Rank31

It is the set of points satisfying a cubic equation in two variables. The tidy version — the one every picture on this page uses — is

y² = x³ + A x + B

together with one extra point, written O, that sits infinitely far up the y-axis. You will see why that point has to be there in a moment.

The only rule is that the curve must be smooth: no corners, no self-crossings. That is controlled by a single number, the discriminant

Δ = −16 (4A³ + 27B²)

When Δ ≠ 0 the curve is an elliptic curve. When Δ = 0 it degenerates into something with a pinch point, and everything that follows falls apart. Drag the sliders and watch the two regimes: when Δ > 0 the real picture has two pieces — a closed oval and an unbounded branch; when Δ < 0 there is just the one branch.

This is the strange and wonderful fact that makes elliptic curves worth a century of attention. Take any two points P and Q on the curve. Draw the straight line through them. Because the equation is a cubic, that line meets the curve in exactly one more place — call it R. Now reflect R across the x-axis. That reflected point is defined to be P + Q.

Three points on a line always sum to O. That single rule forces everything else: O is the identity, the reflection of P is its inverse −P, and a vertical line — which "meets the curve at infinity" — is exactly the statement P + (−P) = O. That is why the point at infinity has to exist. Without it the operation would have holes.

When P = Q there is no line through two distinct points, so you take the tangent instead. That gives doubling, P + P = 2P.

The one hard fact is that this addition is associative: (P + Q) + S = P + (Q + S). It is not obvious from the picture at all. With it, the points of the curve form an abelian group.

Everything above works over the real numbers, where the curve is a smooth line you can trace with a finger. The actual subject — arithmetic geometry — starts when you throw away every point whose coordinates are not rational numbers.

What survives is a scatter of dust on the curve. But the group law is built from addition, subtraction, multiplication and division, so adding two rational points always lands on a third rational point. The dust is closed under the operation. It is a group of its own, written E(ℚ).

Here is the curve y² + y = x³ − x. Of all the curves whose rational points are infinite in number, this one has the smallest conductor — 37 — which makes it the standard first example. Start at P = (0, 0) and keep adding P to itself. Watch the coordinates.

The coordinates blow up quadratically — the number of digits in x(nP) grows like n². That growth is not an accident; it is the defining property of the canonical height,

ĥ(P) = lim_n→∞ h(nP) ⁄ n²

a function satisfying ĥ(nP) = n² ĥ(P) exactly. The naive height you can count off the digits is not quite that — it differs from ĥ by a bounded amount — so the h ⁄ n² figure in the readout wobbles toward the limit rather than descending onto it, the wobble being that bounded error divided by n². Drag the slider and watch it bounce: for this curve and this generator the true value is ĥ = 0.05111140…, and the readout settles within a couple of percent of it by about n = 20. Height is the ruler used for everything that follows, including the "how big is this curve" axis of the record leaderboard.

It also means these points never repeat. P has infinite order.

In 1922 Mordell proved something that had been conjectured by Poincaré: however infinite E(ℚ) is, it is finitely generated. There is a finite list of rational points from which every other rational point can be reached by repeated addition. Weil generalised it in 1929, and the result is now the Mordell–Weil theorem:

E(ℚ) ≅ ℤ^r ⊕ E(ℚ)_tors

Two pieces. On the right, the torsion subgroup: the points of finite order, the ones that cycle back to O if you keep adding them. On the left, r copies of the integers — r independent points of infinite order, each running off forever like the one in the previous section. That r is the rank.

Torsion is completely understood. In 1977 Mazur proved that only fifteen groups can ever appear, and the largest has just 16 elements. Pick a curve below and step around its cycle.

Rank is the opposite story. There is no theorem like Mazur's. Nobody knows which values of r occur, nobody knows whether r is bounded, and for a given curve there is no algorithm guaranteed to compute it. Everything difficult about elliptic curves lives in that one integer.

Ignore torsion for a moment. Then Mordell–Weil says the rational points form a lattice of dimension r: pick generators P₁, …, P_r, and every rational point on the curve is

n₁P₁ + n₂P₂ + ⋯ + n_rP_r

for exactly one choice of integers n_i. Rank 0 means the lattice is a single point and the curve has only finitely many rational solutions. Rank 1 means a line of them. Rank 2 means a plane, rank 3 a 3-D grid — and rank 31 means a 31-dimensional grid of integer coordinates, every cell of which is a different rational solution to one cubic equation.

The colour below is the height of each combination: how many digits that solution needs. Because the canonical height is a positive definite quadratic form in the n_i, the picture is a paraboloid — a bowl. Cheap solutions sit near the middle; everything else costs digits fast.

Finding the rank directly is hard. So here is the sideways attack that has driven the subject since the 1960s: instead of looking for rational solutions, count solutions modulo p for each prime p. That is a finite computation — just try every pair (x, y) in a grid of size p.

Write N_p for the number of solutions mod p, counting O. On average N_p ≈ p + 1, and Hasse proved in the 1930s that the error term

a_p = p + 1 − N_p, |a_p| ≤ 2√p

never gets larger than 2√p. So a_p/2√p always lands in [−1, 1]. For a curve with no rational points to speak of, those values scatter symmetrically about zero.

But a curve with lots of rational points has lots of points mod p too — every rational solution reduces to a solution mod p. High rank pushes N_p up, which pushes a_p down. Drag the rank slider and watch the whole cloud sink.

The curves in that slider are the smallest known curve at each rank from 1 to 31, straight off the record leaderboard, with the conductor-11 curve standing in at rank 0 — so the rank-31 row is the record curve itself, and its a_p values really do hug the floor of the Hasse band.

In the early 1960s Bryan Birch and Peter Swinnerton-Dyer had access to EDSAC in Cambridge, and they used it to do something nobody had done before: compute N_p for thousands of primes on hundreds of curves, and look for a pattern. What they plotted was the running product

∏_{p ≤ X} N_p ⁄ p

over the primes of good reduction, and what they saw was that it grows like C·(log X)^r, where r is the rank. Take logs twice and the rank is a slope. Run it yourself below — the computation is genuinely happening, prime by prime.

Watch what happens when you add the record curve to the plot. It starts below everything else and then climbs straight through them all. Thirty thousand primes is nowhere near enough for the estimate to settle at 31 — (log X)³¹ is a slow thing to see — but the shape of the claim is unmistakable.

That empirical slope is a heuristic, not a proof — but it was enough to suggest the real statement. Package all the a_p into a single analytic function, the L-function of the curve,

L(E, s) = ∏_{p good} (1 − a_pp^−s + p^1−2s)⁻¹

(the finitely many bad primes contribute simpler factors, and the product only converges for s well to the right of 1 — continuing it to s = 1 at all took the modularity theorem)

and the Birch and Swinnerton-Dyer conjecture says the order of vanishing of L at the point s = 1 equals the rank:

ord_s=1 L(E, s) = r

One side is analysis, the other is arithmetic, and there is no visible reason they should agree. It is one of the seven Clay Millennium Prize Problems, still open. What is proved, by Gross–Zagier and Kolyvagin in the 1980s, is the case where the analytic side is 0 or 1 — there the two ranks do match. Above 1, nothing is known in general, in either direction.

Worth knowing: the product you just watched is not simply the conjecture rewritten. Because the Euler product only converges to the right of the critical line, asserting that ∏ N_p/p behaves like (log X)^r is stronger than the vanishing statement — it implies the Riemann hypothesis for L(E, s) along the way. Birch and Swinnerton-Dyer measured the strong version and conjectured the careful one.

That gap is exactly why the record curves come with an asterisk, as we will see.

Nobody has ever proved that a curve of rank 31 exists, in the sense of a general theorem. The only way anyone knows to demonstrate high rank is to build a curve and exhibit the points. So the history of the subject is a ladder of explicit examples.

The early rungs came from clever hand computation. From 1982 the climb is dominated by machine search: Mestre found families of curves forced to contain many rational points by construction; Elkies sliced high-rank elliptic surfaces — K3 surfaces — to produce curves that inherit lots of points from the surface. His 2006 curve of rank ≥ 28 stood untouched for eighteen years.

There is a second axis to the game, and it is the one the ICARM leaderboard is actually built around: not just how high a rank you can reach, but how small a curve you can reach it with. High rank is easy if you allow monstrous coefficients. The interesting question is the trade-off.

Below is every curve on the board — of them — plotted as rank against size. The frontier along the bottom is the current state of the art.

On 20 August 2026 a user calling themselves ranksunbounded uploaded a curve of rank ≥ 30 to the leaderboard, beating Elkies and Klagsbrun's rank-29 curve from 2024. Three days later, on 23 August, curve #302 arrived at rank ≥ 31. Its commentary, verbatim:

BSD + GRH certified to rank 31, found by Claude, Levent Alpöge, and Ava Howell.

That sentence is commentary — free text typed by the submitter. The only field the leaderboard machine-checks is rank_lower_bound = 31, and it checks it properly, which is the subject of the next chapter. Here is the curve. This is the whole equation, every digit.

Nothing about that curve can be drawn. Plot it on any screen and the visible window is a featureless vertical wall — the interesting structure happens at x-coordinates around 10³³, and the 31 witness points have numerators up to 43 digits over denominators of 10.

But you can check it. Verifying that a point lies on the curve is just arithmetic on big integers, and your browser has arbitrary-precision integers built in. Press the button: each of the 31 points gets substituted into the equation, both sides evaluated exactly, and compared.

Those 31 points prove a lower bound and nothing more. To pin the rank down you need both halves of a sandwich, and the two halves are not equally solid.

The lower bound is unconditional. Exhibiting 31 points is not enough on its own — they must be shown to be independent, that is, no combination n₁P₁ + ⋯ + n₃₁P₃₁ collapses to O unless every n_i is zero. The leaderboard certifies this by exact 2-descent: quadratic characters evaluated at good primes build a matrix over 𝔽₂ whose rank is a proven lower bound for the rank of the curve. No floating point is involved anywhere in the decision, so rank ≥ 31 is a theorem.

The upper bound is not. There is no known unconditional way to prove a curve this large has rank at most 31. What can be done is Bober's method: assuming the Generalised Riemann Hypothesis for the curve's L-function, an explicit formula bounds the order of vanishing of L at s = 1 — the analytic rank. Then you need BSD to say that the analytic rank is the actual rank. Neither is a theorem. Hence: BSD + GRH certified.

For the rank-30 curve the argument is on the record: Bober's bound with Δ = 4.25 gives analytic rank ≤ 31, and the root number is +1, so the rank is even — forcing exactly 30.

How far can you get without assumptions? The largest rank ever determined exactly and unconditionally is 20, by Elkies and Klagsbrun in 2020. Assuming GRH alone — no BSD — Klagsbrun, Sherman and Weigandt pinned Elkies' 2006 curve at exactly 28 in 2019. Everything above that is a certified lower bound with a conditional lid.

For most of the twentieth century the working assumption was that ranks are unbounded — that for every r there is a curve of rank r, and the record climbs forever because there is always more room. The pseudonym the rank-30 curve arrived under, ranksunbounded, is a joke about exactly this.

In 2016 Park, Poonen, Voight and Wood built a heuristic model that points the other way. Counting curves by size and modelling their points as random lattice data, their model predicts that only finitely many elliptic curves over ℚ have rank greater than 21. If that is right, the ladder above rank 21 is not a staircase but a short, finite list — and every new record is a rare specimen rather than a step.

So a curve of rank 31 is either a routine rung on an infinite ladder or one of a handful of freaks in the entire universe of cubic equations. The same discovery is evidence for two opposite pictures, depending on which you already believed. That is a genuinely open question, and it is why a record that would be a curiosity in another field is news in this one.

Four days, two records, and the arithmetic still checks out by hand.