# Properties

 Label 5.2.ag_r_abb_ba_ay 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: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + x^{2} + x^{3} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.132091856901$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.309487084859$, $\pm0.780459932197$ 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 $A(\F_{q^r})$ 3 1575 111033 6181875 44988603 1224138825 27759090936 869771266875 38981319187473 1175636550020625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 18 47 42 72 102 199 567 1068

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ac_b_b 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_b_b : 6.0.2369943.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.o_dx_sl. 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.o_dx_sl : 6.0.2369943.1.
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.ac_j_an. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.ac_j_an : 6.0.2369943.1.

## 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_b_b_g_aq $2$ (not in LMFDB) 5.2.ac_b_d_c_ai $2$ (not in LMFDB) 5.2.c_b_ad_c_i $2$ (not in LMFDB) 5.2.c_b_ab_g_q $2$ (not in LMFDB) 5.2.g_r_bb_ba_y $2$ (not in LMFDB) 5.2.a_ab_d_c_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_b_b_g_aq $2$ (not in LMFDB) 5.2.ac_b_d_c_ai $2$ (not in LMFDB) 5.2.c_b_ad_c_i $2$ (not in LMFDB) 5.2.c_b_ab_g_q $2$ (not in LMFDB) 5.2.g_r_bb_ba_y $2$ (not in LMFDB) 5.2.a_ab_d_c_ag $3$ (not in LMFDB) 5.2.ae_h_aj_o_aw $6$ (not in LMFDB) 5.2.a_ab_ad_c_g $6$ (not in LMFDB) 5.2.e_h_j_o_w $6$ (not in LMFDB) 5.2.ae_j_an_q_au $8$ (not in LMFDB) 5.2.ac_ad_j_c_au $8$ (not in LMFDB) 5.2.ac_f_ah_k_am $8$ (not in LMFDB) 5.2.a_b_ab_e_ae $8$ (not in LMFDB) 5.2.a_b_b_e_e $8$ (not in LMFDB) 5.2.c_ad_aj_c_u $8$ (not in LMFDB) 5.2.c_f_h_k_m $8$ (not in LMFDB) 5.2.e_j_n_q_u $8$ (not in LMFDB) 5.2.a_ab_d_c_ag $12$ (not in LMFDB) 5.2.ac_ab_f_e_as $24$ (not in LMFDB) 5.2.ac_d_ad_i_ao $24$ (not in LMFDB) 5.2.c_ab_af_e_s $24$ (not in LMFDB) 5.2.c_d_d_i_o $24$ (not in LMFDB)