TITLE: Torsion subgroups in families of twists of elliptic curves 
QUESTION [15 upvotes]: Here is the short version:

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 $E$?

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. 

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? 

(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.)

REPLY [10 votes]: The maximum number of quadratic twists with $n$-torsion (n odd) that an elliptic curve over a number field $K$ can have is 2, and here is an easy proof. 
You can see this by asking how many twists with $n$-torsion can an elliptic curve which already has $n$-torsion have? If this number is $k$, then clearly an elliptic curve without $n$-torsion can have $k+1$ twists with $n$-torsion.
Suppose now that $E/K,\ E(K)\supset \mathbb Z/ n\mathbb Z$, has 2 twists $E_{d_1}$ and $E_{d_2}$ with $n$-torsion. It follows from what Chris wrote that
$$E(K(\sqrt{d_1},\sqrt{d_2}))[n]\supset E(K)[n] \oplus E_{d_1}(K)[n]  \oplus E_{d_2}(K)[n]\supset (\mathbb Z/ n\mathbb Z)^3,$$
which is clearly impossible.
Note that the answer depends a lot on the number field $K$ and the number $n$ you choose. The possibility of 2 twists exists only over number fields $K$ such that the complete $n$-tosion can be defined over a quadratic extension of $K$. For example, it follows that over $\mathbb Q$, an elliptic curve can have at most 2 twists with 3-torsion, 1 twist with 5 or 7-torison and 0 twists with $p$ torsion for all primes $p>7$.  
So, Giuseppe, your curve $E$ with 5-torsion, as any other curve with 5-torsion, cannot have a twist with 5-torsion over $\mathbb Q$.