Pyromancy Flame

The Pyromancy Flame is a magic tool similar to a Talisman or Catalyst, only it is used to cast Pyromancies.

Obtaining
A Pyromancy Flame is the standard starting equipment for a Pyromancer. It can also be gifted to you if you free Laurentius of the Great Swamp from captivity in the Depths and meet him back at Firelink Shrine, or you can get it from Eingyi in Queelag's Domain.

Upgrades
The Pyromancy Flame does indeed say that it scales with Intelligence, but this does not affect the damage output of cast Pyromancies, like with other Magic equipment. It only affects the R2 melee attack. Instead, the flame can be upgraded at certain NPCs.

There are two levels of upgrades for the flame, the first one, Laurentius' flame or your own flame (equally powerful) and the second one, Quelaana of Izalith's flame. The first one can be upgraded a total of 15 times by Laurentius or Eingyi. After this, the player must travel to Blighttown and fight their way to the lower swamp. There, on the small island with a large column, right outside the entrance to Quelaag, sits Quelaana of Izalith herself. She will offer to upgrade the flame to a stronger version, at a price. This can be done another five times. She will also be selling much stronger versions of the Pyromancies that Laurentius sells.

Keep in mind that fully upgrading the flame, including changing to Quelaana's fire, costs a total of 340500 Souls. This is equivalent to 55 Soul Levels (from Level 1).

The very first level of the flame has a base damage of 40 and Magic adjustment of 100. The level 15 version has a base damage of 40 and Magic adjustment of 190. The first level of Quelaana's flame has a base damage of 60 and Magic adjustment of 196, while the last level has a base damage of 60 and Magic adjustment of 230, making it almost 2,5 times more powerful than the very first flame.