# Properties

 Label 6.2.ai_be_acq_dy_aei_ey Base Field $\F_{2}$ Dimension $6$ Ordinary No $p$-rank $0$ Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $6$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.398391828106$, $\pm0.787778569972$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $0$ Slopes: $[1/4, 1/4, 1/4, 1/4, 1/2, 1/2, 1/2, 1/2, 3/4, 3/4, 3/4, 3/4]$

## Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1 2425 454441 42255625 769227281 80447418025 4133278016393 231405704600625 16960909411229953 1003619006560560625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 1 13 33 25 73 121 209 481 881

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 4.2.ae_g_ae_c 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}($$$\Q(\sqrt{-1})$$$)$ 4.2.ae_g_ae_c : 8.0.18939904.2.
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg 2 $\times$ 4.256.q_ds_glk_jitk. The endomorphism algebra for each factor is:
• 1.256.abg 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
• 4.256.q_ds_glk_jitk : the quaternion algebra over 4.0.1088.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:
 $v$ ($2$,$$\pi$$) ($2$,$$\pi + 1$$) $\operatorname{inv}_v$ $1/2$ $1/2$
where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.\n
All geometric endomorphisms are defined over $\F_{2^{8}}$.
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 $\times$ 4.4.ae_i_ai_e. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 4.4.ae_i_ai_e : 8.0.18939904.2.
• Endomorphism algebra over $\F_{2^{4}}$  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 4.16.a_i_a_q. The endomorphism algebra for each factor is: 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 4.16.a_i_a_q : 8.0.18939904.2.

## 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 6.2.ae_g_ae_g_ay_bw $2$ (not in LMFDB) 6.2.a_ac_ae_g_a_a $2$ (not in LMFDB) 6.2.a_ac_e_g_a_a $2$ (not in LMFDB) 6.2.e_g_e_g_y_bw $2$ (not in LMFDB) 6.2.i_be_cq_dy_ei_ey $2$ (not in LMFDB) 6.2.ac_a_e_ag_ae_u $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 6.2.ae_g_ae_g_ay_bw $2$ (not in LMFDB) 6.2.a_ac_ae_g_a_a $2$ (not in LMFDB) 6.2.a_ac_e_g_a_a $2$ (not in LMFDB) 6.2.e_g_e_g_y_bw $2$ (not in LMFDB) 6.2.i_be_cq_dy_ei_ey $2$ (not in LMFDB) 6.2.ac_a_e_ag_ae_u $3$ (not in LMFDB) 6.2.ae_k_au_bm_acm_ds $4$ (not in LMFDB) 6.2.ae_k_am_g_q_abg $4$ (not in LMFDB) 6.2.a_c_ae_g_ai_q $4$ (not in LMFDB) 6.2.a_c_e_g_i_q $4$ (not in LMFDB) 6.2.e_k_m_g_aq_abg $4$ (not in LMFDB) 6.2.e_k_u_bm_cm_ds $4$ (not in LMFDB) 6.2.ag_q_abc_bq_aci_dg $6$ (not in LMFDB) 6.2.c_a_ae_ag_e_u $6$ (not in LMFDB) 6.2.g_q_bc_bq_ci_dg $6$ (not in LMFDB) 6.2.ag_s_abk_cc_acq_dk $8$ (not in LMFDB) 6.2.ae_c_m_as_ai_bo $8$ (not in LMFDB) 6.2.ae_k_au_be_abo_ce $8$ (not in LMFDB) 6.2.ac_c_ae_g_am_y $8$ (not in LMFDB) 6.2.ac_g_am_w_abk_ce $8$ (not in LMFDB) 6.2.ac_g_ae_g_m_ai $8$ (not in LMFDB) 6.2.a_ac_ae_ac_i_i $8$ (not in LMFDB) 6.2.a_ac_e_ac_ai_i $8$ (not in LMFDB) 6.2.a_g_ae_o_ay_y $8$ (not in LMFDB) 6.2.a_g_e_o_y_y $8$ (not in LMFDB) 6.2.c_c_e_g_m_y $8$ (not in LMFDB) 6.2.c_g_e_g_am_ai $8$ (not in LMFDB) 6.2.c_g_m_w_bk_ce $8$ (not in LMFDB) 6.2.e_c_am_as_i_bo $8$ (not in LMFDB) 6.2.e_k_u_be_bo_ce $8$ (not in LMFDB) 6.2.g_s_bk_cc_cq_dk $8$ (not in LMFDB) 6.2.ac_e_am_s_abc_ca $12$ (not in LMFDB) 6.2.ac_e_ae_c_e_am $12$ (not in LMFDB) 6.2.c_e_e_c_ae_am $12$ (not in LMFDB) 6.2.c_e_m_s_bc_ca $12$ (not in LMFDB) 6.2.ae_e_i_as_ai_bw $16$ (not in LMFDB) 6.2.ae_m_ay_bu_acu_ei $16$ (not in LMFDB) 6.2.ac_a_e_ac_ae_i $16$ (not in LMFDB) 6.2.ac_i_am_be_abk_cu $16$ (not in LMFDB) 6.2.a_ai_a_be_a_acu $16$ (not in LMFDB) 6.2.a_ae_a_o_a_abg $16$ (not in LMFDB) 6.2.a_a_a_ac_a_ai $16$ (not in LMFDB) 6.2.a_a_a_ac_a_i $16$ (not in LMFDB) 6.2.a_e_a_o_a_bg $16$ (not in LMFDB) 6.2.a_i_a_be_a_cu $16$ (not in LMFDB) 6.2.c_a_ae_ac_e_i $16$ (not in LMFDB) 6.2.c_i_m_be_bk_cu $16$ (not in LMFDB) 6.2.e_e_ai_as_i_bw $16$ (not in LMFDB) 6.2.e_m_y_bu_cu_ei $16$ (not in LMFDB) 6.2.ae_e_e_ag_aq_bs $24$ (not in LMFDB) 6.2.ae_i_am_s_abg_ca $24$ (not in LMFDB) 6.2.a_a_ae_c_a_m $24$ (not in LMFDB) 6.2.a_a_e_c_a_m $24$ (not in LMFDB) 6.2.a_e_ae_k_aq_u $24$ (not in LMFDB) 6.2.a_e_e_k_q_u $24$ (not in LMFDB) 6.2.e_e_ae_ag_q_bs $24$ (not in LMFDB) 6.2.e_i_m_s_bg_ca $24$ (not in LMFDB) 6.2.ac_ac_e_g_ae_am $48$ (not in LMFDB) 6.2.ac_g_am_w_abk_ca $48$ (not in LMFDB) 6.2.a_ag_a_w_a_aca $48$ (not in LMFDB) 6.2.a_ac_a_g_a_am $48$ (not in LMFDB) 6.2.a_c_a_g_a_m $48$ (not in LMFDB) 6.2.a_g_a_w_a_ca $48$ (not in LMFDB) 6.2.c_ac_ae_g_e_am $48$ (not in LMFDB) 6.2.c_g_m_w_bk_ca $48$ (not in LMFDB)