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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 211 1861 88831 1046771 12172801 344358281 5598573775 68193052891 1095729526441

Point counts of the (virtual) curve

$r$ 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

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)