https://{YOUR CRUCIBLE URL}/changelog-ajax/{REPOSITORY.NAME}?q=&limit=51&command=branches
That will return something looking like:
{
    totalItems: 5,
    hasQuerySpecified: false,
    items: [
    {
        displayPrimary: "trunk",
        displaySecondary: "Branch",
        id: "trunk",
        isBranch: true,
        isTag: false,
        isCommitter: false
    },
    {
        displayPrimary: "YourBranchName",
        displaySecondary: "Branch",
        id: "YourBranchName",
        isBranch: true,
        isTag: false,
        isCommitter: false
    },
    {
        displayPrimary: "trunk",
        displaySecondary: "Branch",
        id: "trunk",
        isBranch: true,
        isTag: false,
        isCommitter: false
    },
    {
        displayPrimary: "root:",
        displaySecondary: "Branch",
        id: "root:",
        isBranch: true,
        isTag: false,
        isCommitter: false
    },
    {
        displayPrimary: "ANotherBranch",
        displaySecondary: "Branch",
        id: "ANotherBranch",
        isBranch: true,
        isTag: false,
        isCommitter: false
    },
    ]
}