Bill Allombert on Mon, 11 Dec 2023 15:01:32 +0100

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Re: getting strange ellheight() error for number field point and curve

> Thanks, it is not every day that someone compliments me on my nice models
> :)  When there is no global minimal model it is because a certain ideal is
> not principal, and one can choose an ideal in that "obstruction" class.  I
> (i.e. Sage, as I wrote that code) choose the first prime ideal in the class
> (where the curve has good reduction, I think) so that the model given is
> non-minimal at exactly one prime where the discriminant valuation is
> exactly 12 greater than for the minimal model at that prime.

So it is minimal at all primes of bad reduction, which is very nice indeed!

> > What I did was to apply some random [u,r,s,t] transformation, but my choice
> > was not random enough.
> >
> > Are there curves in the LMFDB which do not admit a minimal model ?
> > How to get them ?
> >
> There is not a way to search for these on the website, but on the page
> yu can click on "some interesting
> elliptic curves" where one of the examples is " Good reduction but no
> minimal model" -->

Alas, it is of rank 0

> If you need it I can send more examples.

If you could find one of rank >=1, preferably with one known non-torsion point,
it would be great!