# Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$.

Let $K$ be a number field and let $X$ be a curve over $K$.

Define $f_K(X)$ to be the value of $f$ after base change to $\overline{\mathbf{Q}}$.

Now, a priori, this function $f_K$ is not well-defined. In fact, one has to choose an embedding $K\subset \overline{\mathbf{Q}}$.

I do think that this is independent of the choice of embedding under some mild (or maybe none?) hypthesis on $f$. But why exactly?

Is $f_K$ well-defined? Do isomorphism $K$-curves take the same value?

Let me say that one can ask the same question for any function on some class of varieties over $\overline{k}$, where $k$ is a field.

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In general $f_K$ will depend on the embedding $K \hookrightarrow \overline{\mathbb{Q}}$. Take for instance the real part of the $j$-invariant of an elliptic curve, and consider elliptic curves with $j$-invariants $\pm \sqrt{2}$.