# Properties

 Label 5.2.ag_q_av_i_i 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: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + 3 x^{3} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0889496890695$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.297004294965$, $\pm0.823081333977$ 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 $A(\F_{q^r})$ 2 800 106808 4360000 39237902 1196249600 21135172186 910446480000 35876430274568 1212843550820000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 18 41 37 70 67 209 522 1101

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ac_a_d 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_a_d : 6.0.1539727.2.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.i_bo_fn. 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.i_bo_fn : 6.0.1539727.2.
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.ae_m_az. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.ae_m_az : 6.0.1539727.2.

## 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 5.2.ac_a_d_e_aq $2$ (not in LMFDB) 5.2.ac_a_f_a_ai $2$ (not in LMFDB) 5.2.c_a_af_a_i $2$ (not in LMFDB) 5.2.c_a_ad_e_q $2$ (not in LMFDB) 5.2.g_q_v_i_ai $2$ (not in LMFDB) 5.2.a_ac_d_c_ak $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_a_d_e_aq $2$ (not in LMFDB) 5.2.ac_a_f_a_ai $2$ (not in LMFDB) 5.2.c_a_af_a_i $2$ (not in LMFDB) 5.2.c_a_ad_e_q $2$ (not in LMFDB) 5.2.g_q_v_i_ai $2$ (not in LMFDB) 5.2.a_ac_d_c_ak $3$ (not in LMFDB) 5.2.ac_a_d_e_aq $4$ (not in LMFDB) 5.2.ac_a_f_a_ai $4$ (not in LMFDB) 5.2.c_a_af_a_i $4$ (not in LMFDB) 5.2.c_a_ad_e_q $4$ (not in LMFDB) 5.2.g_q_v_i_ai $4$ (not in LMFDB) 5.2.ae_g_af_g_ak $6$ (not in LMFDB) 5.2.a_ac_ad_c_k $6$ (not in LMFDB) 5.2.e_g_f_g_k $6$ (not in LMFDB) 5.2.ae_i_aj_g_ae $8$ (not in LMFDB) 5.2.ac_ae_l_e_abc $8$ (not in LMFDB) 5.2.ac_e_af_e_ae $8$ (not in LMFDB) 5.2.a_a_ab_c_ae $8$ (not in LMFDB) 5.2.a_a_b_c_e $8$ (not in LMFDB) 5.2.c_ae_al_e_bc $8$ (not in LMFDB) 5.2.c_e_f_e_e $8$ (not in LMFDB) 5.2.e_i_j_g_e $8$ (not in LMFDB) 5.2.ae_g_af_g_ak $12$ (not in LMFDB) 5.2.a_ac_ad_c_k $12$ (not in LMFDB) 5.2.a_ac_d_c_ak $12$ (not in LMFDB) 5.2.e_g_f_g_k $12$ (not in LMFDB) 5.2.ac_ac_h_e_aw $24$ (not in LMFDB) 5.2.ac_c_ab_e_ak $24$ (not in LMFDB) 5.2.c_ac_ah_e_w $24$ (not in LMFDB) 5.2.c_c_b_e_k $24$ (not in LMFDB)