When the BSD Real Period Remembers Elliptic Curves over $\mathbb{Q}$

We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period ${\mathrm{\Omega }}_E=\frac{\mathrm{\Gamma }(1\mathrm{/}4{)}^2}{2\sqrt{\pi }{17}^{1\mathrm{/}4}}$ and the Tate–Shafarevich group $\text{Sha}(E\mathrm{/}\mathbb{Q})\cong \mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}2\mathbb{Z}$ remember the elliptic curve $E:y^2=x^3+17x$. Let $K$ be a number field admitting a real embedding, and fix one such embedding $\sigma \text{\hspace{0.17em}⁣}:K\hookrightarrow \mathbb{R}$. We first prove that, for an elliptic curve $E$ over $K$, the real period of $E$ attached to $\sigma$ determines the $K$-isogeny class of $E$. We then prove that an elliptic curve $E$ over $\mathbb{Q}$ is determined up to isomorphism over $\mathbb{Q}$ by its real period, the number of connected components of $E(\mathbb{R})$, and the real period of the quadratic twist $E^D$ by a negative square-free integer $D\equiv 14$ coprime to the minimal discriminant of $E$. Moreover, over $\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).