# Properties

 Label 5.2.ah_z_acg_dy_afw 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: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - x + x^{2} - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.197201053961$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.652365995579$ 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 $A(\F_{q^r})$ 3 3375 79092 7171875 124264563 1067742000 26487260229 799929421875 25107150425292 1065956954484375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 6 17 50 76 63 94 178 341 966

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 2.2.ab_b 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_b : 4.0.2873.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 3 $\times$ 2.16.j_bx. 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.j_bx : 4.0.2873.1.
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.b_f. The endomorphism algebra for each factor is: 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 2.4.b_f : 4.0.2873.1.

## 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_n_as_o_ai $2$ (not in LMFDB) 5.2.ad_f_ag_o_ay $2$ (not in LMFDB) 5.2.ab_b_c_g_ai $2$ (not in LMFDB) 5.2.b_b_ac_g_i $2$ (not in LMFDB) 5.2.d_f_g_o_y $2$ (not in LMFDB) 5.2.f_n_s_o_i $2$ (not in LMFDB) 5.2.h_z_cg_dy_fw $2$ (not in LMFDB) 5.2.ab_b_c_a_e $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.af_n_as_o_ai $2$ (not in LMFDB) 5.2.ad_f_ag_o_ay $2$ (not in LMFDB) 5.2.ab_b_c_g_ai $2$ (not in LMFDB) 5.2.b_b_ac_g_i $2$ (not in LMFDB) 5.2.d_f_g_o_y $2$ (not in LMFDB) 5.2.f_n_s_o_i $2$ (not in LMFDB) 5.2.h_z_cg_dy_fw $2$ (not in LMFDB) 5.2.ab_b_c_a_e $3$ (not in LMFDB) 5.2.af_n_aba_bw_acy $6$ (not in LMFDB) 5.2.ad_f_ag_i_am $6$ (not in LMFDB) 5.2.ab_b_ag_i_ae $6$ (not in LMFDB) 5.2.b_b_ac_a_ae $6$ (not in LMFDB) 5.2.b_b_g_i_e $6$ (not in LMFDB) 5.2.d_f_g_i_m $6$ (not in LMFDB) 5.2.f_n_ba_bw_cy $6$ (not in LMFDB) 5.2.af_p_abg_cg_adk $8$ (not in LMFDB) 5.2.ad_b_g_ag_a $8$ (not in LMFDB) 5.2.ad_h_ai_k_ai $8$ (not in LMFDB) 5.2.ad_j_as_bi_abw $8$ (not in LMFDB) 5.2.ab_ad_g_c_aq $8$ (not in LMFDB) 5.2.ab_ab_a_ac_i $8$ (not in LMFDB) 5.2.ab_d_ae_k_ai $8$ (not in LMFDB) 5.2.ab_f_ac_k_a $8$ (not in LMFDB) 5.2.ab_h_ai_w_ay $8$ (not in LMFDB) 5.2.b_ad_ag_c_q $8$ (not in LMFDB) 5.2.b_ab_a_ac_ai $8$ (not in LMFDB) 5.2.b_d_e_k_i $8$ (not in LMFDB) 5.2.b_f_c_k_a $8$ (not in LMFDB) 5.2.b_h_i_w_y $8$ (not in LMFDB) 5.2.d_b_ag_ag_a $8$ (not in LMFDB) 5.2.d_h_i_k_i $8$ (not in LMFDB) 5.2.d_j_s_bi_bw $8$ (not in LMFDB) 5.2.f_p_bg_cg_dk $8$ (not in LMFDB) 5.2.ad_d_a_e_am $24$ (not in LMFDB) 5.2.ad_h_aq_bc_abo $24$ (not in LMFDB) 5.2.ad_h_am_y_abk $24$ (not in LMFDB) 5.2.ab_ab_e_e_am $24$ (not in LMFDB) 5.2.ab_b_ac_e_a $24$ (not in LMFDB) 5.2.ab_d_ae_e_ai $24$ (not in LMFDB) 5.2.ab_d_a_i_ae $24$ (not in LMFDB) 5.2.ab_f_ag_q_aq $24$ (not in LMFDB) 5.2.b_ab_ae_e_m $24$ (not in LMFDB) 5.2.b_b_c_e_a $24$ (not in LMFDB) 5.2.b_d_a_i_e $24$ (not in LMFDB) 5.2.b_d_e_e_i $24$ (not in LMFDB) 5.2.b_f_g_q_q $24$ (not in LMFDB) 5.2.d_d_a_e_m $24$ (not in LMFDB) 5.2.d_h_m_y_bk $24$ (not in LMFDB) 5.2.d_h_q_bc_bo $24$ (not in LMFDB)