Here is the short version:



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> Fix an elliptic curve $E/\mathbb{Q}$. How does the torsion structure $E\_d(\mathbb{Q})\_{tors}$ vary, as $E\_d$ runs through the quadratic twists of $E$?
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Here is the longer version:


I have been playing with SAGE this morning. I inserted the elliptic curve ('11a1') $$E : y^2 + y = x^3 - x^2 - 10x - 20$$ which has rational torsion subgroup isomorphic to $\mathbb{Z}/5\mathbb{Z}$. I then computed its quadratic twist $E\_d$ for all squarefree $d$ up to 2000, and observed $E\_d(\mathbb{Q})\_{tors}$ was always trivial. 



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> Can it be that, in this particular family of quadratic twists, all but one of the curves have trivial torsion? Is this a general phenomenon? 
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(I ran this experiment for several other curves $E$ and got the same impression; that all but one of the curves in a family of twists have the same torsion structure.) 



Torsion subgroups in families of twists of elliptic curves 
**Theorem** (originally due to Setzer?): Fix $E/\mathbb{Q}$ with $j(E)$ not 0 or 1728. Then for all but finitely many inequivalent twists $E\_d$, the torsion subgroup $E\_d(\mathbb{Q})\_{tors}$ is isomorphic to $E[2](\mathbb{Q})$, so in particular $E\_d(\mathbb{Q})\_{tors}$ has order 1, 2, or 4. (Probably he also proved it for number fields.)


There's a paper of mine$^1$ with a much more general theorem using the theory of heights. I don't recall Setzer's proof except that it doesn't use heights.


**Theorem**: Let $K$ be a number field and let $A/K$ be an abelian variety with $\mu\_n\subset {\rm Aut}(A)$. (This means we can twist $A$ by $n$'th roots of $d$.) Then every point $P\in A\_d(K)$ satisfies one of the following two conditions:


* $P$ is fixed by a non-trivial $\zeta\in\mu\_n$.
* $\hat h(P) \ge C\_1(A)h^{(n)}(d) - C\_2(A)$.


Here $\hat h$ is the canonical height relative to an ample symmetric divisor, and $h^{(n)}(d)$ is a sort of "$n$'th power free height," say equal to the minimum of $h(du^n)$ for $u\in K^\*$. The constants depend on $A$, but are independent of $d$.


It follows from the theorem that after discarding finitely many $d \in K^\*/{K^\*}^n$, a point in $A\_d(K)$ is either $1-\zeta$ torsion (hence $nP=O$), or its canonical height is positive, and hence it is nontorsion.


Of course, to describe more precisely what happens for the finitely many exceptional $d$ can be a delicate matter, as some of the other answers have indicated. I think it's interesting to see how one can approach the problem via heights or via representation theory.


$^1$ J.H. Silverman, Lower bounds for height functions, *Duke Math. J.* **51** (1984), 395-403.


EDIT: Fixed statement of first theorem. I'd originally written that "for all but finitely many inequivalent twists $E\_d$, the torsion subgroup $E\_d(\mathbb{Q})\_{tors}$ has at most two elements." This is clearly false, since if $E$ has the form $E:y^2=(x-a)(x-b)(x-c)$ with $a,b,c\in\mathbb{Q}$, then $E\_{d}[2](\mathbb{Q})$ has order 4 for every twist.


