Properties

Label 2.16.aj_br
Base Field $\F_{2^{4}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $2$
L-polynomial:  $1 - 9 x + 43 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:  $\pm0.108303609292$, $\pm0.441636942625$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{37})\)
Galois group:  $C_2^2$
Jacobians:  12

This isogeny class is simple but not geometrically 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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 147 66591 16780932 4263222411 1098088275837 281599678788624 72072407321352543 18447199763816467539 4722366482733934692252 1208927448569688921723951

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 262 4097 65050 1047218 16784647 268490636 4295073394 68719476737 1099513109302

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{37})\).
Endomorphism algebra over $\overline{\F}_{2^{4}}$
The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.fmx 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-111}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.j_br$2$2.256.f_aix
2.16.a_af$3$(not in LMFDB)
2.16.j_br$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.j_br$2$2.256.f_aix
2.16.a_af$3$(not in LMFDB)
2.16.j_br$3$(not in LMFDB)
2.16.a_af$6$(not in LMFDB)
2.16.a_f$12$(not in LMFDB)