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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 441 3695139 10270374912 23261199311259 51175237809194541 112420114045738942464 247030727708668032881577 542790679774383601869692475 1192546604047795506691655814144 2620019637063803102463380377310139

Point counts of the (virtual) curve

$r$ 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:
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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)