One of the new items you'll find while out on Patrol in Destiny: The Taken King's Dreadnaught area is the Agonarch Rune. With it, you'll be able to participate in an event that will reward you with a Calcified Fragment and other loot.
To first get the Agonarch Rune, you must defeat a unique set of Taken enemies while on Dreadnaught Patrol. Walk around the area until you spot a glowing, floating white orb surrounded by a circle. Walk into it and Taken enemies will spawn. Kill all the Taken as they spawn in waves until a Major—an enemy with a yellow health bar and possibly even a unique name—spawns. Kill him and the rest of the group and you will be rewarded with an Agonarch Rune.
You'll notice that the Agonarch Rune has room for 7 charges on it. To fill up these charges, you must kill seven groups of these Taken spawns for a full week. It doesn't matter where you do it, as they are scattered throughout the Dreadnaught. Check out Hull Breach, the area by the entry point or look further in at the Mausoleum.
For each kill, you will receive one charge to power up the Agonarch Rune. Fortunately, you can cut down the amount of time it takes to charge up the Agonarch Rune to 3 days by transferring it to your other characters. Each character can charge it up once per day.
With a Charged Agonarch Rune, head to the Mausoleum and go to the far side by the chasm and enter the pathway that leads to the Founts. From there, jump into the level below and make a right. Proceed forward to the other side of the room until you find a small opening in the wall. Crouch through it to enter a room with a glowing terminal on the right. Activate this terminal with the Charged Agonarch Rune to activate the event.
Now leave the room and go towards the large chamber with the big chained platform in the center. A host of Taken bosses will spawn—the ones you killed to first charge the rune. Kill them all and a Zealot Champion will spawn. Kill him too.
Once that's done, you'll be able to place the Charged Agonarch Rune into a chest that appears in the middle to obtain a Calcified Fragment, a Moldering Shard, and some other loot. You'll also receive reputation with Eris Morn.
Be sure to check out our guides on the fastest way to earn Legendary Marks, Infusing weapons and armor, the fastest way to raise your light level, and how the Exotic Blueprint system works. Here's our guide on finding all 50 Calcified Fragments.