It is known that the complex isomorphism class of an elliptic curve $E/\mathbb{C}$ is uniquely determined by its $j$-invariant. One way to define it algebraically for a curve
$$\displaystyle E : y^2 = x^3 + Ax + B, A,B \in \mathbb{C}$$
is by
$$j(E) = 1728 \cdot \frac{4A^3}{4A^3 + 27B^2}.$$
Of course $j$ is also a modular function, and can be expressed as $j(\tau)$, where $\tau$ is a complex number such that $E/\mathbb{C} \cong \mathbb{C}/\langle 1, \tau \rangle$. By reduction theory, we can further assume that $\tau$ is in the Gauss fundamental domain in the upper half-plane: that is, we may assume that $|\Re(\tau)| \leq 1/2$ and $\Im(\tau) \geq 1$.
If we assume that $A,B \in \mathbb{Z}$ (or more generally, simply algebraic integers), then we can attach a modular height to the elliptic curve by simply evaluating the Weil height (as an algebraic number) of the $j$-invariant of $E$, say $h(j(E))$.
When $\gcd(A,B) = 1$, then $h(j(E))$ is typically close to the so-called naive height of $E$, namely $H(E_{A,B}) = \max\{4|A|^3, 27B^2\}$. However, there are cases when the two heights differ significantly.
Another way to measure the size of $j(E)$ with $E$ defined over $\mathbb{Q}$ is to simply take its absolute value. In this case, we see that if we expect $4A^3$ and $4A^3 + 27B^2$ to be roughly the same size, then $|j(E)|$ is typically very small.
Therefore, elliptic curves over $\mathbb{Q}$ whose $j$-invariants have very large absolute value are exceptional, as they correspond to those curves whose discriminants are very small compared to the naive height of the curve.
My question is, what arithmetic information can one glean from a curve $E/\mathbb{Q}$ with $\gcd(A,B) = 1$ (or more generally, taking a minimal element in the twist family) whose $j$-invariant has very large absolute value? Alternatively, such an elliptic curve has a period $\tau$ in the fundamental domain with large imaginary part.
