in LMFDB) 3.13.ag_ak_fi $12$ (not in LMFDB) 3.13.ag_aj_fc $12$ (not in LMFDB) 3.13.ag_m_g $12$ (not in LMFDB) 3.13.ag_o_ag $12$ (not in LMFDB) 3.13.ag_bj_afc $12$ (not in LMFDB) 3.13.ag_bk_afi $12$ (not in LMFDB) 3.13.ae_aj_dk $12$ (not in LMFDB) 3.13.ae_e_bu $12$ (not in LMFDB) 3.13.ae_m_e $12$ (not in LMFDB) 3.13.ae_bk_ado $12$ (not in LMFDB) 3.13.ad_ab_g $12$ (not in LMFDB) 3.13.ad_l_as $12$ (not in LMFDB) 3.13.ac_t_acy $12$ (not in LMFDB) 3.13.ab_af_cg $12$ (not in LMFDB) 3.13.ab_h_bi $12$ (not in LMFDB) 3.13.b_af_acg $12$ (not in LMFDB) 3.13.b_h_abi $12$ (not in LMFDB) 3.13.c_t_cy $12$ (not in LMFDB) 3.13.d_ab_ag $12$ (not in LMFDB) 3.13.d_l_s $12$ (not in LMFDB) 3.13.e_aj_adk $12$ (not in LMFDB) 3.13.e_e_abu $12$ (not in LMFDB) 3.13.e_m_ae $12$ (not in LMFDB) 3.13.e_bk_do $12$ (not in LMFDB) 3.13.g_ak_afi $12$ (not in LMFDB) 3.13.g_aj_afc $12$ (not in LMFDB) 3.13.g_m_ag $12$ (not in LMFDB) 3.13.g_o_g $12$ (not in LMFDB) 3.13.g_bj_fc $12$ (not in LMFDB) 3.13.g_bk_fi $12$ (not in LMFDB) 3.13.i_e_adi $12$ (not in LMFDB) 3.13.j_bv_gs $12$ (not in LMFDB) 3.13.k_cp_ky $12$ (not in LMFDB) 3.13.l_cd_hu $12$ (not in LMFDB) 3.13.n_dn_pi $12$ (not in LMFDB) 3.13.p_ed_sg $12$ (not in LMFDB) 3.13.q_eu_vu $12$ (not in LMFDB) 3.13.u_gq_bfi $12$ (not in LMFDB)