Mix-up in 54bX

[Måns Daniel Larsson]

Hi,

There seems to be some mix-up with the entries for 54bX in 

Best regards,

/Daniel

Professor of Mathematics

Faculty of Technology, Natural Sciences and Maritime Sciences
University of South-Eastern Norway
Kongsberg, Norway

www.numberlab.se

[Edgar Costa]

Would you like to elaborate on what you think is the mix-up?

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[John Cremona]

In particular, for this isogeny class the LMFDB and Cremona numberings of the curves are different.

John Cremona 

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[Edgar Costa]

Hopefully by displaying all the labels it will solve the mix-up:

https://beta.lmfdb.org/EllipticCurve/Q/?hst=List&conductor=54&search_type=List&showcol=cremona_label.cremona_iso

[Måns Daniel Larsson]

Sage produces the following 

Which is not the same as the one in table:

/Daniel

Professor of Mathematics

Faculty of Technology, Natural Sciences and Maritime Sciences
University of South-Eastern Norway
Kongsberg, Norway

www.numberlab.se

[Måns Daniel Larsson]

Ahhh, now I see: there are two different labels. But is that such a good idea? This confused me… I’m maybe not the only one…

/Daniel 

Professor of Mathematics

Faculty of Technology, Natural Sciences and Maritime Sciences
University of South-Eastern Norway
Kongsberg, Norway

www.numberlab.se

2 juni 2023 kl. 16:41 skrev Daniel Larsson <daniel....@usn.no>:

Sage produces the following 

<Skärmavbild 2023-06-02 kl. 16.39.22.png>

Which is not the same as the one in table:

[John Cremona]

On Fri, 2 Jun 2023 at 15:45, Måns Daniel Larsson <Daniel....@usn.no> wrote:

Ahhh, now I see: there are two different labels. But is that such a good idea? This confused me… I’m maybe not the only one…

Yes indeed, it is confusing (and has cost me many hours of my life!).  When I started tabulating elliptic curves I was not very careful about labelling them in a deterministic way, so the Cremona labels for curves of conductor up to a few hundred are fairly random.  Since I published those tables (up to conductor 1000) in 1002, I though I had an obligation not to change the labels.  But when the LMFDB came along it was decided to have a systematic labelling which is deterministic (and for isogeny classes, agrees with the labelling of the associated newforms).  So curves have two labels (at least for conductors up to 500000).

For the isogeny classes, I switched to the LMFDB ordering at that time so there should be no discrepancies beying conductor ~200k (I forget the last difference).  But for curves within each isogeny class, the LMFDB ordering is lexicographic ordering of the coefficients [a1,a2,a3,a4,a6] of a reduced minimal model, which is is easy to implment though the reults are often not nice (look at the curves of conductor 11, for example, and see which one is first).  One good thing about the Cremona labels within an isogeny class is the curve #1 is the optimal one (with respect to Gamma_0(N)).

You can see a description of the labelling system by clicking on the link "elliptic curve labels" under "Learn more" on the rght side of elliptic curve pages, that takes you to http://www.lmfdb.org/EllipticCurve/Q/Labels.

Plenty of people use each of the two labelling systems.  Sage's labels are Cremona labels (E.label() is an alias for E.cremona_label()), same in Magma and gp.

John