Torsion points on $E/\mathbb{Q}$ with large coordinates

Question

Let $E/\mathbb{Q}$ be an elliptic curve with finitely many rational points.

What are some examples where at least one rational point has large coordinates (compared to the height of $E$)?

Answer 1

Generalizing Noam's comment and expanding a bit on anon's comment, if you put your curve in Weierstrass form $$ y^2 = P(x) = x^3+ax^2+bx+c\quad\text{with $a,b,c\in\mathbb Z$,} $$ then the Lutz-Nagell theorem says two things:

• Any torsion point $(x,y)\in E(\mathbb Q)$ satisfies $x,y\in\mathbb Z$.
• For such a point, either $y=0$, which case the point has order 2, or else $y^2$ divides the discriminant of $P(x)$.

In particular, if you put $E$ in short Weierstrass form $$y^2=x^3+Ax+B\quad\text{ with $A,B\in\mathbb Z$},$$ then any torsion point $(x,y)\in E(\mathbb Q)$ of order at least $3$ has integer coordinates satisfying $y^2\mid16(4A^3+27B^2)$. So if the point has large coordinates, then so does $E$.

An alternative version of this is to note that torsion points have canonical height $0$, and the canonical height differs from the Weil height by a quantity bounded in terms of the coefficients. Explicitly, there are absolute constants $c_1>0$ and $c_2$ so that for all elliptic curves as above and all rational points $(x,y)$, we have $$ \Bigl| h([x,y,1]) - \hat h([x,y,1]) \Bigr| \le c_1 h([A^3,B^2,1]) + c_2.$$ In particular, if $(x,y)$ is a torsion point, then $$ h([A^3,B^2,1]) \ge c_1^{-1} h([x,y,1]) - c_1^{-1}c_2. $$ One can find various explicit values for $c_1$ and $c_2$ in the literature.