# Properties

 Label 6.2.ai_bg_adj_hb_amh_sn Base Field $\F_{2}$ Dimension $6$ Ordinary Yes $p$-rank $6$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0635622003031$, $\pm0.123548644961$, $\pm0.165221137389$, $\pm0.365221137389$, $\pm0.456881978294$, $\pm0.663562200303$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary. $p$-rank: $6$ Slopes: $[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]$

## 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 4009 141436 15190101 1005946931 70310098576 7593444454331 391558651249725 17823209075803324 1220422450820459359

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 5 4 13 30 68 198 341 508 1080

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 2.2.ad_f $\times$ 4.2.af_m_au_bd 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 $\F_{2^{30}}$ is 1.1073741824.acgor 2 $\times$ 2.1073741824.dfu_aggenfjn 2 . The endomorphism algebra for each factor is: 1.1073741824.acgor 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 2.1073741824.dfu_aggenfjn 2 : $\mathrm{M}_{2}($4.0.3625.1$)$
All geometric endomorphisms are defined over $\F_{2^{30}}$.
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 $\times$ 4.4.ab_c_ai_z. 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 $\times$ 4.8.af_d_z_adn. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{5}}$  The base change of $A$ to $\F_{2^{5}}$ is 2.32.ad_bj $\times$ 4.32.a_ad_a_bwv. 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 $\times$ 4.64.at_cz_bur_abapj. The endomorphism algebra for each factor is: 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 4.64.at_cz_bur_abapj : 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{10}}$  The base change of $A$ to $\F_{2^{10}}$ is 2.1024.ad_bwv 2 $\times$ 2.1024.cj_dzt. The endomorphism algebra for each factor is: 2.1024.ad_bwv 2 : $\mathrm{M}_{2}($4.0.3625.1$)$ 2.1024.cj_dzt : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{15}}$  The base change of $A$ to $\F_{2^{15}}$ is 2.32768.a_acgor $\times$ 4.32768.a_dfu_a_aggenfjn. 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 6.2.ac_c_ad_d_ab_b $2$ (not in LMFDB) 6.2.c_c_d_d_b_b $2$ (not in LMFDB) 6.2.i_bg_dj_hb_mh_sn $2$ (not in LMFDB) 6.2.af_l_ap_v_abo_cp $3$ (not in LMFDB) 6.2.ac_c_ad_d_ab_b $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ac_c_ad_d_ab_b $2$ (not in LMFDB) 6.2.c_c_d_d_b_b $2$ (not in LMFDB) 6.2.i_bg_dj_hb_mh_sn $2$ (not in LMFDB) 6.2.af_l_ap_v_abo_cp $3$ (not in LMFDB) 6.2.ac_c_ad_d_ab_b $3$ (not in LMFDB) 6.2.ad_c_d_ac_aj_v $5$ (not in LMFDB) 6.2.ad_h_ar_bc_abs_ct $5$ (not in LMFDB) 6.2.ad_h_ah_ac_ba_abx $5$ (not in LMFDB) 6.2.f_l_p_v_bo_cp $6$ (not in LMFDB) 6.2.d_c_ad_ac_j_v $10$ (not in LMFDB) 6.2.d_h_h_ac_aba_abx $10$ (not in LMFDB) 6.2.d_h_r_bc_bs_ct $10$ (not in LMFDB) 6.2.af_n_az_bt_adc_ev $12$ (not in LMFDB) 6.2.f_n_z_bt_dc_ev $12$ (not in LMFDB) 6.2.a_ae_a_q_a_abh $15$ (not in LMFDB) 6.2.a_b_af_b_af_r $15$ (not in LMFDB) 6.2.a_b_f_b_f_r $15$ (not in LMFDB) 6.2.d_c_ad_ac_j_v $15$ (not in LMFDB) 6.2.d_h_h_ac_aba_abx $15$ (not in LMFDB) 6.2.d_h_r_bc_bs_ct $15$ (not in LMFDB) 6.2.ad_i_ap_bc_abt_cr $20$ (not in LMFDB) 6.2.d_i_p_bc_bt_cr $20$ (not in LMFDB) 6.2.a_ac_a_k_a_ap $60$ (not in LMFDB) 6.2.a_c_a_k_a_p $60$ (not in LMFDB) 6.2.a_d_af_f_ap_p $60$ (not in LMFDB) 6.2.a_d_f_f_p_p $60$ (not in LMFDB) 6.2.a_e_a_q_a_bh $60$ (not in LMFDB)