Let a differentiable function \( f \) satisfy the equation. If \( \int_{0}^{36} f\!\left(\frac{t x}{36}\right) dt = 4 \alpha \, f(x) \). If \( y = f(x) \) is a standard parabola passing through the points \( (2, 1) \) and \( (-4, \beta) \), then \( \beta^{\alpha} \) is equal to _____.
