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An integral binary quartic form is said to be locally soluble (resp. soluble)if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$for every place $v$ of $\mathbb{Q}$ (resp. over $\mathbb{Q}$). We consider theproportion of soluble integral binary quartic forms in locally soluble forms.Bhargava showed the proportion is positive when one considers all binaryquartics, and Bhargava--Ho proved the proportion is zero for a subfamily. Inthis paper, we estimate the proportions for some other subfamilies. It relieson results for elliptic curves $y^2=x^3-n^2x$ by Heath-Brown, Xiong--Zaharescuand Smith.

On the proportions of soluble forms in some families of locally soluble binary quartic forms