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Format of elliptic curve database files
Each file consists of lines, one per curve, with certain data fields
on each line, separated by whitespace. The names of these data fields
used here are not standard except for "Cremona label" and "LMFDB
label", and are defined here for local reference only.
Data fields (and metafields combining several fields):
N: Conductor
CLASSCODE: Letter code of isogeny class (a, b, ..., z, ba, bb, ...)
in the Cremona labelling scheme (for many ranges this was
not deterministic and is only defined by the data files
here).
NCURVE: Number of the curve in its isogeny class (1,2,...)
LMFDB_CLASSCODE: Letter code of isogeny class (a, b, ..., z, ba, bb,
...) in the LMFDB numbering. This is by lexicographical
ordering of the vectors of AP over all primes, good and
bad, i.e. by lexicographical order of the integral
coefficients of the L-functions as a Dirichlet series.
LMFDB_NCURVE: Number of the curve in its isogeny class (1,2,...) in
the LMFDB numbering. This is by lexicographical ordering
of the vector [a1,a2,a3,a4,a6] of the reduced Weierstrass
coefficients; since a1,a2,a3 are the same in any class
this is also the lexicographical order of [a4,a6].
ID: Concatenation of (N,CLASSCODE,NCURVE) separated by whitespace,
e.g. 11 a 1
IDX: Concatenation of (N,CLASSCODE,NCURVE) with no whitespace,
e.g. 11a1
CREMONA_LABEL: Concatenation of (N,CLASSCODE,NCURVE) with no
whitespace separation, e.g. 11a1. This is called the
"Cremona label" of the elliptic curve.
LMFDB_LABEL: Concatenation of (N,".",CLASSCODE,NCURVE) with no
whitespace separation) e.g. 11.a1. This is called the
"LMFDB label" of the elliptic curve. It differs from the
Cremona label in the insertion of the "." after the
conductor, as well as the possibly different isogeny class
code and curve number.
NNF: Number of newforms / isogeny classes (for a fixed
N). Non-negative integer.
AI: Reduced minimial Weierstrass coefficients
in format [a1,a2,a3,a4,a6] (integers, no spaces; a1,a3
are in {0,1} and a2 is in {-1,0,1}).
R: Rank. Non-negative integer.
T: Torsion order. Positive integers.
TOR: Torsion structure: [] or [n] or [n1,n2] with n, n1, n2
integers>1 and n1 dividing n2.
TGENS: Torsion generators: 0, 1 or 2 points in format [x:y:z]
with x, y, z coprime integers, no embedded space
GENS: Generators of infinite order:
R points in formst [x:y:z] separated by whitespace
AGENS: Generators of infinite order as a list of [x,y] with x, y
rationals, no embedded space.
AP: a2 a3 a5 a7 a11 (etc.) where each ap is the p'th
eigenvalue of the newform (integers, separated by
whitespace). When p divides N this is the W_p
eigenvalue, shown just as "+" or "-"; else the T_p
eigenvalue.
AP25: AP for all primes<100 (the first 25 primes).
BQ: If there are primes q>100 dividing N this string is
either +(q) or -(q) showing the eigenvalue of W_q and
the value of q; or two such strings separated by
whitespace, if there is more than one such prime
(e.g. N=50038, q=127 and q=197). Else empty.
CP: Product of Tamagawa numbers. Positive integer.
OM Real period. Real number.
L1: L(r)(E,1)/r!. Real number.
REG: Regulator. Real number.
SHA: Analytic order of Sha. Either positive integer, or Real number.
DEG: Degree of modular parametrization. Positive integer.
PDEG: Set of primes dividing DEG. e.g. {p,q} with p, q primes.
CLASS: [E1,E2,...,En] where n=#isogeny class, Ei=AI for i'th
curve. No embedded space.
ISOGMAT: Matrix of isogeny degrees between Ei and Ej, as a list of
lists with no spaces. e.g. [[1,5,5],[5,1,25],[5,25,1]].
XLIST: List of x-coordinates of integral points (integers, no spaces).
GAL_IM: Sutherland code for (non-maximal) Galois image
(string, starting with a prime, e.g. 5Cs.1.1)
GAL_IMS: Sequence of 0 or more GAL_IM separated by whitespace
2IND: Index of image of 2-adic representation in GL_2,Z_2).
Positive integer.
2LEVEL: Level of 2-adic image. Positive integer.
2MATGENS: Matrices generating 2-adic image: list of lists of 4
integers, no embedded space.
2LAB: Rouse label of modular curve associated to 2-adic image. String.
Files:
1. curves: ID AI R T (for curves with NCURVE=1 only)
2. allcurves: ID AI R T
3. allgens: ID AI R TOR TGENS GENS
4. aplist: N CLASSCODE AP25 BQ
5. allbsd: ID AI R T CP OM L1 REG SHA
6. degphi: ID DEG PDEG AI (for curves with NCURVE=1 only)
7. alldegphi: ID AI DEG
8. allisog: ID AI CLASS ISOGMAT
9. intpts: IDX AI XLIST
10. allbigsha: ID AI R T SHA (for curves with SHA>1 only)
11. count: N NNF
12. paricurves: ["ID", AI, AGENS]
13. alllabels: N CLASSCODE NCURVE N LMFDB_CLASSCODE LMFDB_NCURVE
14. galrep: IDX GAL_IMS
15. 2adic: ID AI 2IND 2LEVEL 2MATGENS 2LAB
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