2
votes
1answer
346 views

Question on x coordinates of Mordell Curves where $y^2=x^3+k$ and $k^2 = 1$ mod $24$

In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$. In each case the x coordinates are ...
4
votes
1answer
226 views

Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...
3
votes
0answers
284 views

Rational points and Tesla cards

I'm rapidly approaching 300,000 curves in my ongoing search for Mordell curves of rank >=8. Currently I'm finding that I have a bottleneck in the code that locates rational points on these curves. ...
5
votes
2answers
508 views

12 descent scripts for pari/gp

I'm looking around for scripts to facilitate 12 descent on Mordell curves, preferably in Pari/gp. I understand that Magma implements this feature, but unfortunately this software isn't available to ...
13
votes
3answers
2k views

Rational Points on $y^2=x^3-86069^5$

The analytic rank of the Mordell elliptic curve $y^2=x^3-86069^5$ indicates that it has rank 2. However, deriving a set of generators, and hence the regulator, is proving to be a little bit of an ...
2
votes
1answer
379 views

How many points are there on an elliptic curve reduced at a bad prime?

Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, when $p$ is a good prime, a relation between the number of solutions over $\mathbb{F}_{p^n}$ ...
8
votes
3answers
592 views

Smart elliptic curve rational point search given Reg*#Sha

Hi folks, Let E be a global minimal model of an elliptic curve over QQ, with a 2-torsion point which generates the torsion subgroup, and with Mordell-Weil rank 1 (under BSD). Let RegSha be equal to ...
9
votes
1answer
375 views

Existence of hyperelliptic curve with specific number of points in a family

Hi, the following question was posed to me, it apparently has applications for linear codes. Let n>1, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ ...