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 > https://www.lmfdb.org/EllipticCurve/ yu can click on "some interesting > elliptic curves" where one of the examples is " Good reduction but no > minimal model" --> https://www.lmfdb.org/EllipticCurve/2.2.229.1/1.1/a/1. 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! Cheers Bill.