to obtain a Agonarch Rune you have to play the mini Taken pubic event in the patrolmode on Dreadnaught. On this event you charge also the Agonarch Rune. You receive one point each day for playing and beating this mini event. Currently you can with your 2nd and 3th Char also charge the Agonarch Rune at the same day, so you can receive 3 of 7 points in 1 day! I personally think, it will be patched in the future.
After you have charged the rune, go to the Founts area in Patrol mode on Dreadnaught and use your charged Agonarch Rune at the machine. Now you have to beat all minibosses again and a zealot champion. This step is not so easy and you should bring some friends with for this step. The chest appears after you kill the zealot champion and into the chest is the Calcified Fragment XXXVI.