# Mordell-Weil groups and isogenies of the families of elliptic curves

Mikić, Miljen (2014) Mordell-Weil groups and isogenies of the families of elliptic curves. Doctoral thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this work it is proved that the torsion group of elliptic curves induced by $D(4)$-triples can be either $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$. Therefore, as a special case it is proved that the torsion group of elliptic curves induced by Diophantine triples can be one of these two, as well. Families of elliptic curves induced by Diophantine triples of the form $\{k − 1, k + 1, c_{l}(k)\}$ are examined and the possible form of the torsion group and rank (i.e. Mordell-Weil group) of these curves is determined for the large number of values of $k$ and $l$. Finally, by studying modular curves $X_0(n)$ the number of elliptic curves with cyclic isogeny of degree $n$ over various quartic fields is found.