# Properties

 Label 5.2.ag_s_abk_cf_ade Base Field $\F_{2}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $5$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 17 x^{4} - 24 x^{5} + 32 x^{6} - 32 x^{7} + 16 x^{8} )$ Frobenius angles: $\pm0.0755571399449$, $\pm0.203216343788$, $\pm0.250000000000$, $\pm0.424442860055$, $\pm0.703216343788$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

## 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 1460 31226 2131600 38133362 1116954020 45547817842 962596454400 28494805076114 1124768298875300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 5 9 29 37 65 165 221 405 1025

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 4.2.ae_i_am_r and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.c_b 2 . The endomorphism algebra for each factor is: 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 2.16.c_b 2 : $\mathrm{M}_{2}($4.0.1088.2$)$
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 $\times$ 4.4.a_c_a_b. The endomorphism algebra for each factor is:

## 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_ae_j_ao $2$ (not in LMFDB) 5.2.c_c_e_j_o $2$ (not in LMFDB) 5.2.g_s_bk_cf_de $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ac_c_ae_j_ao $2$ (not in LMFDB) 5.2.c_c_e_j_o $2$ (not in LMFDB) 5.2.g_s_bk_cf_de $2$ (not in LMFDB) 5.2.ag_u_abw_dp_afq $8$ (not in LMFDB) 5.2.ae_k_au_bh_abw $8$ (not in LMFDB) 5.2.ae_m_abc_cb_adc $8$ (not in LMFDB) 5.2.ac_a_e_ad_ac $8$ (not in LMFDB) 5.2.ac_e_ai_n_ao $8$ (not in LMFDB) 5.2.ac_e_ae_f_ac $8$ (not in LMFDB) 5.2.a_a_a_ad_a $8$ (not in LMFDB) 5.2.a_e_a_f_a $8$ (not in LMFDB) 5.2.c_a_ae_ad_c $8$ (not in LMFDB) 5.2.c_e_e_f_c $8$ (not in LMFDB) 5.2.c_e_i_n_o $8$ (not in LMFDB) 5.2.e_k_u_bh_bw $8$ (not in LMFDB) 5.2.e_m_bc_cb_dc $8$ (not in LMFDB) 5.2.g_u_bw_dp_fq $8$ (not in LMFDB) 5.2.ae_h_ae_af_o $24$ (not in LMFDB) 5.2.ac_d_ac_ab_i $24$ (not in LMFDB) 5.2.a_ab_a_d_ac $24$ (not in LMFDB) 5.2.a_ab_a_d_c $24$ (not in LMFDB) 5.2.c_d_c_ab_ai $24$ (not in LMFDB) 5.2.e_h_e_af_ao $24$ (not in LMFDB)