Shinryu was added in patch 4.0 of the Stormblood expansion.
Shinryu becomes available at level 70 as part of the “Stormblood” quest, which is given by Pipin – The Lochs (X:32, Y:22).
The trial has a minimum ilevel of 280 to enter and rewards you with:
Allagan Tomestone of Poetics: 12