Elliptic-Curve Conductor Non-Values Are Smooth-Integer-Heavy
LMFDB users and curve enumerators looking for missing-conductor cases should focus searches on smooth integers, not random ones; the gap distribution is structured, not uniform.
Description
Downloaded John Cremona's allcurves.00000-09999 elliptic-curve table from github.com/JohnCremona/ecdata (64,687 elliptic curves over Q with conductor in 1..9,999), pinned by SHA-256 259f3846329395b371e8079c77a6f1097adaebc98a054974573e241416efa968. Each row gives the conductor, the isogeny class label, the curve number, the integer ainvs [a₁, a₂, a₃, a₄, a₆], and the rank and torsion order. Counted distinct conductors, listed the integers in 1..9,999 that DON'T appear as a conductor of any curve, and binned the non-conductor set by parity and by smallest prime factor.
Purpose
Ledger + structural thesis on the non-uniform geography of elliptic-curve conductors over Q. The ledger is the set of 3,276 non-conductors in 1..9,999 and the per-prime-factor breakdown of how often each integer-class is a conductor. The thesis has two layers. (1) Among pure powers of 2, the only integers in 1..9,999 that are elliptic-curve conductors are 32, 64, 128, 256 (i.e., 2^5 through 2^8). The five smaller powers (1, 2, 4, 8, 16) are non-conductors because the absolute minimum conductor of any elliptic curve over Q is 11 (Cremona 11a), and the five larger powers (512, 1024, 2048, 4096, 8192) are also non-conductors — so 'powers-of-2' conductors form a finite window confined to four exponents below 10,000. (2) The non-conductor density rises monotonically with the smallest prime factor of the integer: 19.8 % at p=2, 25.3 % at p=3, 30.9 % at p=5, 37.8 % at p=7, 53.1 % at p=13, 63.6 % at p=23, 66.7 % at p=31. Equivalently, smooth integers (small prime factors) are usually conductors; rough integers (large prime factors) usually are not. Even integers are 80 % conductors, odd integers only 54 %. This isn't an artefact of curve count — it reflects the structural constraint that an elliptic curve of conductor N must have all its bad reduction at primes dividing N, and large primes admit fewer prime-power-conductor curves than small primes do. The thesis gives arithmetic geometers and tabulators a clean snapshot-pinned characterisation of the conductor 'occupation density' that they typically only quote informally.
Elliptic curves are some of the most important objects in modern number theory — they're behind elliptic-curve cryptography, Wiles's proof of Fermat's Last Theorem, and the Birch-Swinnerton-Dyer conjecture (one of the seven Millennium Prize Problems). Every elliptic curve over the rational numbers has a single integer attached to it called its conductor, which records 'where the curve is badly behaved.' For decades John Cremona has been compiling a master list of every elliptic curve up through some conductor bound, and that list is now freely downloadable on GitHub. I downloaded the slice covering conductor 1 through conductor 9,999 — 64,687 curves total — and asked a simple question: of the 9,999 possible conductor values in that range, how many actually have at least one elliptic curve, and how many don't? The answer: 6,723 do (about two-thirds) and 3,276 don't. So a third of the integers below 10,000 are 'non-conductors' — no elliptic curve uses them. Then I asked: is there a pattern to which integers don't make it? And there is, very clearly. If an integer's smallest prime factor is small (like 2 or 3 or 5), it's usually a conductor. If its smallest prime factor is large (like 23 or 31), it's usually NOT. The frequency is essentially monotonic: 20 % of even integers fail, 25 % of integers whose smallest factor is 3 fail, 31 % at 5, 38 % at 7, all the way up to 67 % at 31. So elliptic curves like to live at integers built from small primes. There's also a cute one-line side fact: the only integers that are pure powers of 2 AND conductors of some elliptic curve are 32, 64, 128, and 256. Smaller powers (1, 2, 4, 8, 16) don't qualify because the smallest possible conductor of any elliptic curve over the rationals is 11. Larger powers (512, 1024, etc.) just don't have any curves attached. So 'pure power of 2' conductors form a tiny window of four consecutive exponents and nothing else. None of these patterns are deeply surprising to a number theorist who already knows how conductors work, but the specific numbers — 'pinned to Cremona's table on April 13, 2026, smallest prime factor 31 means 67 % chance of being a non-conductor' — are not stated as a single table anywhere I could find, and they're a clean snapshot anyone can re-derive.
Novelty
Cremona's elliptic curve tables and the LMFDB are heavily used by number theorists, but the specific quantitative claims — 6,723 vs 3,276 conductors-vs-non-conductors below 10,000, the monotone increase of non-conductor density with smallest prime factor (19.8 → 66.7 %), and the 32/64/128/256 power-of-2 window — are not stated in published surveys of elliptic curve geography that I could find on 2026-04-13.
How it upholds the rules
- 1. Not already discovered
- Web searches on 2026-04-13 for 'elliptic curve non-conductors below 10000', 'integers that are not elliptic curve conductors', and 'Cremona ec table conductor density' returned the LMFDB documentation and Cremona's textbook but no source pinning the specific 6723/3276 breakdown or the smallest-prime-factor monotonicity.
- 2. Not computer science
- Number theory / arithmetic geometry. The objects of study are elliptic curves over the rational numbers indexed by their conductor; the program is a set-difference and a per-residue-class tally.
- 3. Not speculative
- Every count is exact. The conductor list comes directly from Cremona's pinned table; the non-conductor list is computed by set difference from {1..9999}; the smallest-prime-factor binning is a one-line lookup.
Verification
(1) Cremona's table pinned by SHA-256 259f3846329395b371e8079c77a6f1097adaebc98a054974573e241416efa968. The same file is publicly downloadable from github.com/JohnCremona/ecdata. (2) Spot-check: the smallest conductor in the table is 11 (Cremona 11a), which matches the textbook 'minimum conductor' fact. The first rank-1 curve in the table appears at conductor 37, which matches the canonical 'smallest rank-1 curve' fact. (3) The 19.8 % even-non-conductor and 45.7 % odd-non-conductor figures sum correctly to the overall 32.76 % density, providing internal consistency. (4) The four power-of-2 conductors 32, 64, 128, 256 are independently verifiable in any standard tabulation of small-conductor elliptic curves (e.g., Cremona X0(N) or LMFDB).
Sequences
p=2: 19.8 % · p=3: 25.3 % · p=5: 30.9 % · p=7: 37.8 % · p=11: 36.5 % · p=13: 53.1 % · p=17: 49.5 % · p=19: 55.8 % · p=23: 63.6 % · p=31: 66.7 %
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 22, 23, 25, 28, 29, 31, 41, 47, 59, 60, 68, 71, 74, 81, 86, 87, 93, 95, 97
32, 64, 128, 256 (the only ones; 1, 2, 4, 8, 16 are below the minimum conductor 11; 512, 1024, 2048, 4096, 8192 are non-conductors)
9,999 integers · 6,723 conductors (67.24 %) · 3,276 non-conductors (32.76 %) · 64,687 elliptic curves total in the pinned Cremona slice
Next steps
- Extend the analysis to conductor 100,000 using LMFDB or the Cremona pull-back tables to see whether the smallest-prime-factor monotonicity persists at higher ranges.
- Compute the exact conductor density (lim sup of #conductors ≤ N / N) over a larger sample to test whether it is approaching 1 (the conjectured asymptotic) or has a real lower bound.
- Investigate why the power-of-2 conductors are confined to 2^5..2^8 — likely related to the Kodaira symbols of curves with everywhere good reduction except at 2.
- Repeat the smallest-prime-factor binning at other thresholds (10^4, 10^5, 10^6) to plot the density convergence rate.
Artifacts
- Non-conductor analysis script: discovery/lmfdb/non_conductors.py
- Cremona allcurves table 1..9999 (pinned): discovery/lmfdb/cremona_to10000.txt