# Chapter ᚒ(U): The crossroads
*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T19, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*
[Guide for AI readers](index.md) · [Its glossary cards](ch19-cards.md) · [Its references](ch19-refs.md) · [Previous chapter](ch18.md) · [Next chapter](ch20.md)
*A footnote such as [r-19.3] is reference entry 19.3 and links to it. Section 19 of 28; chapters are named by their Ogham letter, never by number.*
ᚒ – [Úr](ch19-cards.md#g-og-ur): associated with the heather, Section 19
## Chapter ᚒ(U): The crossroads
> “It is possible to write endlessly on elliptic curves. (This is not a threat.)”
> — Serge Lang · *Elliptic Curves: Diophantine Analysis*, 1978, opening line of the preface [r-19.22](ch19-refs.md#r-19.22)
Formula: ([X(𝒪)](ch19-cards.md#g-ptinf) + [rank(y² = x³ − x)](ch19-cards.md#g-rank32))/(([Δ(y² = x³ − x)](ch19-cards.md#g-discE) + [j(y² = x³ − x)](ch19-cards.md#g-j1728) + [|E(ℚ)tors|](ch19-cards.md#g-tors4) + [#E(𝔽5)](ch19-cards.md#g-pts5)) {brace: [y² = x³ − x](ch19-cards.md#g-weierstrass)} + ([N(E₃₇)](ch19-cards.md#g-cond37)) {brace: [conductor 37](ch28-cards.md#g-E37)} + ([16](ch19-cards.md#g-mazur16) + [5](ch19-cards.md#g-congruent5)) {brace: a bound and a smallest} )
Key concepts: Elliptic curves: the group law, the point at infinity, rank, and the invariants that name a curve. It includes symbols you may not have seen before, such as: [𝒪](ch19-cards.md#g-ptinf), [Δ](ch19-cards.md#g-discE), [j](ch19-cards.md#g-j1728), and [N](ch19-cards.md#g-cond37). Zero here sits at infinity: the curve’s identity is a point on no ordinary map, and y² = x³ − x has rank 0, as Fermat in effect proved.
Here we are. Leave the moped by the signpost and spread the map between us. An intersection gives us a point; a line drawn through two points can give us another. For an elliptic curve, that simple drawing becomes the beginning of an addition rule. That is why the primes led us here: counting a curve’s points modulo different primes can reveal things the drawing alone does not show. Geometry and arithmetic have arrived at the same junction.
Field notes
An elliptic curve is not an ellipse; the name comes from the integrals associated with ellipses, which lead to this family of curves. Our main example is E: [y² = x³ − x](ch19-cards.md#g-weierstrass). On such a curve, draw a line through two points; the third point where it meets the curve, reflected across the x-axis, is their sum (a tangent handles adding a point to itself). Rational points give rational results, and [this rule makes the curve’s points into a group](ch19-cards.md#g-grouplaw). [r-19.2](ch19-refs.md#r-19.2) Over the complex numbers the curve is [the torus from the gallery of shape](ch12-cards.md#g-genusT).
#### Numerator
[X(𝒪)](ch19-cards.md#g-ptinf) = 0; the group’s identity, the extra point that lives at infinity.
A group needs a point that changes nothing when added. Draw the vertical line through a point and its mirror image across the x-axis. A line should meet the curve three times, but this one meets it only twice, so its third meeting lies off the page. That is 𝒪, the one point at infinity, where every vertical line ends. Add 𝒪 to any point and the drawing rule hands the point back: the line through them is the point’s own vertical line, whose third meeting is its mirror image, and reflecting that returns the point. So 𝒪 is the identity, an added geometric point, not an infinitely large number. [r-19.2](ch19-refs.md#r-19.2) [Projective coordinates](ch19-cards.md#g-projcoord) give it an address. Points become triples (X : Y : Z) that count as equal when scaled, the ordinary plane is the part with Z = 1, and 𝒪 is (0 : 1 : 0), so its [X-coordinate is 0](ch19-cards.md#g-ptinf). That 0 is the address that brings 𝒪 onto the page, not the reason it is the identity. [r-19.1](ch19-refs.md#r-19.1)
[rank(y² = x³ − x)](ch19-cards.md#g-rank32) = 0; [the rank](ch28-cards.md#g-rankE) of our curve, which Fermat in effect settled.
[Mordell’s theorem](ch19-cards.md#g-mordellweil) says all rational points come from finitely many generators. Generators that eventually cycle back to the identity are *torsion*; the number of independent ones that never do is the [rank](ch28-cards.md#g-rankE). [r-19.3](ch19-refs.md#r-19.3) Here the rational points are exactly 𝒪, (0, 0), (1, 0), and (−1, 0), all torsion, so [the rank is 0](ch19-cards.md#g-rank32). That the list stops at four is a theorem, not a failure to find more points: it comes down to what Fermat proved by [infinite descent](ch19-cards.md#g-descent), that no right triangle with rational sides has area 1. [r-19.4](ch19-refs.md#r-19.4) The three points with y = 0 cycle back after two steps: doubling any one of them lands on 𝒪, so each has order 2. [r-19.6](ch19-refs.md#r-19.6)
#### Denominator
The first four numbers describe one curve: its smoothness, which complex curve it is, its rational torsion points, and its points on a five-position clock. A second curve, a torsion bound, and a right-triangle area widen the picture.
y² = x³ − x
[Δ(y² = x³ − x)](ch19-cards.md#g-discE) = 64; the discriminant, the test that makes a cubic an elliptic curve at all.
For y² = x³ + ax + b, Δ = −16(4a³ + 27b²). With a = −1 and b = 0 this is [64](ch19-cards.md#g-discE). Nonzero means x³ − x has three different roots, −1, 0, and 1, so the curve has no sharp point and never crosses itself: it is smooth all the way along, which is what the drawing rule needs. [r-19.5](ch19-refs.md#r-19.5)
[j(y² = x³ − x)](ch19-cards.md#g-j1728) = 1728; [the j-invariant](ch14-cards.md#g-jinv), the number that names a curve.
Over the complex numbers, two curves have the same j exactly when a change of coordinates turns one into the other, however different their equations look. The special value [1728](ch19-cards.md#g-j1728) goes with this curve’s extra symmetry: sending (x, y) to (−x, iy) carries the curve onto itself. [r-19.5](ch19-refs.md#r-19.5) [r-19.14](ch19-refs.md#r-19.14)
[|E(ℚ)tors|](ch19-cards.md#g-tors4) = 4; the torsion points we just listed.
The identity, of order 1, and the three points with y = 0, (0, 0), (1, 0), and (−1, 0), each of order 2. [r-19.6](ch19-refs.md#r-19.6)
[#E(𝔽5)](ch19-cards.md#g-pts5) = 8; the same curve counted on a five-hour clock, the arithmetic of Chapter ᚄ(S).
Modulo 5: for x = 0, 1, and 4, y = 0; for x = 2, y = 1 or 4; for x = 3, y = 2 or 3. That is seven points, plus 𝒪: eight, where the rationals gave four. [r-19.7](ch19-refs.md#r-19.7) Every prime gives the curve a count of its own, and it was in such counts that Birch and Swinnerton-Dyer went looking for clues to the rank. [r-19.12](ch19-refs.md#r-19.12)
conductor 37
[N(E₃₇)](ch19-cards.md#g-cond37) = 37; a second curve, [E₃₇: y² + y = x³ − x](ch28-cards.md#g-E37), the smallest-conductor curve of rank one.
The conductor is an integer encoding the primes where the curve behaves badly, and how badly. [E₃₇](ch28-cards.md#g-E37) has [rank 1](ch28-cards.md#g-rankE), unlike our [rank-zero curve](ch19-cards.md#g-rank32). The point (0, 0) lies on both, but on E₃₇ it never cycles back to 𝒪: adding it to itself again and again gives infinitely many rational points. A small change on the page, with large consequences. [r-19.8](ch19-refs.md#r-19.8)
a bound and a smallest
[16](ch19-cards.md#g-mazur16) = 16; Mazur’s bound, a ceiling on torsion for every elliptic curve over the rationals.
By Mazur’s theorem, no elliptic curve over [ℚ](ch01-cards.md#g-rats) has more than 16 rational points of finite order. [r-19.9](ch19-refs.md#r-19.9)
[5](ch19-cards.md#g-congruent5) = 5; the smallest *congruent number*, the other side of Fermat’s area-1 result.
5 is the smallest positive integer that is the area of a right triangle with rational sides: 3/2, 20/3, and 41/6. Their squares satisfy the right-triangle relation, and half the product of the shorter sides is 5. [r-19.10](ch19-refs.md#r-19.10)
The only fraction is (0 + 0)/1862: 64 + 1728 + 4 + 8 + 37 + 16 + 5 = 1862.
**Fine print.** Scaling (0 : 1 : 0) changes its Y, never its X, so the numerator’s 0 does not depend on how the point is written. The ordinary point (0, 0) of the curve is a different point, not the identity. Birch and Swinnerton-Dyer found evidence that counts modulo primes reveal [the rank](ch28-cards.md#g-rankE); [their conjecture](ch28-cards.md#g-bsdconj) says [the rank](ch28-cards.md#g-rankE) is the order of the zero at s = 1 of the curve’s [L-function](ch19-cards.md#g-LfuncE), an [Euler product](ch18-cards.md#g-eulerprod) like [zeta](ch18-cards.md#g-zeta)’s but built from those counts: a simple zero for rank 1, a double zero for rank 2. [r-19.21](ch19-refs.md#r-19.21) [E₃₇](ch28-cards.md#g-E37) has [rank 1](ch28-cards.md#g-rankE) and [a simple zero there](ch28-cards.md#g-lE37), an established example that returns in the epilogue, not a proof for every curve. [r-19.11](ch19-refs.md#r-19.11) [r-19.12](ch19-refs.md#r-19.12)
What this chapter adds to the formula: (0 + 0) / 1862 = 0
The crossroads has let us add points, count them over different fields, and ask what those counts reveal. These are also the curves that finished [Fermat’s story](ch08-cards.md#g-fermat) from the observation car, Chapter ᚆ(H): a counterexample to his theorem would have produced a curve that the [modularity results](ch19-cards.md#g-modularity) in Wiles’s proof say cannot exist. [r-19.13](ch19-refs.md#r-19.13) We are almost at our destination: the space terminal! It is just through that deep cutting in the hill they made for the road, where the ground has been stripped away in layers. Let’s go off-road for a few minutes; I want to stop at the top of the cutting and have a look at what is left. We have been counting points; the cutting ahead asks how much room points take up, and the answer is stranger than any count.
Watch and read
- [Elliptic Curves - Computerphile](https://www.youtube.com/watch?v=NF1pwjL9-DE) Mike Pound draws the group law, the line through two points and the third point it hits
- [What is... an elliptic curve?](https://www.youtube.com/watch?v=A2KNrgiWquU) the shape, the addition, and why number theorists care, in one sitting
- [Elliptic curve](https://en.wikipedia.org/wiki/Elliptic_curve) Wikipedia
- [Birch and Swinnerton-Dyer conjecture](https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) Wikipedia