  We prove that the irreducible components of the space of framed deformations
of the trivial 2-dimensional mod 2 representation of the absolute Galois group
of Q_2 are in natural bijection with those of the trivial character, confirming
a conjecture of B\"ockle. We deduce from this result that crystalline points
are Zariski dense in that space: this provides the missing ingredient for the
surjectivity of the p-adic local Langlands correspondence for GL_2(Q_p) in the
case p=2 (the result was already known for p\geq 3).
