# Properties

 Label 3.7.an_cz_akc Base Field $\F_{7}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{7}$ Dimension: $3$ L-polynomial: $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )^{2}$ Frobenius angles: $\pm0.106147807505$, $\pm0.227185525829$, $\pm0.227185525829$ 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 $A(\F_{q^r})$ 48 89856 42928704 14821208064 4889951756688 1640360041721856 558709157229483504 191455107206130892800 65691875420472321139392 22538448577696454233249536

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 35 364 2567 17305 118508 823783 5761007 40341028 282464075

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 1.7.ae 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.7.af : $$\Q(\sqrt{-3})$$. 1.7.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{7}$
 The base change of $A$ to $\F_{7^{6}}$ is 1.117649.la 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$
All geometric endomorphisms are defined over $\F_{7^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{7^{2}}$  The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.ac 2 . The endomorphism algebra for each factor is: 1.49.al : $$\Q(\sqrt{-3})$$. 1.49.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{7^{3}}$  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u 2 . The endomorphism algebra for each factor is: 1.343.au : $$\Q(\sqrt{-3})$$. 1.343.u 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

## 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 3.7.af_f_k $2$ (not in LMFDB) 3.7.ad_ad_bm $2$ (not in LMFDB) 3.7.d_ad_abm $2$ (not in LMFDB) 3.7.f_f_ak $2$ (not in LMFDB) 3.7.n_cz_kc $2$ (not in LMFDB) 3.7.ak_by_age $3$ (not in LMFDB) 3.7.ah_bd_ade $3$ (not in LMFDB) 3.7.ah_bg_adz $3$ (not in LMFDB) 3.7.ae_ae_bs $3$ (not in LMFDB) 3.7.ae_f_i $3$ (not in LMFDB) 3.7.ae_u_aca $3$ (not in LMFDB) 3.7.ab_ae_l $3$ (not in LMFDB) 3.7.ab_f_c $3$ (not in LMFDB) 3.7.ab_u_an $3$ (not in LMFDB) 3.7.c_c_i $3$ (not in LMFDB) 3.7.c_o_bg $3$ (not in LMFDB) 3.7.f_ae_acd $3$ (not in LMFDB) 3.7.f_f_ak $3$ (not in LMFDB) 3.7.f_u_cn $3$ (not in LMFDB) 3.7.i_bg_do $3$ (not in LMFDB) 3.7.l_ce_gx $3$ (not in LMFDB) 3.7.o_di_lk $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.af_f_k $2$ (not in LMFDB) 3.7.ad_ad_bm $2$ (not in LMFDB) 3.7.d_ad_abm $2$ (not in LMFDB) 3.7.f_f_ak $2$ (not in LMFDB) 3.7.n_cz_kc $2$ (not in LMFDB) 3.7.ak_by_age $3$ (not in LMFDB) 3.7.ah_bd_ade $3$ (not in LMFDB) 3.7.ah_bg_adz $3$ (not in LMFDB) 3.7.ae_ae_bs $3$ (not in LMFDB) 3.7.ae_f_i $3$ (not in LMFDB) 3.7.ae_u_aca $3$ (not in LMFDB) 3.7.ab_ae_l $3$ (not in LMFDB) 3.7.ab_f_c $3$ (not in LMFDB) 3.7.ab_u_an $3$ (not in LMFDB) 3.7.c_c_i $3$ (not in LMFDB) 3.7.c_o_bg $3$ (not in LMFDB) 3.7.f_ae_acd $3$ (not in LMFDB) 3.7.f_f_ak $3$ (not in LMFDB) 3.7.f_u_cn $3$ (not in LMFDB) 3.7.i_bg_do $3$ (not in LMFDB) 3.7.l_ce_gx $3$ (not in LMFDB) 3.7.o_di_lk $3$ (not in LMFDB) 3.7.af_j_ak $4$ (not in LMFDB) 3.7.f_j_k $4$ (not in LMFDB) 3.7.ap_ds_amx $6$ (not in LMFDB) 3.7.ao_di_alk $6$ (not in LMFDB) 3.7.am_cr_aiy $6$ (not in LMFDB) 3.7.al_ce_agx $6$ (not in LMFDB) 3.7.aj_bk_adx $6$ (not in LMFDB) 3.7.aj_bt_afm $6$ (not in LMFDB) 3.7.ai_bg_ado $6$ (not in LMFDB) 3.7.ag_g_q $6$ (not in LMFDB) 3.7.ag_be_adk $6$ (not in LMFDB) 3.7.af_ae_cd $6$ (not in LMFDB) 3.7.af_u_acn $6$ (not in LMFDB) 3.7.ad_m_abv $6$ (not in LMFDB) 3.7.ad_y_abr $6$ (not in LMFDB) 3.7.ac_c_ai $6$ (not in LMFDB) 3.7.ac_o_abg $6$ (not in LMFDB) 3.7.a_a_au $6$ (not in LMFDB) 3.7.a_a_u $6$ (not in LMFDB) 3.7.b_ae_al $6$ (not in LMFDB) 3.7.b_f_ac $6$ (not in LMFDB) 3.7.b_u_n $6$ (not in LMFDB) 3.7.d_m_bv $6$ (not in LMFDB) 3.7.d_y_br $6$ (not in LMFDB) 3.7.e_ae_abs $6$ (not in LMFDB) 3.7.e_f_ai $6$ (not in LMFDB) 3.7.e_u_ca $6$ (not in LMFDB) 3.7.g_g_aq $6$ (not in LMFDB) 3.7.g_be_dk $6$ (not in LMFDB) 3.7.h_bd_de $6$ (not in LMFDB) 3.7.h_bg_dz $6$ (not in LMFDB) 3.7.j_bk_dx $6$ (not in LMFDB) 3.7.j_bt_fm $6$ (not in LMFDB) 3.7.k_by_ge $6$ (not in LMFDB) 3.7.m_cr_iy $6$ (not in LMFDB) 3.7.p_ds_mx $6$ (not in LMFDB) 3.7.af_ag_cn $12$ (not in LMFDB) 3.7.af_s_acd $12$ (not in LMFDB) 3.7.ae_ag_ca $12$ (not in LMFDB) 3.7.ae_j_ai $12$ (not in LMFDB) 3.7.ae_s_abs $12$ (not in LMFDB) 3.7.ab_ag_n $12$ (not in LMFDB) 3.7.ab_j_ac $12$ (not in LMFDB) 3.7.ab_s_al $12$ (not in LMFDB) 3.7.b_ag_an $12$ (not in LMFDB) 3.7.b_j_c $12$ (not in LMFDB) 3.7.b_s_l $12$ (not in LMFDB) 3.7.e_ag_aca $12$ (not in LMFDB) 3.7.e_j_i $12$ (not in LMFDB) 3.7.e_s_bs $12$ (not in LMFDB) 3.7.f_ag_acn $12$ (not in LMFDB) 3.7.f_s_cd $12$ (not in LMFDB) 3.7.a_a_abl $18$ (not in LMFDB) 3.7.a_a_ar $18$ (not in LMFDB) 3.7.a_a_r $18$ (not in LMFDB) 3.7.a_a_bl $18$ (not in LMFDB)