A gem birthed from a contract with Kyubey. Proof that one is a Puella Magi; it is the source of the girls' magical power.

Episode 6 spoilers:
As titled, it is the the container of their soul.
Episode 7 spoilers:
Stimulating the jewel gives various sensations to its owner.
Episode 8 spoilers:
If a soul gem darkens completely through magic use or negative emotions without regular cleansing through Grief Seeds, it becomes a Grief Seed itself, and its owner transforms into a witch.

Pixiv has a tag for images about the wrongful use of the stimulatory properties of the gem; "ジェム姦 (Gem Rape)"
The Pixpedia article is: ジェム姦
If you want to search for these images on Danbooru, try searching for soul_gem uncommon_stimulation.

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See also