# Properties

 Label 3.13.ar_fe_axt 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: $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.0772104791556$, $\pm0.256122854178$, $\pm0.256122854178$ Angle rank: $1$ (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 $A(\F_{q^r})$ 567 4298427 10946057472 23694103094571 51341099522009787 112420114045738942464 246973196605307985010863 542739386390369424398926875 1192519981354751996352349595904 2620011687322981914904984277550147

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 149 2268 29045 372417 4825292 62725317 815638469 10604381004 137859336029

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah $\times$ 1.13.af 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.13.ah : $$\Q(\sqrt{-3})$$. 1.13.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{13}$
 The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$
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 2 . The endomorphism algebra for each factor is: 1.169.ax : $$\Q(\sqrt{-3})$$. 1.169.b 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{13^{3}}$  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs 2 . The endomorphism algebra for each factor is: 1.2197.acs : $$\Q(\sqrt{-3})$$. 1.2197.cs 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

## 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.ah_o_ah $2$ (not in LMFDB) 3.13.ad_ag_dt $2$ (not in LMFDB) 3.13.d_ag_adt $2$ (not in LMFDB) 3.13.h_o_h $2$ (not in LMFDB) 3.13.r_fe_xt $2$ (not in LMFDB) 3.13.ao_du_aqs $3$ (not in LMFDB) 3.13.al_ct_amc $3$ (not in LMFDB) 3.13.ai_bs_agc $3$ (not in LMFDB) 3.13.af_ak_el $3$ (not in LMFDB) 3.13.af_o_af $3$ (not in LMFDB) 3.13.af_bj_aeg $3$ (not in LMFDB) 3.13.ac_ak_bu $3$ (not in LMFDB) 3.13.ac_o_ac $3$ (not in LMFDB) 3.13.ac_bj_abs $3$ (not in LMFDB) 3.13.b_x_bu $3$ (not in LMFDB) 3.13.e_i_bi $3$ (not in LMFDB) 3.13.h_ak_agf $3$ (not in LMFDB) 3.13.h_bj_fy $3$ (not in LMFDB) 3.13.k_by_hi $3$ (not in LMFDB) 3.13.q_em_tu $3$ (not in LMFDB) 3.13.t_gc_bcl $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ah_o_ah $2$ (not in LMFDB) 3.13.ad_ag_dt $2$ (not in LMFDB) 3.13.d_ag_adt $2$ (not in LMFDB) 3.13.h_o_h $2$ (not in LMFDB) 3.13.r_fe_xt $2$ (not in LMFDB) 3.13.ao_du_aqs $3$ (not in LMFDB) 3.13.al_ct_amc $3$ (not in LMFDB) 3.13.ai_bs_agc $3$ (not in LMFDB) 3.13.af_ak_el $3$ (not in LMFDB) 3.13.af_o_af $3$ (not in LMFDB) 3.13.af_bj_aeg $3$ (not in LMFDB) 3.13.ac_ak_bu $3$ (not in LMFDB) 3.13.ac_o_ac $3$ (not in LMFDB) 3.13.ac_bj_abs $3$ (not in LMFDB) 3.13.b_x_bu $3$ (not in LMFDB) 3.13.e_i_bi $3$ (not in LMFDB) 3.13.h_ak_agf $3$ (not in LMFDB) 3.13.h_bj_fy $3$ (not in LMFDB) 3.13.k_by_hi $3$ (not in LMFDB) 3.13.q_em_tu $3$ (not in LMFDB) 3.13.t_gc_bcl $3$ (not in LMFDB) 3.13.ah_m_h $4$ (not in LMFDB) 3.13.h_m_ah $4$ (not in LMFDB) 3.13.av_he_abif $6$ (not in LMFDB) 3.13.at_gc_abcl $6$ (not in LMFDB) 3.13.aq_em_atu $6$ (not in LMFDB) 3.13.ap_ek_atv $6$ (not in LMFDB) 3.13.am_ci_aig $6$ (not in LMFDB) 3.13.am_dg_any $6$ (not in LMFDB) 3.13.ak_by_ahi $6$ (not in LMFDB) 3.13.aj_s_l $6$ (not in LMFDB) 3.13.aj_cl_aju $6$ (not in LMFDB) 3.13.ah_ak_gf $6$ (not in LMFDB) 3.13.ah_bj_afy $6$ (not in LMFDB) 3.13.ag_bz_agi $6$ (not in LMFDB) 3.13.ae_i_abi $6$ (not in LMFDB) 3.13.ad_ag_dt $6$ (not in LMFDB) 3.13.ad_p_aec $6$ (not in LMFDB) 3.13.ab_x_abu $6$ (not in LMFDB) 3.13.a_a_acs $6$ (not in LMFDB) 3.13.a_a_cs $6$ (not in LMFDB) 3.13.c_ak_abu $6$ (not in LMFDB) 3.13.c_o_c $6$ (not in LMFDB) 3.13.c_bj_bs $6$ (not in LMFDB) 3.13.d_p_ec $6$ (not in LMFDB) 3.13.f_ak_ael $6$ (not in LMFDB) 3.13.f_o_f $6$ (not in LMFDB) 3.13.f_bj_eg $6$ (not in LMFDB) 3.13.g_bz_gi $6$ (not in LMFDB) 3.13.i_bs_gc $6$ (not in LMFDB) 3.13.j_s_al $6$ (not in LMFDB) 3.13.j_cl_ju $6$ (not in LMFDB) 3.13.l_ct_mc $6$ (not in LMFDB) 3.13.m_ci_ig $6$ (not in LMFDB) 3.13.m_dg_ny $6$ (not in LMFDB) 3.13.o_du_qs $6$ (not in LMFDB) 3.13.p_ek_tv $6$ (not in LMFDB) 3.13.v_he_bif $6$ (not in LMFDB) 3.13.ah_aj_fy $12$ (not in LMFDB) 3.13.ah_bk_agf $12$ (not in LMFDB) 3.13.af_aj_eg $12$ (not in LMFDB) 3.13.af_m_f $12$ (not in LMFDB) 3.13.af_bk_ael $12$ (not in LMFDB) 3.13.ac_aj_bs $12$ (not in LMFDB) 3.13.ac_m_c $12$ (not in LMFDB) 3.13.ac_bk_abu $12$ (not in LMFDB) 3.13.c_aj_abs $12$ (not in LMFDB) 3.13.c_m_ac $12$ (not in LMFDB) 3.13.c_bk_bu $12$ (not in LMFDB) 3.13.f_aj_aeg $12$ (not in LMFDB) 3.13.f_m_af $12$ (not in LMFDB) 3.13.f_bk_el $12$ (not in LMFDB) 3.13.h_aj_afy $12$ (not in LMFDB) 3.13.h_bk_gf $12$ (not in LMFDB) 3.13.a_a_adl $18$ (not in LMFDB) 3.13.a_a_at $18$ (not in LMFDB) 3.13.a_a_t $18$ (not in LMFDB) 3.13.a_a_dl $18$ (not in LMFDB)