Properties

Label 2.4.a_ab
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 4 x^{2} )( 1 + 3 x + 4 x^{2} )$
Frobenius angles:  $\pm0.230053456163$, $\pm0.769946543837$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 16 256 4144 82944 1047376 17172736 268460656 4236447744 68719003024 1096996485376

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 15 65 319 1025 4191 16385 64639 262145 1046175

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ad $\times$ 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:
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
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.ag_r$2$2.16.ac_bh
2.4.g_r$2$2.16.ac_bh
2.4.a_b$4$2.256.ck_cer
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.ag_r$2$2.16.ac_bh
2.4.g_r$2$2.16.ac_bh
2.4.a_b$4$2.256.ck_cer
2.4.ad_f$6$(not in LMFDB)
2.4.d_f$6$(not in LMFDB)