Properties

Label 3.13.ap_eg_atb
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:  $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$
Frobenius angles:  $\pm0.0772104791556$, $\pm0.256122854178$, $\pm0.363422825076$
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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 693 4700619 11042583552 23457973758219 51092996893356033 112345575772777611264 247031404823063675979549 542813356177029969244688475 1192548565270092874353721795584 2620003212890448180857709012026739

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 165 2288 28757 370619 4822092 62740103 815749637 10604635184 137858890125

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
The isogeny class factors as 1.13.ah $\times$ 1.13.af $\times$ 1.13.ad 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.afmo $\times$ 1.4826809.atm 2 . The endomorphism algebra for each factor is:
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_bm_aez$2$(not in LMFDB)
3.13.af_k_az$2$(not in LMFDB)
3.13.ab_ac_db$2$(not in LMFDB)
3.13.b_ac_adb$2$(not in LMFDB)
3.13.f_k_z$2$(not in LMFDB)
3.13.j_bm_ez$2$(not in LMFDB)
3.13.p_eg_tb$2$(not in LMFDB)
3.13.am_dc_anq$3$(not in LMFDB)
3.13.ag_bm_aew$3$(not in LMFDB)
3.13.ad_ak_cr$3$(not in LMFDB)
3.13.ad_o_ad$3$(not in LMFDB)
3.13.ad_bj_aco$3$(not in LMFDB)
3.13.a_u_be$3$(not in LMFDB)
3.13.g_ba_ek$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.13.aj_bm_aez$2$(not in LMFDB)
3.13.af_k_az$2$(not in LMFDB)
3.13.ab_ac_db$2$(not in LMFDB)
3.13.b_ac_adb$2$(not in LMFDB)
3.13.f_k_z$2$(not in LMFDB)
3.13.j_bm_ez$2$(not in LMFDB)
3.13.p_eg_tb$2$(not in LMFDB)
3.13.am_dc_anq$3$(not in LMFDB)
3.13.ag_bm_aew$3$(not in LMFDB)
3.13.ad_ak_cr$3$(not in LMFDB)
3.13.ad_o_ad$3$(not in LMFDB)
3.13.ad_bj_aco$3$(not in LMFDB)
3.13.a_u_be$3$(not in LMFDB)
3.13.g_ba_ek$3$(not in LMFDB)
3.13.ar_fa_awr$6$(not in LMFDB)
3.13.an_dq_apx$6$(not in LMFDB)
3.13.al_bu_afj$6$(not in LMFDB)
3.13.ak_cs_ale$6$(not in LMFDB)
3.13.ai_bo_agk$6$(not in LMFDB)
3.13.ah_bi_aed$6$(not in LMFDB)
3.13.ah_cd_ahm$6$(not in LMFDB)
3.13.ag_ba_aek$6$(not in LMFDB)
3.13.ae_bc_acw$6$(not in LMFDB)
3.13.ac_k_adq$6$(not in LMFDB)
3.13.ab_bf_ao$6$(not in LMFDB)
3.13.a_u_abe$6$(not in LMFDB)
3.13.b_ac_adb$6$(not in LMFDB)
3.13.b_bf_o$6$(not in LMFDB)
3.13.c_k_dq$6$(not in LMFDB)
3.13.d_ak_acr$6$(not in LMFDB)
3.13.d_o_d$6$(not in LMFDB)
3.13.d_bj_co$6$(not in LMFDB)
3.13.e_bc_cw$6$(not in LMFDB)
3.13.g_bm_ew$6$(not in LMFDB)
3.13.h_bi_ed$6$(not in LMFDB)
3.13.h_cd_hm$6$(not in LMFDB)
3.13.i_bo_gk$6$(not in LMFDB)
3.13.k_cs_le$6$(not in LMFDB)
3.13.l_bu_fj$6$(not in LMFDB)
3.13.m_dc_nq$6$(not in LMFDB)
3.13.n_dq_px$6$(not in LMFDB)
3.13.r_fa_wr$6$(not in LMFDB)
3.13.ad_aj_co$12$(not in LMFDB)
3.13.ad_m_d$12$(not in LMFDB)
3.13.ad_bk_acr$12$(not in LMFDB)
3.13.d_aj_aco$12$(not in LMFDB)
3.13.d_m_ad$12$(not in LMFDB)
3.13.d_bk_cr$12$(not in LMFDB)