# Properties

 Label 1.251.ae Base Field $\F_{251}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{251}$ Dimension: $1$ L-polynomial: $1 - 4 x + 251 x^{2}$ Frobenius angles: $\pm0.459709415580$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-247})$$ Galois group: $C_2$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 248 63488 15816200 3969015808 996249445528 250058930124800 62764786092544168 15753961207609786368 3954244264051145055800 992515310306108082403328

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 248 63488 15816200 3969015808 996249445528 250058930124800 62764786092544168 15753961207609786368 3954244264051145055800 992515310306108082403328

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{251}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-247})$$.
All geometric endomorphisms are defined over $\F_{251}$.

## 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 1.251.e $2$ (not in LMFDB)