# Properties

 Label 5.2.ah_ba_acq_fg_aih 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: $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 4 x + 9 x^{2} - 15 x^{3} + 18 x^{4} - 16 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0435981566527$, $\pm0.123548644961$, $\pm0.329312442367$, $\pm0.456881978294$, $\pm0.527830414776$ 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 1349 31996 352089 24621781 1208552912 34419317696 1064624536593 39582728628364 1278470195863379

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 8 8 0 21 74 129 248 575 1153

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 2.2.ad_f $\times$ 3.2.ae_j_ap 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.c_ad_an. 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.ab_ag_bb. 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.an_ag_bcp. The endomorphism algebra for each factor is: 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 3.64.an_ag_bcp : $$\Q(\zeta_{7})$$.
• Endomorphism algebra over $\F_{2^{7}}$  The base change of $A$ to $\F_{2^{7}}$ is 1.128.an 3 $\times$ 2.128.bn_yl. The endomorphism algebra for each factor is: 1.128.an 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.edn 3 $\times$ 2.2097152.a_hpued. The endomorphism algebra for each factor is: 1.2097152.edn 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.b_c_c_ac_ab $2$ (not in LMFDB) 5.2.h_ba_cq_fg_ih $2$ (not in LMFDB) 5.2.ae_i_al_n_ar $3$ (not in LMFDB) 5.2.ab_c_ac_ac_b $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.b_c_c_ac_ab $2$ (not in LMFDB) 5.2.h_ba_cq_fg_ih $2$ (not in LMFDB) 5.2.ae_i_al_n_ar $3$ (not in LMFDB) 5.2.ab_c_ac_ac_b $3$ (not in LMFDB) 5.2.e_i_l_n_r $6$ (not in LMFDB) 5.2.a_ac_c_d_af $7$ (not in LMFDB) 5.2.a_f_af_k_at $7$ (not in LMFDB) 5.2.ae_k_at_bf_abv $12$ (not in LMFDB) 5.2.e_k_t_bf_bv $12$ (not in LMFDB) 5.2.ag_q_aba_bh_abr $14$ (not in LMFDB) 5.2.ag_x_acj_eu_ahp $14$ (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.a_ac_ac_d_f $14$ (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.ad_f_al_t_az $21$ (not in LMFDB) 5.2.a_ab_af_e_f $21$ (not in LMFDB) 5.2.d_b_ae_g_z $21$ (not in LMFDB) 5.2.d_f_b_al_az $21$ (not in LMFDB) 5.2.d_i_k_n_l $21$ (not in LMFDB) 5.2.g_q_ba_bh_br $21$ (not in LMFDB) 5.2.g_x_cj_eu_hp $21$ (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.ad_b_e_g_az $42$ (not in LMFDB) 5.2.ad_f_ab_al_z $42$ (not in LMFDB) 5.2.ad_i_ak_n_al $42$ (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_f_e_af $42$ (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_l_t_z $42$ (not in LMFDB) 5.2.f_n_z_bn_cd $42$ (not in LMFDB) 5.2.ad_d_ac_k_ax $84$ (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_d_c_k_x $84$ (not in LMFDB) 5.2.d_k_q_bf_bl $84$ (not in LMFDB)