# Properties

 Label 5.2.ah_bb_acv_ft_ajd 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: $( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )^{2}$ Frobenius angles: $\pm0.123548644961$, $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$, $\pm0.456881978294$ 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 $A(\F_{q^r})$ 2 2888 80864 467856 20317462 1868281856 69047017342 1408741551648 35385170776928 1200850669604168

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 10 14 2 16 100 220 322 518 1090

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ab $\times$ 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.ab : $$\Q(\sqrt{-7})$$. 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 $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l 4 . The endomorphism algebra for each factor is: 1.64.aj : $$\Q(\sqrt{-7})$$. 1.64.l 4 : $\mathrm{M}_{4}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 2.4.b_ad 2 . The endomorphism algebra for each factor is: 1.4.d : $$\Q(\sqrt{-7})$$. 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.f $\times$ 2.8.a_l 2 . The endomorphism algebra for each factor is: 1.8.f : $$\Q(\sqrt{-7})$$. 2.8.a_l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3}, \sqrt{5})$$$)$

## 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.af_p_abj_cn_adv $2$ (not in LMFDB) 5.2.ab_d_ab_ab_d $2$ (not in LMFDB) 5.2.b_d_b_ab_ad $2$ (not in LMFDB) 5.2.f_p_bj_cn_dv $2$ (not in LMFDB) 5.2.h_bb_cv_ft_jd $2$ (not in LMFDB) 5.2.ae_j_an_o_ap $3$ (not in LMFDB) 5.2.ab_a_c_f_aj $3$ (not in LMFDB) 5.2.ab_d_ab_ab_d $3$ (not in LMFDB) 5.2.c_d_f_i_j $3$ (not in LMFDB) 5.2.f_p_bj_cn_dv $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.af_p_abj_cn_adv $2$ (not in LMFDB) 5.2.ab_d_ab_ab_d $2$ (not in LMFDB) 5.2.b_d_b_ab_ad $2$ (not in LMFDB) 5.2.f_p_bj_cn_dv $2$ (not in LMFDB) 5.2.h_bb_cv_ft_jd $2$ (not in LMFDB) 5.2.ae_j_an_o_ap $3$ (not in LMFDB) 5.2.ab_a_c_f_aj $3$ (not in LMFDB) 5.2.ab_d_ab_ab_d $3$ (not in LMFDB) 5.2.c_d_f_i_j $3$ (not in LMFDB) 5.2.f_p_bj_cn_dv $3$ (not in LMFDB) 5.2.ab_b_b_af_d $4$ (not in LMFDB) 5.2.b_b_ab_af_ad $4$ (not in LMFDB) 5.2.ac_h_ai_t_ar $5$ (not in LMFDB) 5.2.d_c_ad_ag_ah $5$ (not in LMFDB) 5.2.ac_d_af_i_aj $6$ (not in LMFDB) 5.2.b_a_ac_f_j $6$ (not in LMFDB) 5.2.e_j_n_o_p $6$ (not in LMFDB) 5.2.af_k_af_au_bx $10$ (not in LMFDB) 5.2.ad_c_d_ag_h $10$ (not in LMFDB) 5.2.a_f_a_p_ab $10$ (not in LMFDB) 5.2.a_f_a_p_b $10$ (not in LMFDB) 5.2.c_h_i_t_r $10$ (not in LMFDB) 5.2.f_k_f_au_abx $10$ (not in LMFDB) 5.2.ae_l_av_bi_abx $12$ (not in LMFDB) 5.2.ac_f_aj_q_ax $12$ (not in LMFDB) 5.2.ab_c_a_h_ah $12$ (not in LMFDB) 5.2.ab_e_ac_n_aj $12$ (not in LMFDB) 5.2.b_c_a_h_h $12$ (not in LMFDB) 5.2.b_e_c_n_j $12$ (not in LMFDB) 5.2.c_f_j_q_x $12$ (not in LMFDB) 5.2.e_l_v_bi_bx $12$ (not in LMFDB) 5.2.ag_u_abw_dm_afj $15$ (not in LMFDB) 5.2.af_k_af_au_bx $15$ (not in LMFDB) 5.2.a_f_a_p_ab $15$ (not in LMFDB) 5.2.e_k_w_bo_cj $15$ (not in LMFDB) 5.2.ab_c_a_ah_h $24$ (not in LMFDB) 5.2.b_c_a_ah_ah $24$ (not in LMFDB) 5.2.ae_k_aw_bo_acj $30$ (not in LMFDB) 5.2.g_u_bw_dm_fj $30$ (not in LMFDB)