What was the name of Beethoven's famous piano piece for Elise?

Bagatelle No. 25 in A minor for solo piano, commonly known as "Für Elise", is one of Ludwig van Beethoven's most popular compositions. It was not published during his lifetime but discovered 40 years after his death, and may be termed either a Bagatelle or an Albumblatt. The identity of "Elise" is unknown; researchers have suggested Therese Malfatti, Elisabeth Röckel, and Elise Barensfeld.