# Properties

 Label 4.2.af_m_au_bd Base Field $\F_{2}$ Dimension $4$ Ordinary Yes $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{2}$ Dimension: $4$ L-polynomial: $1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8}$ Frobenius angles: $\pm0.0635622003031$, $\pm0.165221137389$, $\pm0.365221137389$, $\pm0.663562200303$ Angle rank: $2$ (numerical) Number field: 8.0.13140625.1 Galois group: $C_2^2:C_4$

This isogeny class is simple but not geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1 211 1861 88831 1046771 12172801 344358281 5598573775 68193052891 1095729526441

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 4 4 20 33 46 159 324 508 1019

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The endomorphism algebra of this simple isogeny class is 8.0.13140625.1.
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{10}}$ is 2.1024.ad_bwv 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.3625.1$)$
All geometric endomorphisms are defined over $\F_{2^{10}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ab_c_ai_z and its endomorphism algebra is 8.0.13140625.1.
• Endomorphism algebra over $\F_{2^{5}}$  The base change of $A$ to $\F_{2^{5}}$ is the simple isogeny class 4.32.a_ad_a_bwv and its endomorphism algebra is 8.0.13140625.1.

## 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 4.2.f_m_u_bd $2$ 4.4.ab_c_ai_z 4.2.a_ad_a_j $5$ (not in LMFDB) 4.2.a_c_af_ab $5$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.2.f_m_u_bd $2$ 4.4.ab_c_ai_z 4.2.a_ad_a_j $5$ (not in LMFDB) 4.2.a_c_af_ab $5$ (not in LMFDB) 4.2.a_c_f_ab $5$ (not in LMFDB) 4.2.f_m_u_bd $5$ (not in LMFDB) 4.2.a_d_a_j $20$ (not in LMFDB)