We construct an elliptic curve of rank at least 6 over $Q(t)$ with three non-trivial rational points of order 2.

Pour-El and Richards [5] discussed computable smooth functions with non-computable first derivatives. We show that a similar result holds in the case of Sobolev spaces by giving a non-computable $\mathcal{H}^1(0,1)$-element which, however, is computable in any of larger Sobolev spaces $\mathcal{H}^s(0,1)$ for any computable $s$, $0 \le s < 1$.

In this paper, we explicitly compute defining polynomials of the first layer of anti-cyclotomic $\mathbf{Z}_3$-extension of imaginary quadratic fields of class number 1.