Each Eidolon has a specific type of Ore that affects their attack power. Generally speaking attack power increases with each type of Ore in your inventory. The exceptions are Phoenix, Odin and Madeen which are covered below.

Ramuh : 32 + Peridots
Leviathan: 59 + Aquamarine
Ifrit : 42 + Topaz
Shiva : 36 + Opal
Atomos : 30 + Amethyst (Each Amethyst will result in an extra 1% of HP taken off the target. Damage is capped at 9999. Atmos doesn't work on bosses. Note that is effects Maximum HP, not current HP)
Bahamut: 88 + Garnet
Ark : 106 + Lapis Lazuli
Fenrir : 42 + Sapphire
Fenrir : 44 + Sapphire (There are two formula here for the two different attacks Fenrir can use).

Odin is the opposite in this regard. With him, the more Ore you have in your inventory, the less damage he will do with his sword attack (obtained by having the Odin's Sword ability equipped). The formula is 145 - Ore. Conversely, the more Ore you have the higher chance his normal Instant Death attack has of occuring, with every two ore's adding an extra 1% chance.

Carbuncle doesn't have a bonus, however he does gain different abilities depending on whether or not Eiko has one of the ore's equipped;

-- : Ruby Light = Reflect (+ Protect if full animation)
Emerald : Emerald Light = Haste (+ Protect if full animation)
Diamond : Diamond Light = Vanish (+ Protect if full animation)
Moonstone: Pearl Light = Shell (+ Protect if full animation)

Also Phoenix doesn't have any bonuses, but does gain a bonus from Phoenix Pinions. If Eiko is in your party, Phoenix has a % chance to appear upon the party being wiped out, equal to [(# of Phoenix Pinions / 256) * 100].

Madeen doesn't get a bonus from any Ore, but does get a bonus based on Eiko's level.

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