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^...
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0
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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.
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- 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 ...
- 1,133
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 ...
- 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}_{...
- 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 ...
- 41
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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