# Properties

 Label 3.7.ak_cb_agq 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: $( 1 - 2 x + 7 x^{2} )( 1 - 4 x + 7 x^{2} )^{2}$ Frobenius angles: $\pm0.227185525829$, $\pm0.227185525829$, $\pm0.376624142786$ 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 $A(\F_{q^r})$ 96 138240 50083488 14952038400 4812388807776 1629844913249280 558369785712578208 191472530091304550400 65678429853734041509984 22533826216321205439667200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 56 418 2588 17038 117752 823282 5761532 40332766 282406136

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.ae 2 $\times$ 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: 1.7.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.7.ac : $$\Q(\sqrt{-6})$$.
All geometric endomorphisms are defined over $\F_{7}$.

## 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.ag_v_aca $2$ (not in LMFDB) 3.7.ac_f_e $2$ (not in LMFDB) 3.7.c_f_ae $2$ (not in LMFDB) 3.7.g_v_ca $2$ (not in LMFDB) 3.7.k_cb_gq $2$ (not in LMFDB) 3.7.ah_bj_aec $3$ (not in LMFDB) 3.7.ae_ba_acg $3$ (not in LMFDB) 3.7.ab_ab_ba $3$ (not in LMFDB) 3.7.c_i_bm $3$ (not in LMFDB) 3.7.i_ba_ck $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ag_v_aca $2$ (not in LMFDB) 3.7.ac_f_e $2$ (not in LMFDB) 3.7.c_f_ae $2$ (not in LMFDB) 3.7.g_v_ca $2$ (not in LMFDB) 3.7.k_cb_gq $2$ (not in LMFDB) 3.7.ah_bj_aec $3$ (not in LMFDB) 3.7.ae_ba_acg $3$ (not in LMFDB) 3.7.ab_ab_ba $3$ (not in LMFDB) 3.7.c_i_bm $3$ (not in LMFDB) 3.7.i_ba_ck $3$ (not in LMFDB) 3.7.ac_j_ae $4$ (not in LMFDB) 3.7.c_j_e $4$ (not in LMFDB) 3.7.am_co_aik $6$ (not in LMFDB) 3.7.al_ch_ahm $6$ (not in LMFDB) 3.7.ai_ba_ack $6$ (not in LMFDB) 3.7.ai_bm_aes $6$ (not in LMFDB) 3.7.ah_x_acg $6$ (not in LMFDB) 3.7.ag_y_acw $6$ (not in LMFDB) 3.7.af_x_ack $6$ (not in LMFDB) 3.7.ae_o_abu $6$ (not in LMFDB) 3.7.ad_d_ac $6$ (not in LMFDB) 3.7.ad_p_abi $6$ (not in LMFDB) 3.7.ac_ae_w $6$ (not in LMFDB) 3.7.ac_i_abm $6$ (not in LMFDB) 3.7.ac_u_aba $6$ (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.b_ab_aba $6$ (not in LMFDB) 3.7.b_l_w $6$ (not in LMFDB) 3.7.c_ae_aw $6$ (not in LMFDB) 3.7.c_u_ba $6$ (not in LMFDB) 3.7.d_d_c $6$ (not in LMFDB) 3.7.d_p_bi $6$ (not in LMFDB) 3.7.e_o_bu $6$ (not in LMFDB) 3.7.e_ba_cg $6$ (not in LMFDB) 3.7.f_x_ck $6$ (not in LMFDB) 3.7.g_y_cw $6$ (not in LMFDB) 3.7.h_x_cg $6$ (not in LMFDB) 3.7.h_bj_ec $6$ (not in LMFDB) 3.7.i_bm_es $6$ (not in LMFDB) 3.7.l_ch_hm $6$ (not in LMFDB) 3.7.m_co_ik $6$ (not in LMFDB) 3.7.ac_ag_ba $12$ (not in LMFDB) 3.7.ac_s_aw $12$ (not in LMFDB) 3.7.c_ag_aba $12$ (not in LMFDB) 3.7.c_s_w $12$ (not in LMFDB)