|Jira Agile 6.3
Basically the API is /api/rank/before (or /api/rank/after) you can access it for example: http://JIRA_Host/api/rank/before
The parameters for /before are:
For /after its rankAfterId/rankAfterKey but they are otherwise the same. You can trigger REST calls using curl, or a Google REST console, or a native plugin, such as the atlassian developer toolbox. Please learn more from our API's by using our REST API browser from the atlassian developer toolbox. Once that's installed log into JIRA > Administration > Scroll down > REST API Browser. There are more general examples using JIRA here and further details and examples here.