# Properties

 Label 5.2.ag_q_aba_bh_abr Base Field $\F_{2}$ Dimension $5$ Ordinary Yes $p$-rank $5$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0992589862044$, $\pm0.123548644961$, $\pm0.186455299510$, $\pm0.456881978294$, $\pm0.757883870938$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $5$ Slopes: $[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1 551 22876 1393479 33501421 2168004272 63138363688 1149554681487 39154418373364 1183443090553781

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 3 21 32 109 207 269 570 1076

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 2.2.ad_f $\times$ 3.2.ad_c_b 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 $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{42}}$ is 1.4398046511104.ahxvrd 3 $\times$ 1.4398046511104.hpued 2 . The endomorphism algebra for each factor is: 1.4398046511104.ahxvrd 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 1.4398046511104.hpued 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{42}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 2.4.b_ad $\times$ 3.4.af_s_abp. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 2.8.a_l $\times$ 3.8.ag_bd_adf. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.l 2 $\times$ 3.64.w_lx_ekt. The endomorphism algebra for each factor is: 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 3.64.w_lx_ekt : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{7}}$  The base change of $A$ to $\F_{2^{7}}$ is 1.128.n 3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is: 1.128.n 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 2.128.bn_yl : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{14}}$  The base change of $A$ to $\F_{2^{14}}$ is 1.16384.dj 3 $\times$ 2.16384.ajr_cqyz. The endomorphism algebra for each factor is: 1.16384.dj 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 2.16384.ajr_cqyz : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{21}}$  The base change of $A$ to $\F_{2^{21}}$ is 1.2097152.aedn 3 $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is: 1.2097152.aedn 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-7})$$$)$ 2.2097152.a_hpued : $$\Q(\sqrt{-3}, \sqrt{5})$$.

## 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 5.2.a_ac_ac_d_f $2$ (not in LMFDB) 5.2.a_ac_c_d_af $2$ (not in LMFDB) 5.2.g_q_ba_bh_br $2$ (not in LMFDB) 5.2.ad_b_e_g_az $3$ (not in LMFDB) 5.2.a_ac_ac_d_f $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.a_ac_ac_d_f $2$ (not in LMFDB) 5.2.a_ac_c_d_af $2$ (not in LMFDB) 5.2.g_q_ba_bh_br $2$ (not in LMFDB) 5.2.ad_b_e_g_az $3$ (not in LMFDB) 5.2.a_ac_ac_d_f $3$ (not in LMFDB) 5.2.d_b_ae_g_z $6$ (not in LMFDB) 5.2.ag_x_acj_eu_ahp $7$ (not in LMFDB) 5.2.b_c_c_ac_ab $7$ (not in LMFDB) 5.2.ad_d_ac_k_ax $12$ (not in LMFDB) 5.2.d_d_c_k_x $12$ (not in LMFDB) 5.2.ah_ba_acq_fg_aih $14$ (not in LMFDB) 5.2.ae_n_abd_cc_adf $14$ (not in LMFDB) 5.2.ac_h_an_y_abl $14$ (not in LMFDB) 5.2.ab_c_ac_ac_b $14$ (not in LMFDB) 5.2.a_f_af_k_at $14$ (not in LMFDB) 5.2.a_f_f_k_t $14$ (not in LMFDB) 5.2.c_h_n_y_bl $14$ (not in LMFDB) 5.2.e_n_bd_cc_df $14$ (not in LMFDB) 5.2.g_x_cj_eu_hp $14$ (not in LMFDB) 5.2.h_ba_cq_fg_ih $14$ (not in LMFDB) 5.2.ad_f_ab_al_z $21$ (not in LMFDB) 5.2.ad_i_ak_n_al $21$ (not in LMFDB) 5.2.a_ab_f_e_af $21$ (not in LMFDB) 5.2.a_f_f_k_t $21$ (not in LMFDB) 5.2.d_f_l_t_z $21$ (not in LMFDB) 5.2.e_i_l_n_r $21$ (not in LMFDB) 5.2.h_ba_cq_fg_ih $21$ (not in LMFDB) 5.2.ae_h_af_ag_t $28$ (not in LMFDB) 5.2.ac_b_ab_a_f $28$ (not in LMFDB) 5.2.c_b_b_a_af $28$ (not in LMFDB) 5.2.e_h_f_ag_at $28$ (not in LMFDB) 5.2.af_n_az_bn_acd $42$ (not in LMFDB) 5.2.ae_i_al_n_ar $42$ (not in LMFDB) 5.2.ad_f_al_t_az $42$ (not in LMFDB) 5.2.ac_b_ab_g_an $42$ (not in LMFDB) 5.2.ab_b_b_ad_h $42$ (not in LMFDB) 5.2.ab_e_ac_j_af $42$ (not in LMFDB) 5.2.a_ab_af_e_f $42$ (not in LMFDB) 5.2.b_b_ab_ad_ah $42$ (not in LMFDB) 5.2.b_e_c_j_f $42$ (not in LMFDB) 5.2.c_b_b_g_n $42$ (not in LMFDB) 5.2.d_f_b_al_az $42$ (not in LMFDB) 5.2.d_i_k_n_l $42$ (not in LMFDB) 5.2.f_n_z_bn_cd $42$ (not in LMFDB) 5.2.ae_k_at_bf_abv $84$ (not in LMFDB) 5.2.ad_k_aq_bf_abl $84$ (not in LMFDB) 5.2.ac_d_af_k_at $84$ (not in LMFDB) 5.2.ab_ac_e_d_al $84$ (not in LMFDB) 5.2.ab_a_c_b_af $84$ (not in LMFDB) 5.2.ab_g_ae_t_al $84$ (not in LMFDB) 5.2.a_b_af_e_af $84$ (not in LMFDB) 5.2.a_b_f_e_f $84$ (not in LMFDB) 5.2.b_ac_ae_d_l $84$ (not in LMFDB) 5.2.b_a_ac_b_f $84$ (not in LMFDB) 5.2.b_g_e_t_l $84$ (not in LMFDB) 5.2.c_d_f_k_t $84$ (not in LMFDB) 5.2.d_k_q_bf_bl $84$ (not in LMFDB) 5.2.e_k_t_bf_bv $84$ (not in LMFDB)