Properties

Label 1.128.aq
Base Field $\F_{2^{7}}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{7}}$
Dimension:  $1$
L-polynomial:  $1 - 16 x + 128 x^{2}$
Frobenius angles:  $\pm0.250000000000$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$
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
$A(\F_{q^r})$ 113 16385 2099201 268468225 34360000513 4398046511105 562949919866881 72057593501057025 9223372032559808513 1180591620717411303425

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 113 16385 2099201 268468225 34360000513 4398046511105 562949919866881 72057593501057025 9223372032559808513 1180591620717411303425

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{7}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).
Endomorphism algebra over $\overline{\F}_{2^{7}}$
The base change of $A$ to $\F_{2^{28}}$ is the simple isogeny class 1.268435456.bwmi and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{28}}$.
Remainder of endomorphism lattice by field

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{7}}$.

SubfieldPrimitive Model
$\F_{2}$1.2.ac

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.128.q$2$(not in LMFDB)
1.128.a$8$(not in LMFDB)