  We provide the first example of a finitely presented (in fact, type
$F_\infty$) group with elements whose stable commutator length is algebraic and
irrational, answering a question of Calegari. Our example is the lift to the
real line of the \emph{golden ratio Thompson group} $T_\tau$: the circle
analogue of the Cleary's golden ratio Thompson group $F_\tau$ which acts on the
interval.
