  A theorem of Serre states that almost all plane conics over $\mathbb{Q}$ have
no rational point. We prove an analogue of this for families of conics
parametrised by elliptic curves using elliptic divisibility sequences and a
version of the Selberg sieve for elliptic curves. We also give more general
results for specialisations of Brauer groups, which yields applications to norm
form equations.
