Properties

Label 1.16.i
Base Field $\F_{2^{4}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $1$
L-polynomial:  $( 1 + 4 x )^{2}$
Frobenius angles:  $1$, $1$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial
Jacobians:  1

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 25 225 4225 65025 1050625 16769025 268468225 4294836225 68720001025 1099509530625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 25 225 4225 65025 1050625 16769025 268468225 4294836225 68720001025 1099509530625

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.

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}$1.2.ac
$\F_{2}$1.2.c

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
1.16.ai$2$1.256.abg
1.16.ae$3$(not in LMFDB)
1.16.a$4$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.16.ai$2$1.256.abg
1.16.ae$3$(not in LMFDB)
1.16.a$4$(not in LMFDB)
1.16.e$6$(not in LMFDB)