# Properties

 Label 5.2.ah_z_acj_el_agw Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 50 x^{5} + 52 x^{6} - 40 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.00978468837242$, $\pm0.190215311628$, $\pm0.250000000000$, $\pm0.409784688372$, $\pm0.609784688372$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

## 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 1205 24583 1090525 38101136 918297965 28940657243 1030791493125 27499042761151 885180832006400

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 6 8 18 41 54 108 242 386 781

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 4.2.af_n_az_bn 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 $\F_{2^{20}}$ is 1.1048576.acmj 4 $\times$ 1.1048576.dau. The endomorphism algebra for each factor is: 1.1048576.acmj 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$ 1.1048576.dau : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{20}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 4.4.b_ad_ah_f. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 4.16.ah_bh_aep_lt. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 4.16.ah_bh_aep_lt : $$\Q(\zeta_{15})$$.
• Endomorphism algebra over $\F_{2^{5}}$  The base change of $A$ to $\F_{2^{5}}$ is 1.32.i $\times$ 2.32.a_acj 2 . The endomorphism algebra for each factor is: 1.32.i : $$\Q(\sqrt{-1})$$. 2.32.a_acj 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{10}}$  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acj 4 $\times$ 1.1024.a. The endomorphism algebra for each factor is: 1.1024.acj 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$ 1.1024.a : $$\Q(\sqrt{-1})$$.

## 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.ad_f_aj_p_aw $2$ (not in LMFDB) 5.2.d_f_j_p_w $2$ (not in LMFDB) 5.2.h_z_cj_el_gw $2$ (not in LMFDB) 5.2.ab_e_ae_n_ak $3$ (not in LMFDB) 5.2.c_ac_ah_b_o $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_f_aj_p_aw $2$ (not in LMFDB) 5.2.d_f_j_p_w $2$ (not in LMFDB) 5.2.h_z_cj_el_gw $2$ (not in LMFDB) 5.2.ab_e_ae_n_ak $3$ (not in LMFDB) 5.2.c_ac_ah_b_o $3$ (not in LMFDB) 5.2.ac_a_e_f_as $5$ (not in LMFDB) 5.2.d_f_j_p_w $5$ (not in LMFDB) 5.2.ag_o_aj_abb_cs $6$ (not in LMFDB) 5.2.ad_i_am_v_aba $6$ (not in LMFDB) 5.2.ac_ac_h_b_ao $6$ (not in LMFDB) 5.2.b_e_e_n_k $6$ (not in LMFDB) 5.2.d_i_m_v_ba $6$ (not in LMFDB) 5.2.g_o_j_abb_acs $6$ (not in LMFDB) 5.2.af_p_abj_cn_adw $8$ (not in LMFDB) 5.2.f_p_bj_cn_dw $8$ (not in LMFDB) 5.2.c_a_ae_f_s $10$ (not in LMFDB) 5.2.ai_bh_ado_hj_alu $15$ (not in LMFDB) 5.2.ag_o_aj_abb_cs $15$ (not in LMFDB) 5.2.af_m_ar_r_as $15$ (not in LMFDB) 5.2.ad_i_am_v_aba $15$ (not in LMFDB) 5.2.ac_d_ac_ab_g $15$ (not in LMFDB) 5.2.ab_e_ae_n_ak $15$ (not in LMFDB) 5.2.b_a_b_f_g $15$ (not in LMFDB) 5.2.e_j_q_x_be $15$ (not in LMFDB) 5.2.ac_c_a_h_ao $20$ (not in LMFDB) 5.2.ac_e_ae_n_as $20$ (not in LMFDB) 5.2.c_c_a_h_o $20$ (not in LMFDB) 5.2.c_e_e_n_s $20$ (not in LMFDB) 5.2.ae_g_ab_an_bc $24$ (not in LMFDB) 5.2.ab_g_ae_r_ai $24$ (not in LMFDB) 5.2.b_g_e_r_i $24$ (not in LMFDB) 5.2.e_g_b_an_abc $24$ (not in LMFDB) 5.2.ae_j_aq_x_abe $30$ (not in LMFDB) 5.2.ab_a_ab_f_ag $30$ (not in LMFDB) 5.2.ab_e_ae_n_ak $30$ (not in LMFDB) 5.2.c_d_c_ab_ag $30$ (not in LMFDB) 5.2.f_m_r_r_s $30$ (not in LMFDB) 5.2.i_bh_do_hj_lu $30$ (not in LMFDB) 5.2.ac_c_a_ah_o $40$ (not in LMFDB) 5.2.a_a_a_f_a $40$ (not in LMFDB) 5.2.a_c_a_ah_a $40$ (not in LMFDB) 5.2.a_c_a_h_a $40$ (not in LMFDB) 5.2.a_e_a_n_a $40$ (not in LMFDB) 5.2.c_c_a_ah_ao $40$ (not in LMFDB) 5.2.af_o_abb_br_ack $60$ (not in LMFDB) 5.2.ac_b_c_af_g $60$ (not in LMFDB) 5.2.ab_c_ad_h_ak $60$ (not in LMFDB) 5.2.b_c_d_h_k $60$ (not in LMFDB) 5.2.c_b_ac_af_ag $60$ (not in LMFDB) 5.2.f_o_bb_br_ck $60$ (not in LMFDB) 5.2.ag_v_acc_ed_agm $120$ (not in LMFDB) 5.2.ad_g_aj_l_am $120$ (not in LMFDB) 5.2.ad_i_ap_z_abk $120$ (not in LMFDB) 5.2.a_b_a_af_a $120$ (not in LMFDB) 5.2.a_d_a_ab_a $120$ (not in LMFDB) 5.2.d_g_j_l_m $120$ (not in LMFDB) 5.2.d_i_p_z_bk $120$ (not in LMFDB) 5.2.g_v_cc_ed_gm $120$ (not in LMFDB)