Temperature is perhaps the primary factor, but there are many others; -40 (and a few other numbers in s similar range (eg, -35, -37, etc) are sometimes thrown out as being the point after which (below which) ice formation will occur without condensation nuclei or surface tension disruption on the water droplets, but that's over-simplified. Were it not so, the government(s) wouldn't have spent ridiculous sums of money over decades attempting to eliminate tell-tale contrails.

The oversimplification stem from the principle that all water vapor below a certain point freezes, indicating that regardless of other factors, the condensation trail should form at or below that absolute value....most hold that at -40 and below, all water moisture is at or below freezing in the atmosphere, which isn't necessarily true, nor does it necessarily mean a visible contrail will form. One of the byproducts of combustion of fuel in an engine is water vapor, and in high concentrations in the exhaust stream, it can be visible; in many cases, rapidly freezing and forming a visible contrail. The key isn't necessarily ice formation, as much as being visible. There are two types (broadly speaking) of condensation trails; we frequently see the visible formation of a trail around propellers, wingtips, etc, in high humidity, where air pressure drops enough to cause condensation; this typically dissipates or evaporates shortly after it forms and doesn't remain visible; it is also a form of condensation trail, and often takes place in temperatures well above freezing.

Fuel doesn't burn cleanly; it has material which is also expelled into the atmosphere as smoke or very small particles which form the nucleus of condensation drops, which can start the condensation process; water vapor coalesces around these small particles, aiding in condensation and increasing droplet size; these are called "condensation nuclei." In freezing conditions, the ability of a water droplet to remain supercooled, or liquid at a temperature less than freezing, depends on the chemical composition of the water, as well as the surface tension of a droplet that forms; anything that breaks that surface tension or changes the freezing point of the water droplet will affect the temperature at which the droplet turns to an ice crystal (often making it visible, and reflective). The concept of all droplets freezing and no longer remaining supercooled at or below a certain temperature is why -40 is often thrown out as the threshold for contrail formation, but again, that's not a correct assertion, and contrails frequently do not form at colder temperatures, and frequently form at warmer temperatures, too.