# Properties

 Label 5.2.ag_r_abg_bx_acs 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: $( 1 - 2 x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$ Frobenius angles: $\pm0.0516399385854$, $\pm0.123548644961$, $\pm0.250000000000$, $\pm0.456881978294$, $\pm0.718306605252$ Angle rank: $2$ (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 $A(\F_{q^r})$ 1 665 15808 1107225 29590151 1177379840 45108385589 917031425625 32442795501376 1266342321457325

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 3 19 27 69 165 211 471 1143

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 2.2.ad_f $\times$ 2.2.ab_ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 $\times$ 1.4096.bv 2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is: 1.4096.h 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.4096.bv 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 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.ad_f $\times$ 2.4.b_ad. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af 2 $\times$ 1.8.e $\times$ 2.8.a_l. The endomorphism algebra for each factor is: 1.8.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.8.e : $$\Q(\sqrt{-1})$$. 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.i $\times$ 2.16.ah_bh $\times$ 2.16.b_ap. The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.ah_bh : $$\Q(\sqrt{-3}, \sqrt{5})$$. 2.16.b_ap : $$\Q(\sqrt{-3}, \sqrt{-7})$$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj 2 $\times$ 1.64.a $\times$ 1.64.l 2 . The endomorphism algebra for each factor is: 1.64.aj 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$ 1.64.a : $$\Q(\sqrt{-1})$$. 1.64.l 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$

## 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_h_ac_ap_bi $2$ (not in LMFDB) 5.2.ac_b_ae_j_ak $2$ (not in LMFDB) 5.2.a_ab_ac_b_c $2$ (not in LMFDB) 5.2.a_ab_c_b_ac $2$ (not in LMFDB) 5.2.c_b_e_j_k $2$ (not in LMFDB) 5.2.e_h_c_ap_abi $2$ (not in LMFDB) 5.2.g_r_bg_bx_cs $2$ (not in LMFDB) 5.2.ad_c_b_h_aw $3$ (not in LMFDB) 5.2.ad_i_ar_bf_abu $3$ (not in LMFDB) 5.2.a_ab_ac_b_c $3$ (not in LMFDB) 5.2.a_c_ac_h_c $3$ (not in LMFDB) 5.2.d_i_n_t_ba $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ae_h_ac_ap_bi $2$ (not in LMFDB) 5.2.ac_b_ae_j_ak $2$ (not in LMFDB) 5.2.a_ab_ac_b_c $2$ (not in LMFDB) 5.2.a_ab_c_b_ac $2$ (not in LMFDB) 5.2.c_b_e_j_k $2$ (not in LMFDB) 5.2.e_h_c_ap_abi $2$ (not in LMFDB) 5.2.g_r_bg_bx_cs $2$ (not in LMFDB) 5.2.ad_c_b_h_aw $3$ (not in LMFDB) 5.2.ad_i_ar_bf_abu $3$ (not in LMFDB) 5.2.a_ab_ac_b_c $3$ (not in LMFDB) 5.2.a_c_ac_h_c $3$ (not in LMFDB) 5.2.d_i_n_t_ba $3$ (not in LMFDB) 5.2.ah_bc_acz_gd_aju $6$ (not in LMFDB) 5.2.af_q_abl_cr_aec $6$ (not in LMFDB) 5.2.ae_k_ao_p_ao $6$ (not in LMFDB) 5.2.ad_i_an_t_aba $6$ (not in LMFDB) 5.2.ac_e_ae_j_ak $6$ (not in LMFDB) 5.2.ab_ac_h_d_ao $6$ (not in LMFDB) 5.2.ab_e_af_j_ao $6$ (not in LMFDB) 5.2.ab_e_b_d_k $6$ (not in LMFDB) 5.2.a_c_c_h_ac $6$ (not in LMFDB) 5.2.b_ac_ah_d_o $6$ (not in LMFDB) 5.2.b_e_ab_d_ak $6$ (not in LMFDB) 5.2.b_e_f_j_o $6$ (not in LMFDB) 5.2.c_e_e_j_k $6$ (not in LMFDB) 5.2.d_c_ab_h_w $6$ (not in LMFDB) 5.2.d_i_r_bf_bu $6$ (not in LMFDB) 5.2.e_k_o_p_o $6$ (not in LMFDB) 5.2.f_q_bl_cr_ec $6$ (not in LMFDB) 5.2.h_bc_cz_gd_ju $6$ (not in LMFDB) 5.2.ae_j_as_bd_abo $8$ (not in LMFDB) 5.2.ac_d_a_ah_q $8$ (not in LMFDB) 5.2.c_d_a_ah_aq $8$ (not in LMFDB) 5.2.e_j_s_bd_bo $8$ (not in LMFDB) 5.2.af_k_ah_aj_ba $12$ (not in LMFDB) 5.2.ae_m_aw_bl_aby $12$ (not in LMFDB) 5.2.ad_e_af_n_aba $12$ (not in LMFDB) 5.2.ac_ac_i_d_aw $12$ (not in LMFDB) 5.2.ac_a_e_b_ak $12$ (not in LMFDB) 5.2.ac_g_ai_t_aw $12$ (not in LMFDB) 5.2.ab_ac_b_d_ac $12$ (not in LMFDB) 5.2.ab_a_f_b_ac $12$ (not in LMFDB) 5.2.a_e_ac_n_ac $12$ (not in LMFDB) 5.2.a_e_c_n_c $12$ (not in LMFDB) 5.2.b_ac_ab_d_c $12$ (not in LMFDB) 5.2.b_a_af_b_c $12$ (not in LMFDB) 5.2.c_ac_ai_d_w $12$ (not in LMFDB) 5.2.c_a_ae_b_k $12$ (not in LMFDB) 5.2.c_g_i_t_w $12$ (not in LMFDB) 5.2.d_e_f_n_ba $12$ (not in LMFDB) 5.2.e_m_w_bl_by $12$ (not in LMFDB) 5.2.f_k_h_aj_aba $12$ (not in LMFDB) 5.2.af_s_abt_dl_afk $24$ (not in LMFDB) 5.2.ad_e_ad_ad_m $24$ (not in LMFDB) 5.2.ad_k_av_bn_aci $24$ (not in LMFDB) 5.2.ac_g_ag_l_ai $24$ (not in LMFDB) 5.2.ac_i_ak_z_ay $24$ (not in LMFDB) 5.2.ab_a_ad_f_ae $24$ (not in LMFDB) 5.2.ab_c_af_h_am $24$ (not in LMFDB) 5.2.ab_g_aj_r_abc $24$ (not in LMFDB) 5.2.a_ac_a_d_a $24$ (not in LMFDB) 5.2.a_a_a_b_a $24$ (not in LMFDB) 5.2.a_e_a_j_a $24$ (not in LMFDB) 5.2.a_g_a_t_a $24$ (not in LMFDB) 5.2.b_a_d_f_e $24$ (not in LMFDB) 5.2.b_c_f_h_m $24$ (not in LMFDB) 5.2.b_g_j_r_bc $24$ (not in LMFDB) 5.2.c_g_g_l_i $24$ (not in LMFDB) 5.2.c_i_k_z_y $24$ (not in LMFDB) 5.2.d_e_d_ad_am $24$ (not in LMFDB) 5.2.d_k_v_bn_ci $24$ (not in LMFDB) 5.2.f_s_bt_dl_fk $24$ (not in LMFDB)