One of those rare quests with no navpoints. Check the map above for all thirteen artifacts, though only eight are required for mission completion.
Get the task itself from Sage Fleera in High Mithrava.
She'll send you hunting for Remnant tech that's been pushed to the surface of Havarl- they look like data cores that have fallen on their sides, but their most distinctive feature is the waves of electricity that ripple off them. Look for those flashes, and look extra hard near the glowing blue grass that tends to hide such flashes. The artifact locations are:
At the Pelaav Research Station, underneath the shuttle platform (this is the same platform where the cowgirl trader Cody eventually shows up)
Amidst some rocks north of the research station, near the researcher's house from Forgotten Stars.
Up on a jutting rock formation north of the final monolith.
Near the Roekaar base on the far side of the Chasm bridge, ground level, to the south.
On a clump of rocks just south of the cul-de-sac that has Thaldyr's house.
On a rock near the water, in the ravine north of the Ancient Courtyard.
On a rocky outcropping south and down-the-wall from the minivault which holds the remains of Zorai's previous heir.
On the same level as the minivault entrance, just outside the cave in the southwest corner of this area.
Just southwest of the Vault entrance.
There's rectangular Remnant platform south of the Forward Station in th Ancient Courtyard. The artifact is in the northwest corner of this platform, guarded by Roekaar.
Almost directly above the Forward Station at the north end of the Chasm of the Builders.
From the Tempest's landing spot, hug the south wall and move west to discover this one.
At the base of clump of rocks, to the south of the eastern end of the bridge across the chasm.
Between the two Remnant spires southwest of the research station (you'll bee in this area during the start of your search for the Adrenaline Fusion Mod).
You only need eight to complete the mission, but the others can still be gathered for Remnant research data.