Properties

Label 2.4.ab_ae
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:  $( 1 - 2 x )^{2}( 1 + 3 x + 4 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.769946543837$
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
$A(\F_{q^r})$ 8 144 2744 64800 930248 16447536 265676888 4232347200 68712424424 1096109357904

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 8 40 256 904 4016 16216 64576 262120 1045328

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{2}}$.

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.ah_u$2$2.16.aj_bo
2.4.b_ae$2$2.16.aj_bo
2.4.h_u$2$2.16.aj_bo
2.4.f_o$3$2.64.az_km
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ah_u$2$2.16.aj_bo
2.4.b_ae$2$2.16.aj_bo
2.4.h_u$2$2.16.aj_bo
2.4.f_o$3$2.64.az_km
2.4.ad_i$4$2.256.ab_asm
2.4.d_i$4$2.256.ab_asm
2.4.af_o$6$(not in LMFDB)
2.4.ab_c$6$(not in LMFDB)
2.4.b_c$6$(not in LMFDB)