John Cremona on Mon, 11 Dec 2023 14:47:23 +0100

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

On Mon, 11 Dec 2023 at 12:03, Bill Allombert <> wrote:
On Mon, Dec 11, 2023 at 10:22:47AM +0000, John Cremona wrote:
> You can help yourself at  Note that
> if you go to the home page of any curve, for example (random!)
>, on the right-hand
> side you can see a link "Download all code to PariGP" which then shows you
> something like
> which
> gives you the code to create that curve in gp and compute some of the
> quantities there.  I think that with recent (and perhaps not so recent)
> advances in pari, there are more pieces of code which we could include.

Indeed, this is exactly what I did! However the LMFDB does not list points for
all curves of positive rank, and all the curves in the LMFDB are given
by nice models that does not trigger this bug.

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.
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" -->

If you need it I can send more examples.


> By the way, as ellbsd() was mentioned, I have a couple of comments: first
> that you cannot rely on having a global minimal model, as they do not
> always exist, so rather than having functions which give an incorrect
> answer when a model is not globally minimal, it is better (where possible)
> to adjust by the "difference" between the given model and a minimal model
> (which do of course exist locally).

Indeed, that is what PARI was attempting to do, except it assumed the model
was integral.