# Properties

 Label 3.13.at_gc_abcl 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: $( 1 - 5 x + 13 x^{2} )( 1 - 7 x + 13 x^{2} )^{2}$ Frobenius angles: $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.256122854178$ 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 $A(\F_{q^r})$ 441 3695139 10270374912 23261199311259 51175237809194541 112420114045738942464 247030727708668032881577 542790679774383601869692475 1192546604047795506691655814144 2620019637063803102463380377310139

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 125 2128 28517 371215 4825292 62739931 815715557 10604617744 137859754325

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ah 2 $\times$ 1.13.af 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.13.af : $$\Q(\sqrt{-3})$$.
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 2 $\times$ 1.169.b. The endomorphism algebra for each factor is: 1.169.ax 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.169.b : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{13^{3}}$  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs 2 $\times$ 1.2197.cs. The endomorphism algebra for each factor is: 1.2197.acs 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.2197.cs : $$\Q(\sqrt{-3})$$.

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