## New answers tagged elliptic-curves

8
votes

Accepted

### 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^...

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.
...

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 ...

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 ...

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}_{...

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 ...

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 ...

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