# Properties

 Label 3.7.al_ch_ahm 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: $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )( 1 - 2 x + 7 x^{2} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.227185525829$, $\pm0.376624142786$ 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 $A(\F_{q^r})$ 72 112320 44579808 14251161600 4752812767032 1629844913249280 559565852387925672 191748632567132160000 65719392352139385778464 22538219724640995377745600

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 47 378 2471 16827 117752 825045 5769839 40357926 282461207

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 1.7.ae $\times$ 1.7.ac 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 $\overline{\F}_{7}$
 The base change of $A$ to $\F_{7^{6}}$ is 1.117649.asc $\times$ 1.117649.la 2 . The endomorphism algebra for each factor is: 1.117649.asc : $$\Q(\sqrt{-6})$$. 1.117649.la 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
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 $\times$ 1.49.k. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{7^{3}}$  The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u $\times$ 1.343.bi. 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 3.7.ah_x_acg $2$ (not in LMFDB) 3.7.ad_d_ac $2$ (not in LMFDB) 3.7.ab_ab_ba $2$ (not in LMFDB) 3.7.b_ab_aba $2$ (not in LMFDB) 3.7.d_d_c $2$ (not in LMFDB) 3.7.h_x_cg $2$ (not in LMFDB) 3.7.l_ch_hm $2$ (not in LMFDB) 3.7.ai_bm_aes $3$ (not in LMFDB) 3.7.af_x_ack $3$ (not in LMFDB) 3.7.ac_ae_w $3$ (not in LMFDB) 3.7.ac_f_e $3$ (not in LMFDB) 3.7.ac_u_aba $3$ (not in LMFDB) 3.7.b_l_w $3$ (not in LMFDB) 3.7.e_o_bu $3$ (not in LMFDB) 3.7.h_x_cg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ah_x_acg $2$ (not in LMFDB) 3.7.ad_d_ac $2$ (not in LMFDB) 3.7.ab_ab_ba $2$ (not in LMFDB) 3.7.b_ab_aba $2$ (not in LMFDB) 3.7.d_d_c $2$ (not in LMFDB) 3.7.h_x_cg $2$ (not in LMFDB) 3.7.l_ch_hm $2$ (not in LMFDB) 3.7.ai_bm_aes $3$ (not in LMFDB) 3.7.af_x_ack $3$ (not in LMFDB) 3.7.ac_ae_w $3$ (not in LMFDB) 3.7.ac_f_e $3$ (not in LMFDB) 3.7.ac_u_aba $3$ (not in LMFDB) 3.7.b_l_w $3$ (not in LMFDB) 3.7.e_o_bu $3$ (not in LMFDB) 3.7.h_x_cg $3$ (not in LMFDB) 3.7.am_co_aik $6$ (not in LMFDB) 3.7.ak_cb_agq $6$ (not in LMFDB) 3.7.ai_ba_ack $6$ (not in LMFDB) 3.7.ah_bj_aec $6$ (not in LMFDB) 3.7.ag_v_aca $6$ (not in LMFDB) 3.7.ag_y_acw $6$ (not in LMFDB) 3.7.ae_o_abu $6$ (not in LMFDB) 3.7.ae_ba_acg $6$ (not in LMFDB) 3.7.ad_p_abi $6$ (not in LMFDB) 3.7.ac_i_abm $6$ (not in LMFDB) 3.7.ab_l_aw $6$ (not in LMFDB) 3.7.a_s_ac $6$ (not in LMFDB) 3.7.a_s_c $6$ (not in LMFDB) 3.7.c_ae_aw $6$ (not in LMFDB) 3.7.c_f_ae $6$ (not in LMFDB) 3.7.c_i_bm $6$ (not in LMFDB) 3.7.c_u_ba $6$ (not in LMFDB) 3.7.d_p_bi $6$ (not in LMFDB) 3.7.e_ba_cg $6$ (not in LMFDB) 3.7.f_x_ck $6$ (not in LMFDB) 3.7.g_v_ca $6$ (not in LMFDB) 3.7.g_y_cw $6$ (not in LMFDB) 3.7.h_bj_ec $6$ (not in LMFDB) 3.7.i_ba_ck $6$ (not in LMFDB) 3.7.i_bm_es $6$ (not in LMFDB) 3.7.k_cb_gq $6$ (not in LMFDB) 3.7.m_co_ik $6$ (not in LMFDB) 3.7.ac_ag_ba $12$ (not in LMFDB) 3.7.ac_j_ae $12$ (not in LMFDB) 3.7.ac_s_aw $12$ (not in LMFDB) 3.7.c_ag_aba $12$ (not in LMFDB) 3.7.c_j_e $12$ (not in LMFDB) 3.7.c_s_w $12$ (not in LMFDB)