Elliptic curves by rank

Elliptic curves over Q with explicit Mordell–Weil generators, in three separate collections.

Every curve is given by its global minimal model [a1,a2,a3,a4,a6], and every generator is a point [x,y] on that model. Coordinates and conductors are exact; read them as integers and rationals, not as floating point numbers. The three collections are kept apart, and a curve can appear in more than one of them (for example, the small rank 4 curves of the census are also among the search curves). To read a curve in PARI/GP: E = ellinit([a1,a2,a3,a4,a6]);

1. All curves of rank ≥ 3 with |Δ| ≤ 10¹³

These files list every elliptic curve over Q of rank at least 3 whose minimal model has discriminant Δ and invariant c₄ with |Δ| ≤ 10¹³ and c₄ ≤ 1.44·10¹²: 2,302,875,461 of the 78,220,212,957 curves in that region. The bound on c₄ is needed because |Δ| ≤ 10¹³ alone allows arbitrarily large c₄ near c₄³ = c₆²; of the 65,652 curves in Cremona's tables with |Δ| ≤ 499,999, only three have larger c₄. There is one file per rank, sorted by conductor; since the conductor divides the discriminant, every conductor here is at most 10¹³. The region is not complete by conductor: curves of small conductor can have a large discriminant.

What the ranks rest on. rank_lo is the number of independent points listed, an unconditional lower bound, and rank_hi is the 2-descent upper bound, also unconditional. The status column says how the listed rank is established:

So the rank is proven exactly for 99.973% of the listed curves. The S curves at rank 3 also have analytic rank exactly 3, assuming GRH.

Completeness. Every curve in the region that is not listed has analytic rank at most 2, assuming the Generalized Riemann Hypothesis for its L-function. Its rank is then at most 2: by the theorems of Gross–Zagier and Kolyvagin when the analytic rank is at most 1, and by the Birch and Swinnerton-Dyer conjecture when it is 2.

Check against Cremona's tables. All 8,561 curves of rank ≥ 3 in John Cremona's database (conductor ≤ 499,999) that lie in the region are listed here with the same rank, and every listed curve of conductor ≤ 499,999 is one of them.

Format

Tab-separated with a header line, compressed with zstd: zstdcat census_rank3.tsv.zst | head. The rank 3 file holds 2.1 billion lines; read it as a stream (for example pandas.read_csv(..., sep="\t", compression="zstd", chunksize=10**7)) rather than all at once.

Torsion orders of the listed curves: trivial 2,302,410,005; order 2: 464,750; order 3: 646; order 4: 60.

2. Curves of rank 4 to 15 from searches

Every curve here has proven rank. For each one the listed generators are verified to lie on the curve and to be independent, which gives the rank as a lower bound, and a 2-descent upper bound matches it. Every model is the minimal model, and no curve appears twice.

16,947,302 curves in total. Every figure in the table links to the ten smallest curves of that rank in that category, with their a-invariants, exact conductors and full generators. The four minima in each row are taken independently, so they are generally attained by four different curves. Heights are natural logarithms. The naive height is log max(|c4|³, c6²), and abs(disc) is the absolute value of the minimal discriminant. The Faltings height is the unstable one, normalized as h_Fal(E) = −(1/2) log |Im(conj(ω₁) ω₂)|, with ω₁, ω₂ a basis of the period lattice of the minimal model, the normalization used by the Elliptic Curve Rank Leaderboard. It is negative for some rank 4 curves.

Format

Tab-separated with a header line, gzipped, one curve per line, sorted by ascending conductor; zcat, pandas, R and awk read them directly.

Points may have rational coordinates: 32.5% of these curves have a generator with a denominator, and denominators reach 35 digits. Most conductors exceed 2⁵³. No curve appears twice, but some are isogenous to each other (isogenous curves share rank, conductor and L-function): there are 32,651 isogenous pairs, almost all at ranks 6 and 7.

By torsion subgroup

The curves above with a nontrivial torsion subgroup, one file per group, with the same columns plus torsion (the group) and torsion_generators (points generating the torsion subgroup). Every curve in these files also appears in its rank_r file.

Smallest log N by rank and torsion group:

3. Curves from parametrized families

Curves obtained by specializing parametrized families of elliptic curves (elliptic surfaces over Q(t)). A curve is included only when its rank is larger than the generic rank of the family it comes from. There is one file per torsion subgroup, with the same columns as the torsion files above; for trivial torsion torsion_generators is []. Every curve has proven rank, is given by its global minimal model, and appears once.

Also here

generators_scatterplots.pdf (1.1 MB): notes on what the Mordell–Weil generators of these curves look like when suitably normalized.