Can you create a droplet from Ubuntu 14.04 x64 or with Docker application using API?

Both images are returned by the /images API call:

    {
      "id":3240036,
      "name":"Ubuntu 14.04 x64",
      "distribution":"Ubuntu",
      "slug":"ubuntu-14-04-x64",
      "public":true,
      "regions":[
        "nyc1",
        "ams1",
        "sfo1",
        "nyc2",
        "ams2",
        "sgp1",
        "lon1"
      ],
      "created_at":"2014-04-18T15:59:36Z"
    }

    {
      "id":4970653,
      "name":"Docker 1.1.1 on Ubuntu 14.04",
      "distribution":"Ubuntu",
      "slug":null,
      "public":true,
      "regions":[
        "nyc1",
        "ams1",
        "sfo1",
        "nyc2",
        "ams2",
        "sgp1",
        "lon1"
      ],
      "created_at":"2014-07-16T20:00:38Z"
    }

At the very bottom of the response there are pagination links (it might be different for your account depending on how many snapshots you have), so just keep changing the URL until you get the image you need:

  "links":{
    "pages":{
      "last":"https://api.digitalocean.com/v2/images?page=3",
      "next":"https://api.digitalocean.com/v2/images?page=2"
    }

You’re most likely going to find the Docker image in the last page as it was just updated a few seconds ago. You can also use the image’s slug instead of its ID: ubuntu-14-04-x64.

P.S.: Since you’re using curl, you can use the Authorization header instead of HTTP basic auth:

curl -H "Authorization: Bearer token" https://api.digitalocean.com/v2/images