Tables for the paper "A lot of fudge around A + B = C"

News

We (David Broadhurst and me) now have much more extensive tables, including new records.
The paper is being rewritten. In the meantime our new tables are available here to download.

New record

The elliptic curve y2 + x y = x3 - 3963439038325921420 x + 2914023365721768114379552400
has conductor 766275510 and Tamagawa product 17915904, leading to a record Tamagawa quality of 2.4641718426.
There are two different abc-triples from which this curve originates, they are
a, b = 1514143744, 12973390625 with quality 1.14369,
a, b = 2051393825, 48174982144 with quality 1.20447 (twisting by -1).

New tables

Format used in the files:
[Tamagawa quality, Tamagawa product, nfac, conductor, Weierstrass coefficients, origins]
where "nfac" lists primes with exponents > 1 in the conductor,
and "origins " lists originating abc-triples with twist used, in the format [twist, a, b].
When a = b = 0, the origin is not an abc-triple but the LMFDB-database.

Small file with all 253 curves with Tamagawa quality > 2.2 coming from abc-triples with quality > 1.1 or from the LMFDB database:
output_all_pari_qua_above_2.2.txt

Big files (each ≈ 200 MB, total ≈ 3.5 GB), containing all 8884725 unique curves (at most one million per file) with
Tamagawa quality > 1.5 coming from abc-triples with quality > 1.15 or from the LMDB-database, or
Tamagawa quality > 2.0 coming from abc-triples with quality > 1.1:
curves (sorted for Tamagawa quality) curves (sorted for Tamagawa product)
output_all_pari_qua_1.txt (195 MB) output_all_pari_tam_1.txt (226 MB)
output_all_pari_qua_2.txt (199 MB) output_all_pari_tam_2.txt (218 MB)
output_all_pari_qua_3.txt (200 MB) output_all_pari_tam_3.txt (214 MB)
output_all_pari_qua_4.txt (201 MB) output_all_pari_tam_4.txt (209 MB)
output_all_pari_qua_5.txt (202 MB) output_all_pari_tam_5.txt (205 MB)
output_all_pari_qua_6.txt (203 MB) output_all_pari_tam_6.txt (200 MB)
output_all_pari_qua_7.txt (204 MB) output_all_pari_tam_7.txt (193 MB)
output_all_pari_qua_8.txt (204 MB) output_all_pari_tam_8.txt (181 MB)
output_all_pari_qua_9.txt (180 MB) output_all_pari_tam_9.txt (138 MB)

Authors

David Broadhurst, Open University, Milton Keynes, United Kingdom.
Benne de Weger, Eindhoven University of Technology, Eindhoven, The Netherlands.
March 9, 2023

Tamagawa in Japanese characters

Old stuff below:

Paper

A lot of fudge around A + B = C, version 1.0, January 7, 2023.

Abstract

This paper reports on experiments searching for elliptic curves over ℚ with large Tamagawa products. The main idea is to look among curves related to good abc-triples.

Tables

database abc-triples curves (sorted for Tamagawa quality) curves (sorted for Tamagawa product)
Cremona output_cremona_qua.txt output_cremona_tam.txt
high quality input_high_quality.txt output_high_quality_qua.txt output_high_quality_tam.txt
medium quality input_medium_quality.txt output_medium_quality_qua.txt output_medium_quality_tam.txt
high merit input_high_merit.txt output_high_merit_qua.txt output_high_merit_tam.txt
unbeaten input_unbeaten.txt output_unbeaten_qua.txt output_unbeaten_tam.txt
triples from triples input_triples_from_triples.txt output_triples_from_triples_qua.txt output_triples_from_triples_tam.txt
all together output_all_qua.txt output_all_tam.txt
Note that only in the "all together" files the curves are given in Weierstrass minimal model. In the other files this is not necessarily so.
Also note that some curves occur in more than one database. So the "all together" files are smaller than all files they are created from together.

Figure (high resolution)


(click to download)

Links

John Cremona's Database
LMFDB
Bart de Smit's abc-triples website

Author

Benne de Weger, Eindhoven University of Technology
January 7, 2023