# Properties

 Label 5.2.ag_q_au_e_q Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{3}( 1 - 2 x^{2} + 4 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.833333333333$ Angle rank: $0$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular.

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

## 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 1125 177957 6890625 68438553 1801814625 23457176529 848931890625 29286667687329 1061866923890625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 21 49 57 97 81 193 417 961

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 3 $\times$ 2.2.a_ac 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.a_ac : $$\Q(\sqrt{-2}, \sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 5 and its endomorphism algebra is $\mathrm{M}_{5}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
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.ac 2 $\times$ 1.4.a 3 . The endomorphism algebra for each factor is: 1.4.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.4.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$
• Endomorphism algebra over $\F_{2^{3}}$  The base change of $A$ to $\F_{2^{3}}$ is 1.8.a 2 $\times$ 1.8.e 3 . The endomorphism algebra for each factor is: 1.8.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.8.e 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.e 2 $\times$ 1.16.i 3 . The endomorphism algebra for each factor is: 1.16.e 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.16.i 3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{6}}$  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a 3 $\times$ 1.64.q 2 . The endomorphism algebra for each factor is: 1.64.a 3 : $\mathrm{M}_{3}($$$\Q(\sqrt{-1})$$$)$ 1.64.q 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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 $\times$ 1.256.q 2 . 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 $\infty$. 1.256.q 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{2^{12}}$  The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey 2 $\times$ 1.4096.ey 3 . The endomorphism algebra for each factor is: 1.4096.aey 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.4096.ey 3 : $\mathrm{M}_{3}(B)$, 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.ac_a_e_e_aq $2$ (not in LMFDB) 5.2.c_a_ae_e_q $2$ (not in LMFDB) 5.2.g_q_u_e_aq $2$ (not in LMFDB) 5.2.ag_w_ace_ei_agu $3$ (not in LMFDB) 5.2.a_ac_e_e_ai $3$ (not in LMFDB) 5.2.a_e_e_e_q $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_a_e_e_aq $2$ (not in LMFDB) 5.2.c_a_ae_e_q $2$ (not in LMFDB) 5.2.g_q_u_e_aq $2$ (not in LMFDB) 5.2.ag_w_ace_ei_agu $3$ (not in LMFDB) 5.2.a_ac_e_e_ai $3$ (not in LMFDB) 5.2.a_e_e_e_q $3$ (not in LMFDB) 5.2.ag_u_abs_cy_aei $4$ (not in LMFDB) 5.2.ac_e_ae_m_aq $4$ (not in LMFDB) 5.2.c_e_e_m_q $4$ (not in LMFDB) 5.2.g_u_bs_cy_ei $4$ (not in LMFDB) 5.2.ae_g_ae_e_ai $6$ (not in LMFDB) 5.2.ae_m_abc_ca_adc $6$ (not in LMFDB) 5.2.ac_g_ai_q_aq $6$ (not in LMFDB) 5.2.a_ac_ae_e_i $6$ (not in LMFDB) 5.2.a_e_ae_e_aq $6$ (not in LMFDB) 5.2.c_g_i_q_q $6$ (not in LMFDB) 5.2.e_g_e_e_i $6$ (not in LMFDB) 5.2.e_m_bc_ca_dc $6$ (not in LMFDB) 5.2.g_w_ce_ei_gu $6$ (not in LMFDB) 5.2.ai_bg_adg_gi_ajw $8$ (not in LMFDB) 5.2.ag_u_abw_do_afo $8$ (not in LMFDB) 5.2.ae_e_e_am_q $8$ (not in LMFDB) 5.2.ae_i_am_u_abg $8$ (not in LMFDB) 5.2.ae_i_ai_e_a $8$ (not in LMFDB) 5.2.ae_i_ae_am_bg $8$ (not in LMFDB) 5.2.ae_m_ay_bs_acm $8$ (not in LMFDB) 5.2.ac_ae_m_e_abg $8$ (not in LMFDB) 5.2.ac_a_a_ae_q $8$ (not in LMFDB) 5.2.ac_a_e_ae_a $8$ (not in LMFDB) 5.2.ac_e_ai_m_aq $8$ (not in LMFDB) 5.2.ac_e_ae_e_a $8$ (not in LMFDB) 5.2.ac_e_a_ae_q $8$ (not in LMFDB) 5.2.ac_i_aq_bc_abw $8$ (not in LMFDB) 5.2.ac_i_am_bc_abg $8$ (not in LMFDB) 5.2.a_ae_ae_e_q $8$ (not in LMFDB) 5.2.a_ae_a_e_a $8$ (not in LMFDB) 5.2.a_ae_e_e_aq $8$ (not in LMFDB) 5.2.a_a_ae_e_a $8$ (not in LMFDB) 5.2.a_a_a_ae_a $8$ (not in LMFDB) 5.2.a_a_a_e_a $8$ (not in LMFDB) 5.2.a_a_e_e_a $8$ (not in LMFDB) 5.2.a_e_a_e_a $8$ (not in LMFDB) 5.2.a_e_a_m_a $8$ (not in LMFDB) 5.2.a_i_a_bc_a $8$ (not in LMFDB) 5.2.c_ae_am_e_bg $8$ (not in LMFDB) 5.2.c_a_ae_ae_a $8$ (not in LMFDB) 5.2.c_a_a_ae_aq $8$ (not in LMFDB) 5.2.c_e_a_ae_aq $8$ (not in LMFDB) 5.2.c_e_e_e_a $8$ (not in LMFDB) 5.2.c_e_i_m_q $8$ (not in LMFDB) 5.2.c_i_m_bc_bg $8$ (not in LMFDB) 5.2.c_i_q_bc_bw $8$ (not in LMFDB) 5.2.e_e_ae_am_aq $8$ (not in LMFDB) 5.2.e_i_e_am_abg $8$ (not in LMFDB) 5.2.e_i_i_e_a $8$ (not in LMFDB) 5.2.e_i_m_u_bg $8$ (not in LMFDB) 5.2.e_m_y_bs_cm $8$ (not in LMFDB) 5.2.g_u_bw_do_fo $8$ (not in LMFDB) 5.2.i_bg_dg_gi_jw $8$ (not in LMFDB) 5.2.ag_o_ai_abg_dc $12$ (not in LMFDB) 5.2.ae_e_e_am_q $12$ (not in LMFDB) 5.2.ae_k_au_bk_ace $12$ (not in LMFDB) 5.2.ac_ac_i_a_aq $12$ (not in LMFDB) 5.2.a_ae_ae_e_q $12$ (not in LMFDB) 5.2.a_ae_e_e_aq $12$ (not in LMFDB) 5.2.a_c_ae_e_ai $12$ (not in LMFDB) 5.2.a_c_e_e_i $12$ (not in LMFDB) 5.2.c_ac_ai_a_q $12$ (not in LMFDB) 5.2.e_e_ae_am_aq $12$ (not in LMFDB) 5.2.e_k_u_bk_ce $12$ (not in LMFDB) 5.2.g_o_i_abg_adc $12$ (not in LMFDB) 5.2.ak_by_age_nw_awu $24$ (not in LMFDB) 5.2.ai_bi_ads_hs_ami $24$ (not in LMFDB) 5.2.ag_s_abo_cy_aeq $24$ (not in LMFDB) 5.2.ag_s_abg_bo_abw $24$ (not in LMFDB) 5.2.ae_g_a_aq_bg $24$ (not in LMFDB) 5.2.ae_k_ay_bs_acm $24$ (not in LMFDB) 5.2.ae_k_aq_y_abg $24$ (not in LMFDB) 5.2.ae_o_abg_cm_ads $24$ (not in LMFDB) 5.2.ac_ag_q_i_abw $24$ (not in LMFDB) 5.2.ac_ac_i_e_ay $24$ (not in LMFDB) 5.2.ac_c_ai_m_ai $24$ (not in LMFDB) 5.2.ac_c_ae_e_a $24$ (not in LMFDB) 5.2.ac_c_a_ai_q $24$ (not in LMFDB) 5.2.ac_c_a_ae_i $24$ (not in LMFDB) 5.2.ac_c_a_e_ai $24$ (not in LMFDB) 5.2.ac_c_a_i_aq $24$ (not in LMFDB) 5.2.ac_g_am_u_abg $24$ (not in LMFDB) 5.2.ac_g_ai_u_ay $24$ (not in LMFDB) 5.2.ac_k_aq_bo_abw $24$ (not in LMFDB) 5.2.a_ag_a_i_a $24$ (not in LMFDB) 5.2.a_ac_a_a_a $24$ (not in LMFDB) 5.2.a_ac_a_e_a $24$ (not in LMFDB) 5.2.a_c_a_ai_a $24$ (not in LMFDB) 5.2.a_c_a_ae_a $24$ (not in LMFDB) 5.2.a_c_a_e_a $24$ (not in LMFDB) 5.2.a_c_a_i_a $24$ (not in LMFDB) 5.2.a_g_a_q_a $24$ (not in LMFDB) 5.2.a_g_a_u_a $24$ (not in LMFDB) 5.2.a_k_a_bo_a $24$ (not in LMFDB) 5.2.c_ag_aq_i_bw $24$ (not in LMFDB) 5.2.c_ac_ai_e_y $24$ (not in LMFDB) 5.2.c_c_a_ai_aq $24$ (not in LMFDB) 5.2.c_c_a_ae_ai $24$ (not in LMFDB) 5.2.c_c_a_e_i $24$ (not in LMFDB) 5.2.c_c_a_i_q $24$ (not in LMFDB) 5.2.c_c_e_e_a $24$ (not in LMFDB) 5.2.c_c_i_m_i $24$ (not in LMFDB) 5.2.c_g_i_u_y $24$ (not in LMFDB) 5.2.c_g_m_u_bg $24$ (not in LMFDB) 5.2.c_k_q_bo_bw $24$ (not in LMFDB) 5.2.e_g_a_aq_abg $24$ (not in LMFDB) 5.2.e_k_q_y_bg $24$ (not in LMFDB) 5.2.e_k_y_bs_cm $24$ (not in LMFDB) 5.2.e_o_bg_cm_ds $24$ (not in LMFDB) 5.2.g_s_bg_bo_bw $24$ (not in LMFDB) 5.2.g_s_bo_cy_eq $24$ (not in LMFDB) 5.2.g_w_ce_ei_gu $24$ (not in LMFDB) 5.2.i_bi_ds_hs_mi $24$ (not in LMFDB) 5.2.k_by_ge_nw_wu $24$ (not in LMFDB) 5.2.ac_c_a_a_a $48$ (not in LMFDB) 5.2.a_c_a_a_a $48$ (not in LMFDB) 5.2.c_c_a_a_a $48$ (not in LMFDB) 5.2.ae_i_ai_a_i $120$ (not in LMFDB) 5.2.ac_a_e_a_ai $120$ (not in LMFDB) 5.2.ac_e_ae_a_a $120$ (not in LMFDB) 5.2.ac_e_ae_i_ai $120$ (not in LMFDB) 5.2.a_a_a_a_ai $120$ (not in LMFDB) 5.2.a_a_a_a_a $120$ (not in LMFDB) 5.2.a_a_a_a_i $120$ (not in LMFDB) 5.2.a_e_a_i_a $120$ (not in LMFDB) 5.2.c_a_ae_a_i $120$ (not in LMFDB) 5.2.c_e_e_a_a $120$ (not in LMFDB) 5.2.c_e_e_i_i $120$ (not in LMFDB) 5.2.e_i_i_a_ai $120$ (not in LMFDB)