# Properties

 Label 5.2.ah_z_ach_ec_age 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: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 7 x^{3} + 10 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.105278500939$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.316838792568$, $\pm0.641249159631$ 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 2300 51038 3910000 72017402 763018100 26336547722 1076814000000 34196345458262 1244370684807500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 6 14 38 56 42 94 254 500 1126

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ad_f_ah 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.ad_f_ah : 6.0.679024.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.f_br_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.f_br_fn : 6.0.679024.1.
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.b_d_af. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.b_d_af : 6.0.679024.1.

## 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.ad_f_ah_o_ay $2$ (not in LMFDB) 5.2.ab_b_d_c_a $2$ (not in LMFDB) 5.2.b_b_ad_c_a $2$ (not in LMFDB) 5.2.d_f_h_o_y $2$ (not in LMFDB) 5.2.h_z_ch_ec_ge $2$ (not in LMFDB) 5.2.ab_b_b_ac_c $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_f_ah_o_ay $2$ (not in LMFDB) 5.2.ab_b_d_c_a $2$ (not in LMFDB) 5.2.b_b_ad_c_a $2$ (not in LMFDB) 5.2.d_f_h_o_y $2$ (not in LMFDB) 5.2.h_z_ch_ec_ge $2$ (not in LMFDB) 5.2.ab_b_b_ac_c $3$ (not in LMFDB) 5.2.h_z_ch_ec_ge $4$ (not in LMFDB) 5.2.af_n_abb_by_ada $6$ (not in LMFDB) 5.2.b_b_ab_ac_ac $6$ (not in LMFDB) 5.2.f_n_bb_by_da $6$ (not in LMFDB) 5.2.af_p_abh_ci_ado $8$ (not in LMFDB) 5.2.ad_b_f_ag_e $8$ (not in LMFDB) 5.2.ad_j_at_bi_aca $8$ (not in LMFDB) 5.2.ab_d_af_i_am $8$ (not in LMFDB) 5.2.b_d_f_i_m $8$ (not in LMFDB) 5.2.d_b_af_ag_ae $8$ (not in LMFDB) 5.2.d_j_t_bi_ca $8$ (not in LMFDB) 5.2.f_p_bh_ci_do $8$ (not in LMFDB) 5.2.ad_d_ab_e_ak $24$ (not in LMFDB) 5.2.ad_h_an_y_abm $24$ (not in LMFDB) 5.2.d_d_b_e_k $24$ (not in LMFDB) 5.2.d_h_n_y_bm $24$ (not in LMFDB)