Properties

Label 2.16.ah_bh
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:  $1 - 7 x + 33 x^{2} - 112 x^{3} + 256 x^{4}$
Frobenius angles:  $\pm0.172472086823$, $\pm0.494194579844$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  29

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 29 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 171 69939 16842816 4280336739 1101267647451 281748310329600 72065710292698251 18446394710506428099 4722354708206963052096 1208926591225993760495859

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 10 274 4111 65314 1050250 16793503 268465690 4294885954 68719305391 1099512329554

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{2^{4}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{12}}$.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

SubfieldPrimitive Model
$\F_{2}$2.2.ad_f
$\F_{2}$2.2.d_f

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.h_bh$2$2.256.r_bh
2.16.o_dd$3$(not in LMFDB)
2.16.ao_dd$6$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.16.h_bh$2$2.256.r_bh
2.16.o_dd$3$(not in LMFDB)
2.16.ao_dd$6$(not in LMFDB)
2.16.a_ar$6$(not in LMFDB)
2.16.a_r$12$(not in LMFDB)