# Properties

 Label 5.2.ah_ba_acn_eu_ahk 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: $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 3 x + 6 x^{2} - 9 x^{3} + 12 x^{4} - 12 x^{5} + 8 x^{6} )$ Frobenius angles: $\pm0.147012170705$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.341962716420$, $\pm0.600633654388$ 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 3825 86697 3538125 78877563 994848075 26085841176 1135175563125 36400837975749 1083629820189375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 8 17 36 61 59 94 268 530 983

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac 2 $\times$ 3.2.ad_g_aj 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_g_aj : 6.0.465831.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 3.16.d_bq_db. 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.d_bq_db : 6.0.465831.1.
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.d_g_h. The endomorphism algebra for each factor is: 1.4.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-1})$$$)$ 3.4.d_g_h : 6.0.465831.1.

## 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_g_aj_q_ay $2$ (not in LMFDB) 5.2.ab_c_b_e_a $2$ (not in LMFDB) 5.2.b_c_ab_e_a $2$ (not in LMFDB) 5.2.d_g_j_q_y $2$ (not in LMFDB) 5.2.h_ba_cn_eu_hk $2$ (not in LMFDB) 5.2.ab_c_b_ac_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.2.ad_g_aj_q_ay $2$ (not in LMFDB) 5.2.ab_c_b_e_a $2$ (not in LMFDB) 5.2.b_c_ab_e_a $2$ (not in LMFDB) 5.2.d_g_j_q_y $2$ (not in LMFDB) 5.2.h_ba_cn_eu_hk $2$ (not in LMFDB) 5.2.ab_c_b_ac_g $3$ (not in LMFDB) 5.2.af_o_abf_cg_adm $6$ (not in LMFDB) 5.2.b_c_ab_ac_ag $6$ (not in LMFDB) 5.2.f_o_bf_cg_dm $6$ (not in LMFDB) 5.2.af_q_abl_cs_aee $8$ (not in LMFDB) 5.2.ad_c_d_ai_m $8$ (not in LMFDB) 5.2.ad_k_av_bo_aci $8$ (not in LMFDB) 5.2.ab_e_af_k_am $8$ (not in LMFDB) 5.2.b_e_f_k_m $8$ (not in LMFDB) 5.2.d_c_ad_ai_am $8$ (not in LMFDB) 5.2.d_k_v_bo_ci $8$ (not in LMFDB) 5.2.f_q_bl_cs_ee $8$ (not in LMFDB) 5.2.ad_e_ad_e_ag $24$ (not in LMFDB) 5.2.ad_i_ap_bc_abq $24$ (not in LMFDB) 5.2.d_e_d_e_g $24$ (not in LMFDB) 5.2.d_i_p_bc_bq $24$ (not in LMFDB)