The main goal of this rune page is to gain bot lane prio early. This is why we take PTA for the burst, Coup de Grace to add on our aggressiveness, and Triumph to stay alive and gain money. We also take triumph over Presence of Mind due to our build having 2 mana items: Essence Reaver, which refunds mana, and Manamune. These 2 mana items are enough to keep us from going OOM (out of mana). Hence, Presence of Mind is not needed BUT it depends on preference. If you feel like you run out of mana in lane fast and that biscuits aren't enough, then Presence of Mind is still a good choice.
We take Longsword + 3 Pots (refillable if enemy has no all-in or poke threat) to get as much AD as possible, and to not delay our build.
Quick Ezreal AD ADC Build
Summoner Spells
Starting Items
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Conqueror? First Strike?
The rule of thumb when going Conqueror is to answer these questions:
1. Are there many melees?
2. Will fights be extended?
if the answer to these is yes, then Conqueror is a good choice.
Conqueror can also be played with the lethality build but it is primarily paired with the Triforce build. Because it is primarily played with Triforce, we take Sorcery secondary to make up for lost ability haste.
Regarding First Strike, I haven't played it enough to give it a detailed response on. First Strike in my opinion is not a good rune for Ezreal though because it's outperformed by Precision keystones and is only really useful for cheesy nukes with R and W/E (probably). Also, poke lanes are running rampant in bot lane so it can easily get put on CD. Another thing is that even if you proc it constantly in the early game, you can't follow up with enough damage to make it any meaningful (probably).
Vs. Melee
Summoner Spells & Items Loadout
Keep in mind that your gameplan/enemy draft can still affect your decision in choosing summoner spells (e.g taking cleanse-flash w/ bard because of enemy cc threat like lux/ashe).
Summoner Spells
Item Builds
Skill Path
Standard skill priority with Ezreal, don't go anything else unless you're playing AP.