# Properties

 Label 3.13.as_fr_abai 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: $( 1 - 6 x + 13 x^{2} )^{3}$ Frobenius angles: $\pm0.187167041811$, $\pm0.187167041811$, $\pm0.187167041811$ 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 $A(\F_{q^r})$ 512 4096000 10882013696 23887872000000 51681708060918272 112740186517737472000 247169660119103650167296 542801726578599395328000000 1192495236320772680785653842432 2619956699878396329300777472000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 140 2252 29276 374876 4839020 62775212 815732156 10604160956 137856442700

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The isogeny class factors as 1.13.ag 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$
All geometric endomorphisms are defined over $\F_{13}$.

## 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.ag_d_ci $2$ (not in LMFDB) 3.13.g_d_aci $2$ (not in LMFDB) 3.13.s_fr_bai $2$ (not in LMFDB) 3.13.a_a_s $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.13.ag_d_ci $2$ (not in LMFDB) 3.13.g_d_aci $2$ (not in LMFDB) 3.13.s_fr_bai $2$ (not in LMFDB) 3.13.a_a_s $3$ (not in LMFDB) 3.13.aq_et_avo $4$ (not in LMFDB) 3.13.ao_dz_ars $4$ (not in LMFDB) 3.13.am_dj_aom $4$ (not in LMFDB) 3.13.ai_bb_acm $4$ (not in LMFDB) 3.13.ag_x_aci $4$ (not in LMFDB) 3.13.ae_d_bo $4$ (not in LMFDB) 3.13.ae_x_abo $4$ (not in LMFDB) 3.13.ac_h_bs $4$ (not in LMFDB) 3.13.c_h_abs $4$ (not in LMFDB) 3.13.e_d_abo $4$ (not in LMFDB) 3.13.e_x_bo $4$ (not in LMFDB) 3.13.g_x_ci $4$ (not in LMFDB) 3.13.i_bb_cm $4$ (not in LMFDB) 3.13.m_dj_om $4$ (not in LMFDB) 3.13.o_dz_rs $4$ (not in LMFDB) 3.13.q_et_vo $4$ (not in LMFDB) 3.13.am_cu_ali $6$ (not in LMFDB) 3.13.a_a_as $6$ (not in LMFDB) 3.13.m_cu_li $6$ (not in LMFDB) 3.13.ag_al_fo $8$ (not in LMFDB) 3.13.ag_bl_afo $8$ (not in LMFDB) 3.13.ae_al_ds $8$ (not in LMFDB) 3.13.ae_bl_ads $8$ (not in LMFDB) 3.13.e_al_ads $8$ (not in LMFDB) 3.13.e_bl_ds $8$ (not in LMFDB) 3.13.g_al_afo $8$ (not in LMFDB) 3.13.g_bl_fo $8$ (not in LMFDB) 3.13.ak_bo_aes $12$ (not in LMFDB) 3.13.ak_ci_ajo $12$ (not in LMFDB) 3.13.ai_bg_aem $12$ (not in LMFDB) 3.13.ac_ai_di $12$ (not in LMFDB) 3.13.ac_m_acm $12$ (not in LMFDB) 3.13.a_a_ado $12$ (not in LMFDB) 3.13.a_a_do $12$ (not in LMFDB) 3.13.c_ai_adi $12$ (not in LMFDB) 3.13.c_m_cm $12$ (not in LMFDB) 3.13.i_bg_em $12$ (not in LMFDB) 3.13.k_bo_es $12$ (not in LMFDB) 3.13.k_ci_jo $12$ (not in LMFDB)