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8 votes
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Cubic twist of elliptic curves and its rank

There is a formula but it involves both cubic twists. Let $E: y^2 = x^3+B$ be an elliptic curve over $\mathbb{Q}$ with $j=0$ as the one in the question. Let $D$ be a cubefree integer. Set $E_1: y^2=x^...
Chris Wuthrich's user avatar
0 votes

On Iwasawa theory of elliptic curves in $\mathrm{PGL}_2(\mathbb{Z}_p)$-extension

It seems that I have obtained some positive results, but I couldn't believe my "solution" is correct since I haven't completely followed Greenberg's hints. Throughout, $p$ is an odd prime. ...
Hetong Xu's user avatar
  • 507
9 votes

When are two elliptic curves with zero j invariant isogenous?

The curves $E_B$ and $E_C$ are isogenous over $\mathbb{Q}$ if and only if $C=u^6B$ or $C=-27u^6B$ for some $u\in\mathbb{Q}^\times$. In other words, up to isomorphism there are exactly two curves ...
Jonathan Love's user avatar
2 votes

When are two elliptic curves with zero j invariant isogenous?

Two elliptic curves are isogoneous if and only if they generate isomorphic Galois representation. So after computing the conductor and then the Sturm bound you know how many $a_p$ to look at to ...
Watson Ladd's user avatar
  • 2,399
5 votes

Upper bound on number of integral solutions of elliptic curves

First, the bound that you cite has been significantly improved by Alpöge and Ho. Here is Theorem 1.1 therein. Let $A,B\in\mathbb{Z}$ satisfy $\Delta_{A,B}:=-16(4A^3+16B^2)\neq 0$. If $\mathscr{E}_{...
2734364041's user avatar
  • 5,014
4 votes

Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

There are likely more-elementary approaches, but here is an argument using some basic function-field arithmetic. Let's work a bit more generally than in the original question, considering a base field ...
user516477's user avatar
1 vote

Upper bound on number of integral solutions of elliptic curves

You can take any $\epsilon>0$ you like. For example, by taking $\epsilon = 0.1118 - 0.1117...$ we get a bound $C|Disc(E)|^{0.1118}$ for some constant $C$. This is an "upper bound which is not ...
Bogdan Grechuk's user avatar

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