# Properties

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

## Invariants

 Base field: $\F_{2^{7}}$ Dimension: $1$ L-polynomial: $1 + 16 x + 128 x^{2}$ Frobenius angles: $\pm0.750000000000$ 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:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 145 16385 2095105 268468225 34359476225 4398046511105 562949986975745 72057593501057025 9223372041149743105 1180591620717411303425

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 145 16385 2095105 268468225 34359476225 4398046511105 562949986975745 72057593501057025 9223372041149743105 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
• Endomorphism algebra over $\F_{2^{14}}$  The base change of $A$ to $\F_{2^{14}}$ is the simple isogeny class 1.16384.a and its endomorphism algebra is $$\Q(\sqrt{-1})$$.

## Base change

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

 Subfield Primitive Model $\F_{2}$ 1.2.c

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.128.aq $2$ (not in LMFDB) 1.128.a $8$ (not in LMFDB)