On the rank of elliptic curves over $\mathbf{Q}(\sqrt{-3})$ with torsion groups $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/3\mathbf{Z}$ and $\mathbf{Z}/3\mathbf{Z} \times \mathbf{Z}/6\mathbf{Z}$
نویسندگان
چکیده
منابع مشابه
On the rank of certain parametrized elliptic curves
In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
متن کاملOn the rank of elliptic curves over Q(i) with torsion group Z/4Z× Z/4Z
We construct an elliptic curve over Q(i) with torsion group Z/4Z× Z/4Z and rank equal to 7 and a family of elliptic curves with the same torsion group and rank ≥ 2.
متن کاملTorsion of elliptic curves over cubic fields
Although it is not known which groups can appear as torsion groups of elliptic curves over cubic number fields, it is known which groups can appear for infinitely many non-isomorphic curves. We denote the set of these groups as S. In this paper we deal with three problems concerning the torsion of elliptic curves over cubic fields. First, we study the possible torsion groups of elliptic curves ...
متن کاملOn the torsion of rational elliptic curves over quartic fields
Let E be an elliptic curve defined over Q and let G = E(Q)tors be the associated torsion subgroup. We study, for a given G, which possible groups G ⊆ H could appear such that H = E(K)tors, for [K : Q] = 4 and H is one of the possible torsion structures that occur infinitely often as torsion structures of elliptic curves defined over quartic number fields. Let K be a number field, and let E be a...
متن کاملHigh rank elliptic curves with torsion Z/4Z
Working over the field Q(t), Kihara constructed an elliptic curve with torsion group Z/4Z and five independent rational points, showing the rank is at least five. Following his approach, we give a new infinite family of elliptic curves with torsion group Z/4Z and rank at least five. This matches the current record for such curves. In addition, we give specific examples of these curves with high...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 2011
ISSN: 0386-2194
DOI: 10.3792/pjaa.87.61