# Properties

 Label 5.2.ag_s_abh_bs_ace Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.161334789180$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.327009058845$, $\pm0.739882802642$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 4 2600 100048 6890000 53126324 1040499200 32604824284 849412980000 35849268493168 1174955063165000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 5 18 49 47 62 123 193 522 1065

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ac_c_ab 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}($$$\Q(\sqrt{-1})$$$)$ 3.2.ac_c_ab : 6.0.2464727.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.q_ey_yp. 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 $\infty$. 3.16.q_ey_yp : 6.0.2464727.1.
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 2 $\times$ 3.4.a_i_ab. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.a_i_ab : 6.0.2464727.1.

## 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.ac_c_ab_i_aq $2$ (not in LMFDB) 5.2.ac_c_b_e_ai $2$ (not in LMFDB) 5.2.c_c_ab_e_i $2$ (not in LMFDB) 5.2.c_c_b_i_q $2$ (not in LMFDB) 5.2.g_s_bh_bs_ce $2$ (not in LMFDB) 5.2.a_a_d_c_ac $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_c_ab_i_aq $2$ (not in LMFDB) 5.2.ac_c_b_e_ai $2$ (not in LMFDB) 5.2.c_c_ab_e_i $2$ (not in LMFDB) 5.2.c_c_b_i_q $2$ (not in LMFDB) 5.2.g_s_bh_bs_ce $2$ (not in LMFDB) 5.2.a_a_d_c_ac $3$ (not in LMFDB) 5.2.ae_i_an_w_abi $6$ (not in LMFDB) 5.2.a_a_ad_c_c $6$ (not in LMFDB) 5.2.e_i_n_w_bi $6$ (not in LMFDB) 5.2.ae_k_ar_ba_abk $8$ (not in LMFDB) 5.2.ac_ac_h_a_am $8$ (not in LMFDB) 5.2.ac_g_aj_q_au $8$ (not in LMFDB) 5.2.a_c_ab_g_ae $8$ (not in LMFDB) 5.2.a_c_b_g_e $8$ (not in LMFDB) 5.2.c_ac_ah_a_m $8$ (not in LMFDB) 5.2.c_g_j_q_u $8$ (not in LMFDB) 5.2.e_k_r_ba_bk $8$ (not in LMFDB) 5.2.a_a_ad_c_c $12$ (not in LMFDB) 5.2.ac_a_d_e_ao $24$ (not in LMFDB) 5.2.ac_e_af_m_as $24$ (not in LMFDB) 5.2.c_a_ad_e_o $24$ (not in LMFDB) 5.2.c_e_f_m_s $24$ (not in LMFDB)