  Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that
admits a $k$-rational isogeny of prime degree $p$. We study the question of
finding a uniform bound on $p$ that depends only on $d$, and obtain, under a
certain condition on the signature of the isogeny, such a uniform bound by
explicitly constructing nonzero integers that $p$ must divide. As a corollary
we find a uniform bound on torsion points defined over unramified extensions of
the base field, generalising Merel's Uniform Boundedness result for torsion.
