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32 votes
4 answers
5k views

Why is this "the first elliptic curve in nature"?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as "The first elliptic curve in nature". It has minimal Weierstraß equation $$ y^2 + y = x^3 - x^2. $$ My guess is that there is some ...
David Roberts's user avatar
  • 35.5k
26 votes
2 answers
1k views

Why do the $2$-Selmer ranks of $y^2 = x^3 + p^3 $ and $y^2 = x^3 - p^3 $ agree?

I was playing around with sage, when I found that the Mordell-Weil ranks (over $\mathbb{Q}$) of the elliptic curves $y^2=x^3+p^3$ and $y^2=x^3-p^3 $ almost always agree, for $p$ prime. The first few ...
Milton's user avatar
  • 582
4 votes
4 answers
609 views

Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ \prod_{i=1}^...
Igor Rivin's user avatar
  • 96.4k
3 votes
1 answer
164 views

Existence of Pillai equations with Catalan type solutions?

In Catalan's conjecture we have $$x^m-y^n=1$$ having solution $(3,2,1,1)$ and $(3,2,2,3)$. Call $$ax^m-by^n=k$$ to be Pillai Diophantine equation. Is it true no Pillai Diophantine equation exists ...
Turbo's user avatar
  • 13.9k
1 vote
0 answers
143 views

A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is ...
roydiptajit's user avatar
0 votes
1 answer
107 views

Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?

Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
Dimitri Koshelev's user avatar
-5 votes
1 answer
150 views

On Mordell equation $y^2=x^3+k$ [closed]

Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not? Please Could you tell me about a good review papers about such equation.
Alpha's user avatar
  • 17