Properties

Label 2.4.af_o
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 4 x^{2} )( 1 - 2 x + 4 x^{2} )$
Frobenius angles:  $\pm0.230053456163$, $\pm0.333333333333$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

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
$A(\F_{q^r})$ 6 336 5994 78624 1074426 16447536 264956586 4282334784 68725531674 1099326896976

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 20 90 304 1050 4016 16170 65344 262170 1048400

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ad $\times$ 1.4.ac 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 $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.j $\times$ 1.64.q. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ab_c$2$2.16.d_bc
2.4.b_c$2$2.16.d_bc
2.4.f_o$2$2.16.d_bc
2.4.b_ae$3$2.64.z_km
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ab_c$2$2.16.d_bc
2.4.b_c$2$2.16.d_bc
2.4.f_o$2$2.16.d_bc
2.4.b_ae$3$2.64.z_km
2.4.ah_u$6$(not in LMFDB)
2.4.ab_ae$6$(not in LMFDB)
2.4.ab_c$6$(not in LMFDB)
2.4.h_u$6$(not in LMFDB)
2.4.ad_i$12$(not in LMFDB)
2.4.d_i$12$(not in LMFDB)