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We consider the proportion of genus one curves over $\mathbb{Q}$ of the form$z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or moregenerally of the form $z^2+h(x,y)z=f(x,y)$ where also$h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have pointseverywhere locally. We show that the proportion of these curves that arelocally soluble, computed as a product of local densities, is approximately75.96%. We prove that the local density at a prime $p$ is given by a fixeddegree-$9$ rational function of $p$ for all odd $p$ (and for the generalisedequation, the same rational function gives the local density at every prime).An additional analysis is carried out to estimate rigorously the local densityat the real place.

The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point