# 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

## 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.

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

 $r$ 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: 1.4826809.afmo : $$\Q(\sqrt{-43})$$. 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.r. 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.dm. 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.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.
 Twist Extension Degree Common 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)