# Properties

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

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.0693533547550$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.339907131295$, $\pm0.770553776540$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

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

## 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 1100 65234 3740000 28143302 896967500 26886431938 844596720000 38112217374998 1149689065827500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 15 39 27 51 95 191 555 1043

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ac_b_a 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_a : 6.0.2580992.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.g_r_ce. 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.g_r_ce : 6.0.2580992.1.
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_f_am. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.ac_f_am : 6.0.2580992.1.

## 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_a_g_aq $2$ (not in LMFDB) 5.2.ac_b_e_ac_a $2$ (not in LMFDB) 5.2.c_b_ae_ac_a $2$ (not in LMFDB) 5.2.c_b_a_g_q $2$ (not in LMFDB) 5.2.g_r_bc_be_bg $2$ (not in LMFDB) 5.2.a_ab_c_a_ai $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_b_a_g_aq $2$ (not in LMFDB) 5.2.ac_b_e_ac_a $2$ (not in LMFDB) 5.2.c_b_ae_ac_a $2$ (not in LMFDB) 5.2.c_b_a_g_q $2$ (not in LMFDB) 5.2.g_r_bc_be_bg $2$ (not in LMFDB) 5.2.a_ab_c_a_ai $3$ (not in LMFDB) 5.2.ae_h_ak_q_ay $6$ (not in LMFDB) 5.2.a_ab_ac_a_i $6$ (not in LMFDB) 5.2.e_h_k_q_y $6$ (not in LMFDB) 5.2.ae_j_ao_s_ay $8$ (not in LMFDB) 5.2.ac_ad_i_c_aq $8$ (not in LMFDB) 5.2.ac_f_ai_k_aq $8$ (not in LMFDB) 5.2.a_b_ac_c_ai $8$ (not in LMFDB) 5.2.a_b_c_c_i $8$ (not in LMFDB) 5.2.c_ad_ai_c_q $8$ (not in LMFDB) 5.2.c_f_i_k_q $8$ (not in LMFDB) 5.2.e_j_o_s_y $8$ (not in LMFDB) 5.2.a_ab_ac_a_i $12$ (not in LMFDB) 5.2.ac_ab_e_e_aq $24$ (not in LMFDB) 5.2.ac_d_ae_i_aq $24$ (not in LMFDB) 5.2.c_ab_ae_e_q $24$ (not in LMFDB) 5.2.c_d_e_i_q $24$ (not in LMFDB)