It's easy to determine if a User is a member of a Team if you know the id
:
GET /teams/:id/members/:user
But how can one easily determine the ID of the special "Owners" team that every Organization has?
As far as I can tell, the only way is to retrieve a full list of all Teams (which I assume may be multiple pages?) and walk through them until you find one with the name "Owners".
This is doable of course, but it's uncharacteristically inconvenient for GitHub's otherwise fantastic API. ;)
For what it's worth, I've tried the following (with no luck):
GET /orgs/:organization/teams?name=Owners # Lists all teams
GET /orgs/:organization/owners # 404
Just to be clear, I've made sure to use a token associated with the user that owns the organization in question, so there shouldn't be any authorization issues.