Properties

Label 2.16.ah_bh
Base field $\F_{2^{4}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple no
Primitive no
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:  $29$
Isomorphism classes:  23

This isogeny class is simple but not geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $171$ $69939$ $16842816$ $4280336739$ $1101267647451$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 29 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

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 $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$

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)