Properties

Label 2.4.ad_e
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:  $( 1 - 2 x )^{2}( 1 + x + 4 x^{2} )$
Frobenius angles:  $0$, $0$, $\pm0.580430623255$
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
$A(\F_{q^r})$ 6 216 2646 54000 1043646 16288776 260225286 4276044000 68655500046 1095615396216

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 16 38 208 1022 3976 15878 65248 261902 1044856

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ae $\times$ 1.4.b 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.af_m$2$2.16.ab_ay
2.4.d_e$2$2.16.ab_ay
2.4.f_m$2$2.16.ab_ay
2.4.d_k$3$2.64.abb_ls
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.af_m$2$2.16.ab_ay
2.4.d_e$2$2.16.ab_ay
2.4.f_m$2$2.16.ab_ay
2.4.d_k$3$2.64.abb_ls
2.4.ab_i$4$2.256.abx_boq
2.4.b_i$4$2.256.abx_boq
2.4.ad_k$6$(not in LMFDB)
2.4.ab_g$6$(not in LMFDB)
2.4.b_g$6$(not in LMFDB)