Given an elliptic curve $E$ over a number field $K$ using an affine equation $f(x,y)=0$ with coefficients in $K$, sometimes it is possible to find an invertible change of coordinates that cause the coefficients of the equation to lie in a strictly smaller subfield $K'\subsetneq K$. In this case we say that $E$ is the base-change of an elliptic curve over $K'$. One necessary condition for something to be a base-change is of course that the $j$-invariant but this is not sufficient.

Are there any techniques or invariants or algorithms that guarantee that (the $K$-isomorphism class) of an elliptic curve with rational $j$-invariant is a NOT base change of one from a strictly smaller sub-field? Even in some special forms of $E$, maybe for CM-curves, or maybe some specific kinds of number fields, say real or imaginary quadratic fields etc?


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