Also in the paper labelled DFKcyclicfields, the result of Theorem 2.1 gives a value for the algebraic value of the central $L$-value of an elliptic curve $L$-function twisted by a character of odd prime order (called $k$). The analytic value of Sha (divided by the corresponding value over $\QQ$) is just the product of these values over all the Galois conjugates, and so os the norm from $\QQ(\chi_k)$ to $\QQ$ of this value. Since the $n_E(\chi)$ all lie in the real subfield, this shows that this norm is $K$ $\times$ a square in $\QQ$ for all curves for which the sign $w_E = -1$, and a square if $w_E = +1$. You might want to verify this numerical in examples. This would say that for any $E$ of sign $w_E = -1$, you should find a non-square She (extra $k$-factor) for any character of order $k.$ Best, Hershy