Carmine is a red colorant made from cochineal, the dried bodies of female Dactylopius coccus insects. E120 is its European additive number, and that one number covers cochineal, carminic acid, and carmines alike. Its international food additive code is INS 120.
Each name on a label points to part of the same product. Cochineal is the insect and its dried bodies. Carminic acid is the red compound inside them, and carmine is the finished pigment made from that compound. Natural Red 4 is its name in the Color Index, the classification system for dyes and pigments, and 75470 is its number there, which cosmetic labels print as CI 75470.
Carmine colors foods, drinks, medicines, and cosmetics in shades from pink to deep red and plum. In food it turns up in fruit yogurts, red and pink soft drinks and juice drinks, gummy sweets, ice cream, and red snack coatings, as well as in some liqueurs and brightly colored cereals. US rules allow it in foods generally, in ingested and externally applied drugs, and in cosmetics including products for the eye area.
Two synthetic reds carry names that echo carmine and cochineal. Carmoisine (E122) is a laboratory azo dye, and so is Ponceau 4R (E124), which is also called Cochineal Red A. Both count as vegan by composition. E120 is the number that marks the insect pigment.