Properties

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

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 4 x^{2} )( 1 + 4 x^{2} )$
Frobenius angles:  $\pm0.230053456163$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  1

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 10 400 4810 64800 1109050 17508400 267042730 4232347200 68458118170 1100399410000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 24 74 256 1082 4272 16298 64576 261146 1049424

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ad $\times$ 1.4.a 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 $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{4}}$.

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.d_i$2$2.16.h_y
2.4.ah_u$4$2.256.ab_asm
2.4.ab_ae$4$2.256.ab_asm
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.d_i$2$2.16.h_y
2.4.ah_u$4$2.256.ab_asm
2.4.ab_ae$4$2.256.ab_asm
2.4.b_ae$4$2.256.ab_asm
2.4.h_u$4$2.256.ab_asm
2.4.af_o$12$(not in LMFDB)
2.4.ab_c$12$(not in LMFDB)
2.4.b_c$12$(not in LMFDB)
2.4.f_o$12$(not in LMFDB)