# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 + 2 x^{2} )( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.5$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $2$ Slopes: $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$

## 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 $A(\F_{q^r})$ 3 4275 115596 961875 53309553 1976691600 36322429251 796648921875 31020820875252 1274331545994375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 10 17 18 46 103 136 178 449 1150

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 1.2.a $\times$ 2.2.ad_f 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.2.a : $$\Q(\sqrt{-2})$$. 2.2.ad_f : $$\Q(\sqrt{-3}, \sqrt{5})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 3 $\times$ 1.16777216.mbf 2 . The endomorphism algebra for each factor is: 1.16777216.amdc 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16777216.mbf 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
All geometric endomorphisms are defined over $\F_{2^{24}}$.
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 2 $\times$ 1.4.e $\times$ 2.4.b_ad. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.4.e : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.4.b_ad : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.a $\times$ 1.8.e 2 $\times$ 2.8.a_l. The endomorphism algebra for each factor is: 1.8.a : $$\Q(\sqrt{-2})$$. 1.8.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 2.8.a_l : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i 2 $\times$ 2.16.ah_bh. The endomorphism algebra for each factor is: 1.16.ai : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 2 $\times$ 1.64.l 2 $\times$ 1.64.q. The endomorphism algebra for each factor is: 1.64.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.64.q : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{8}}$  The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 3 $\times$ 2.256.r_bh. The endomorphism algebra for each factor is: 1.256.abg 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.256.r_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$.
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.h 2 $\times$ 1.4096.ey 2 . The endomorphism algebra for each factor is: 1.4096.aey : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.ey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.

## 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_h_am_s_ay $2$ (not in LMFDB) 5.2.ab_d_a_c_i $2$ (not in LMFDB) 5.2.b_d_a_c_ai $2$ (not in LMFDB) 5.2.d_h_m_s_y $2$ (not in LMFDB) 5.2.h_bb_cu_fq_iy $2$ (not in LMFDB) 5.2.ae_j_am_o_aq $3$ (not in LMFDB) 5.2.ab_d_a_ae_i $3$ (not in LMFDB) 5.2.ab_d_a_c_i $3$ (not in LMFDB) 5.2.c_d_g_i_i $3$ (not in LMFDB) 5.2.f_p_bk_cq_ea $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_h_am_s_ay $2$ (not in LMFDB) 5.2.ab_d_a_c_i $2$ (not in LMFDB) 5.2.b_d_a_c_ai $2$ (not in LMFDB) 5.2.d_h_m_s_y $2$ (not in LMFDB) 5.2.h_bb_cu_fq_iy $2$ (not in LMFDB) 5.2.ae_j_am_o_aq $3$ (not in LMFDB) 5.2.ab_d_a_ae_i $3$ (not in LMFDB) 5.2.ab_d_a_c_i $3$ (not in LMFDB) 5.2.c_d_g_i_i $3$ (not in LMFDB) 5.2.f_p_bk_cq_ea $3$ (not in LMFDB) 5.2.af_p_abk_cq_aea $6$ (not in LMFDB) 5.2.ac_d_ag_i_ai $6$ (not in LMFDB) 5.2.a_b_a_g_a $6$ (not in LMFDB) 5.2.b_d_a_ae_ai $6$ (not in LMFDB) 5.2.e_j_m_o_q $6$ (not in LMFDB) 5.2.aj_bp_aes_kc_aqi $8$ (not in LMFDB) 5.2.af_j_ac_aw_bw $8$ (not in LMFDB) 5.2.af_n_aw_be_abo $8$ (not in LMFDB) 5.2.af_r_abq_de_aey $8$ (not in LMFDB) 5.2.ad_d_a_ak_y $8$ (not in LMFDB) 5.2.ad_f_ac_ac_i $8$ (not in LMFDB) 5.2.ad_l_ay_bu_acu $8$ (not in LMFDB) 5.2.ab_ad_c_c_a $8$ (not in LMFDB) 5.2.ab_b_ac_g_ai $8$ (not in LMFDB) 5.2.ab_f_ag_k_aq $8$ (not in LMFDB) 5.2.b_ad_ac_c_a $8$ (not in LMFDB) 5.2.b_b_c_g_i $8$ (not in LMFDB) 5.2.b_f_g_k_q $8$ (not in LMFDB) 5.2.d_d_a_ak_ay $8$ (not in LMFDB) 5.2.d_f_c_ac_ai $8$ (not in LMFDB) 5.2.d_l_y_bu_cu $8$ (not in LMFDB) 5.2.f_j_c_aw_abw $8$ (not in LMFDB) 5.2.f_n_w_be_bo $8$ (not in LMFDB) 5.2.f_r_bq_de_ey $8$ (not in LMFDB) 5.2.j_bp_es_kc_qi $8$ (not in LMFDB) 5.2.ae_l_au_bi_abw $12$ (not in LMFDB) 5.2.ac_f_ak_q_ay $12$ (not in LMFDB) 5.2.a_d_a_k_a $12$ (not in LMFDB) 5.2.c_f_k_q_y $12$ (not in LMFDB) 5.2.e_l_u_bi_bw $12$ (not in LMFDB) 5.2.ah_z_ack_eq_ahg $24$ (not in LMFDB) 5.2.ag_r_aba_w_aq $24$ (not in LMFDB) 5.2.ag_t_abm_cg_adc $24$ (not in LMFDB) 5.2.af_l_am_e_e $24$ (not in LMFDB) 5.2.af_p_abg_ce_adg $24$ (not in LMFDB) 5.2.ae_h_ai_m_au $24$ (not in LMFDB) 5.2.ae_j_aq_bc_abs $24$ (not in LMFDB) 5.2.ad_f_ak_q_au $24$ (not in LMFDB) 5.2.ad_f_ag_e_a $24$ (not in LMFDB) 5.2.ad_f_ac_ai_u $24$ (not in LMFDB) 5.2.ad_j_as_bg_abw $24$ (not in LMFDB) 5.2.ac_ad_k_c_ay $24$ (not in LMFDB) 5.2.ac_ab_g_ac_ai $24$ (not in LMFDB) 5.2.ac_ab_g_e_au $24$ (not in LMFDB) 5.2.ac_b_c_e_am $24$ (not in LMFDB) 5.2.ac_b_c_g_aq $24$ (not in LMFDB) 5.2.ac_d_ac_i_am $24$ (not in LMFDB) 5.2.ac_d_ac_k_aq $24$ (not in LMFDB) 5.2.ac_f_ag_k_ai $24$ (not in LMFDB) 5.2.ac_f_ag_q_au $24$ (not in LMFDB) 5.2.ac_h_ak_w_ay $24$ (not in LMFDB) 5.2.ab_ab_a_e_ae $24$ (not in LMFDB) 5.2.ab_b_ac_a_e $24$ (not in LMFDB) 5.2.ab_d_ae_i_am $24$ (not in LMFDB) 5.2.a_ad_a_c_a $24$ (not in LMFDB) 5.2.a_ab_ae_e_e $24$ (not in LMFDB) 5.2.a_ab_a_ac_a $24$ (not in LMFDB) 5.2.a_ab_a_e_a $24$ (not in LMFDB) 5.2.a_ab_e_e_ae $24$ (not in LMFDB) 5.2.a_b_ae_e_ae $24$ (not in LMFDB) 5.2.a_b_a_e_a $24$ (not in LMFDB) 5.2.a_b_e_e_e $24$ (not in LMFDB) 5.2.a_d_a_i_a $24$ (not in LMFDB) 5.2.a_f_a_k_a $24$ (not in LMFDB) 5.2.a_f_a_q_a $24$ (not in LMFDB) 5.2.a_h_a_w_a $24$ (not in LMFDB) 5.2.b_ab_a_e_e $24$ (not in LMFDB) 5.2.b_b_c_a_ae $24$ (not in LMFDB) 5.2.b_d_e_i_m $24$ (not in LMFDB) 5.2.c_ad_ak_c_y $24$ (not in LMFDB) 5.2.c_ab_ag_ac_i $24$ (not in LMFDB) 5.2.c_ab_ag_e_u $24$ (not in LMFDB) 5.2.c_b_ac_e_m $24$ (not in LMFDB) 5.2.c_b_ac_g_q $24$ (not in LMFDB) 5.2.c_d_c_i_m $24$ (not in LMFDB) 5.2.c_d_c_k_q $24$ (not in LMFDB) 5.2.c_f_g_k_i $24$ (not in LMFDB) 5.2.c_f_g_q_u $24$ (not in LMFDB) 5.2.c_h_k_w_y $24$ (not in LMFDB) 5.2.d_f_c_ai_au $24$ (not in LMFDB) 5.2.d_f_g_e_a $24$ (not in LMFDB) 5.2.d_f_k_q_u $24$ (not in LMFDB) 5.2.d_j_s_bg_bw $24$ (not in LMFDB) 5.2.e_h_i_m_u $24$ (not in LMFDB) 5.2.e_j_q_bc_bs $24$ (not in LMFDB) 5.2.f_l_m_e_ae $24$ (not in LMFDB) 5.2.f_p_bg_ce_dg $24$ (not in LMFDB) 5.2.g_r_ba_w_q $24$ (not in LMFDB) 5.2.g_t_bm_cg_dc $24$ (not in LMFDB) 5.2.h_z_ck_eq_hg $24$ (not in LMFDB)