# Properties

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

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## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$ Frobenius angles: $\pm0.123548644961$, $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.456881978294$ 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 1805 75088 731025 37864361 2168541440 54945865913 1100579337225 32857658578576 1271561917711025

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 7 13 11 35 109 191 259 481 1147

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ad_f 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.2.ac : $$\Q(\sqrt{-1})$$. 2.2.ad_f 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 4 $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.h 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{12}}$.
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 $\times$ 2.4.b_ad 2 . The endomorphism algebra for each factor is: 1.4.a : $$\Q(\sqrt{-1})$$. 2.4.b_ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e $\times$ 2.8.a_l 2 . The endomorphism algebra for each factor is: 1.8.e : $$\Q(\sqrt{-1})$$. 2.8.a_l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.ah_bh 2 . The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.l 4 . The endomorphism algebra for each factor is: 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.l 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$

## 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.ae_j_aq_x_abe $2$ (not in LMFDB) 5.2.ac_d_ac_ab_g $2$ (not in LMFDB) 5.2.c_d_c_ab_ag $2$ (not in LMFDB) 5.2.e_j_q_x_be $2$ (not in LMFDB) 5.2.i_bh_do_hj_lu $2$ (not in LMFDB) 5.2.af_m_ar_r_as $3$ (not in LMFDB) 5.2.ac_a_e_f_as $3$ (not in LMFDB) 5.2.ac_d_ac_ab_g $3$ (not in LMFDB) 5.2.b_a_b_f_g $3$ (not in LMFDB) 5.2.e_j_q_x_be $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ae_j_aq_x_abe $2$ (not in LMFDB) 5.2.ac_d_ac_ab_g $2$ (not in LMFDB) 5.2.c_d_c_ab_ag $2$ (not in LMFDB) 5.2.e_j_q_x_be $2$ (not in LMFDB) 5.2.i_bh_do_hj_lu $2$ (not in LMFDB) 5.2.af_m_ar_r_as $3$ (not in LMFDB) 5.2.ac_a_e_f_as $3$ (not in LMFDB) 5.2.ac_d_ac_ab_g $3$ (not in LMFDB) 5.2.b_a_b_f_g $3$ (not in LMFDB) 5.2.e_j_q_x_be $3$ (not in LMFDB) 5.2.ac_b_c_af_g $4$ (not in LMFDB) 5.2.c_b_ac_af_ag $4$ (not in LMFDB) 5.2.ad_i_am_v_aba $5$ (not in LMFDB) 5.2.c_ac_ah_b_o $5$ (not in LMFDB) 5.2.ab_a_ab_f_ag $6$ (not in LMFDB) 5.2.c_a_ae_f_s $6$ (not in LMFDB) 5.2.f_m_r_r_s $6$ (not in LMFDB) 5.2.ag_v_acc_ed_agm $8$ (not in LMFDB) 5.2.a_b_a_af_a $8$ (not in LMFDB) 5.2.a_d_a_ab_a $8$ (not in LMFDB) 5.2.g_v_cc_ed_gm $8$ (not in LMFDB) 5.2.ag_o_aj_abb_cs $10$ (not in LMFDB) 5.2.ac_ac_h_b_ao $10$ (not in LMFDB) 5.2.ab_e_ae_n_ak $10$ (not in LMFDB) 5.2.b_e_e_n_k $10$ (not in LMFDB) 5.2.d_i_m_v_ba $10$ (not in LMFDB) 5.2.g_o_j_abb_acs $10$ (not in LMFDB) 5.2.af_o_abb_br_ack $12$ (not in LMFDB) 5.2.ac_c_a_h_ao $12$ (not in LMFDB) 5.2.ac_e_ae_n_as $12$ (not in LMFDB) 5.2.ab_c_ad_h_ak $12$ (not in LMFDB) 5.2.b_c_d_h_k $12$ (not in LMFDB) 5.2.c_c_a_h_o $12$ (not in LMFDB) 5.2.c_e_e_n_s $12$ (not in LMFDB) 5.2.f_o_bb_br_ck $12$ (not in LMFDB) 5.2.ah_z_acj_el_agw $15$ (not in LMFDB) 5.2.ag_o_aj_abb_cs $15$ (not in LMFDB) 5.2.ab_e_ae_n_ak $15$ (not in LMFDB) 5.2.d_f_j_p_w $15$ (not in LMFDB) 5.2.ad_g_aj_l_am $24$ (not in LMFDB) 5.2.ad_i_ap_z_abk $24$ (not in LMFDB) 5.2.ac_c_a_ah_o $24$ (not in LMFDB) 5.2.a_a_a_f_a $24$ (not in LMFDB) 5.2.a_c_a_ah_a $24$ (not in LMFDB) 5.2.a_c_a_h_a $24$ (not in LMFDB) 5.2.a_e_a_n_a $24$ (not in LMFDB) 5.2.c_c_a_ah_ao $24$ (not in LMFDB) 5.2.d_g_j_l_m $24$ (not in LMFDB) 5.2.d_i_p_z_bk $24$ (not in LMFDB) 5.2.ad_f_aj_p_aw $30$ (not in LMFDB) 5.2.h_z_cj_el_gw $30$ (not in LMFDB) 5.2.ae_g_ab_an_bc $40$ (not in LMFDB) 5.2.ab_g_ae_r_ai $40$ (not in LMFDB) 5.2.b_g_e_r_i $40$ (not in LMFDB) 5.2.e_g_b_an_abc $40$ (not in LMFDB) 5.2.af_p_abj_cn_adw $120$ (not in LMFDB) 5.2.f_p_bj_cn_dw $120$ (not in LMFDB)