Imaginary quadratic fields F with X₀(15)(F) finite

Caraiani and Newton have proven that if F is an imaginary quadratic number field such that X₀(15) has rank $0$ over F, then every elliptic curve over F is modular. This paper is concerned with the quadratic fields F=Q(√-p) for a prime number p. We give explicit conditions on p under which the rank is $0$, and prove that these conditions are satisfied for $87,5%$ of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank $0$ follows from a $4$-descent over Q on the quadratic twist X₀(15)₋ₚ. To prove this, we perform two consecutive $2$-descents and prove this gives rank bounds equivalent to those obtained from a $4$-descent using visualisation techniques for Sha[2]. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right.