The closest serial # to your 13604 - serial #13586. However, serial # 13586 was listed in two different volumes of my SRS Research books as having been issued to two different units?
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One unit was Company E, 7th Indiana Volunteer Cavalry in March, 1865, and the other unit was . . . Company D, 6th Illinois Volunteer Cavalry, 1865 (no day or month listed).
Check for the same serial # under the forestock on underside of barrel.
Not sure if this info helps.
Two Flints