# Properties

 Label 1.269.aba Base Field $\F_{269}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{269}$ Dimension: $1$ L-polynomial: $1 - 26 x + 269 x^{2}$ Frobenius angles: $\pm0.208714400160$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$ Jacobians: 8

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

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

## Point counts

This isogeny class contains the Jacobians of 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 244 72224 19468516 5236240000 1408517103764 378890495711264 101921536072759556 27416893177258560000 7375144265954985253684 1983913807582052682723104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 244 72224 19468516 5236240000 1408517103764 378890495711264 101921536072759556 27416893177258560000 7375144265954985253684 1983913807582052682723104

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{269}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{269}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.269.ba $2$ (not in LMFDB) 1.269.au $4$ (not in LMFDB) 1.269.u $4$ (not in LMFDB)