# Properties

 Label 5.2.ah_bb_acs_fi_aii 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: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - x + 3 x^{2} - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.306143893905$, $\pm0.570118980449$ Angle rank: $2$ (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 $A(\F_{q^r})$ 5 6875 175760 4296875 94766375 966680000 16672674835 754913671875 33661378209680 1156445919921875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 10 23 42 66 55 38 162 491 1050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 2.2.ab_d 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 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 2.2.ab_d : 4.0.1025.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 2.16.b_b. The endomorphism algebra for each factor is: 1.16.i 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.b_b : 4.0.1025.1.
All geometric endomorphisms are defined over $\F_{2^{4}}$.
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 3 $\times$ 2.4.f_n. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 2.4.f_n : 4.0.1025.1.

## 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.af_p_abe_by_acu $2$ (not in LMFDB) 5.2.ad_h_ak_s_ay $2$ (not in LMFDB) 5.2.ab_d_ac_k_ai $2$ (not in LMFDB) 5.2.b_d_c_k_i $2$ (not in LMFDB) 5.2.d_h_k_s_y $2$ (not in LMFDB) 5.2.f_p_be_by_cu $2$ (not in LMFDB) 5.2.h_bb_cs_fi_ii $2$ (not in LMFDB) 5.2.ab_d_c_a_m $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.af_p_abe_by_acu $2$ (not in LMFDB) 5.2.ad_h_ak_s_ay $2$ (not in LMFDB) 5.2.ab_d_ac_k_ai $2$ (not in LMFDB) 5.2.b_d_c_k_i $2$ (not in LMFDB) 5.2.d_h_k_s_y $2$ (not in LMFDB) 5.2.f_p_be_by_cu $2$ (not in LMFDB) 5.2.h_bb_cs_fi_ii $2$ (not in LMFDB) 5.2.ab_d_c_a_m $3$ (not in LMFDB) 5.2.af_p_abi_cm_adw $6$ (not in LMFDB) 5.2.ad_h_ao_y_abk $6$ (not in LMFDB) 5.2.ab_d_ag_i_am $6$ (not in LMFDB) 5.2.b_d_ac_a_am $6$ (not in LMFDB) 5.2.b_d_g_i_m $6$ (not in LMFDB) 5.2.d_h_o_y_bk $6$ (not in LMFDB) 5.2.f_p_bi_cm_dw $6$ (not in LMFDB) 5.2.af_r_abo_da_aeq $8$ (not in LMFDB) 5.2.ad_d_c_ak_q $8$ (not in LMFDB) 5.2.ad_j_aq_be_abo $8$ (not in LMFDB) 5.2.ad_l_aw_bu_acm $8$ (not in LMFDB) 5.2.ab_ab_c_ac_a $8$ (not in LMFDB) 5.2.ab_b_a_ag_i $8$ (not in LMFDB) 5.2.ab_f_ae_o_ai $8$ (not in LMFDB) 5.2.ab_h_ag_w_aq $8$ (not in LMFDB) 5.2.ab_j_ai_bi_ay $8$ (not in LMFDB) 5.2.b_ab_ac_ac_a $8$ (not in LMFDB) 5.2.b_b_a_ag_ai $8$ (not in LMFDB) 5.2.b_f_e_o_i $8$ (not in LMFDB) 5.2.b_h_g_w_q $8$ (not in LMFDB) 5.2.b_j_i_bi_y $8$ (not in LMFDB) 5.2.d_d_ac_ak_aq $8$ (not in LMFDB) 5.2.d_j_q_be_bo $8$ (not in LMFDB) 5.2.d_l_w_bu_cm $8$ (not in LMFDB) 5.2.f_r_bo_da_eq $8$ (not in LMFDB) 5.2.af_p_abi_cm_adw $12$ (not in LMFDB) 5.2.ad_f_ae_e_ae $24$ (not in LMFDB) 5.2.ad_j_au_bk_ace $24$ (not in LMFDB) 5.2.ad_j_aq_bg_abs $24$ (not in LMFDB) 5.2.ab_b_a_e_ae $24$ (not in LMFDB) 5.2.ab_d_ac_e_a $24$ (not in LMFDB) 5.2.ab_f_ai_m_ay $24$ (not in LMFDB) 5.2.ab_f_ae_q_am $24$ (not in LMFDB) 5.2.ab_h_ag_y_aq $24$ (not in LMFDB) 5.2.b_b_a_e_e $24$ (not in LMFDB) 5.2.b_d_c_e_a $24$ (not in LMFDB) 5.2.b_f_e_q_m $24$ (not in LMFDB) 5.2.b_f_i_m_y $24$ (not in LMFDB) 5.2.b_h_g_y_q $24$ (not in LMFDB) 5.2.d_f_e_e_e $24$ (not in LMFDB) 5.2.d_j_q_bg_bs $24$ (not in LMFDB) 5.2.d_j_u_bk_ce $24$ (not in LMFDB)