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Results tagged with elliptic-curves
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user 4140
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
7
votes
The rank of $y^2=x^3\pm i$
The two curves are isomorphic via: $(x,y) \mapsto (-x,i y)$, so we only need to do one of them. Let
$$
E \; : \; y^2=x^3+i.
$$
By inspection, this has a point $P=(-i,1+i)$. In fact $P$ has infinite or …
- 3,142
17
votes
The rank of a class elliptic curves
Although Junkie has answered the question, I'd like to point out that in the case of parametrized families of elliptic curves (such as this) it is often easy to find an explicit subfamily with positiv …
- 3,142
10
votes
Accepted
Trivial Weil-Châtelet group
Weil-Chatelet groups are huge. A theorem of Shafarevich states that if $n \ge 2$ and if $E$ is an elliptic curve (or an abelian variety) over a number field $k$ then $H^1(G_k,E)$ has infinitely many e …
- 3,142
5
votes
Accepted
Does the modified Szpiro conjecture require minimal model?
The conductor does not depend on the model. The $c_4$ and $c_6$ do. In fact as $E$ varies among the different integral models for the same elliptic curve, the left-hand side of the inequality differs …
- 3,142