In an attempt to take the castle, Itakura Shigemasa was killed. More shogunate troops under Matsudaira Nobutsuna, Itakura's replacement, soon arrived. However, the rebels at Hara Castle resisted the siege for months and caused the shogunate heavy losses. Both sides had a hard time fighting in winter conditions. On February 3, 1638, a rebel raid killed 2,000 warriors from the Hizen Domain. However, despite this minor victory, the rebels slowly ran out of food, ammunition and other provisions. By April 1638, there were over 27,000 rebels facing about 125,000 shogunate soldiers. Desperate rebels mounted an assault against them on April 4 and were forced to withdraw. Captured survivors and the fortress' rumored sole traitor, Yamada Emosaku, revealed the fortress was out of food and gunpowder. On April 12, 1638, troops under the command of the Kuroda clan of Hizen stormed the fortress and captured the outer defenses. The rebels continued to hold out and caused heavy casualties until they were routed on April 15.

Answer this question based on the article: How many days after troops under the command of the Kuroda clan of Hizen storm the fortress and capture the outer defenses were the rebels routed?
3