Elliptic Curve — Definition, Formula & Examples

An elliptic curve is a smooth, non-singular curve defined by a cubic equation in two variables, typically written in the form

y^2 = x^3 + ax + b

. Despite the name, elliptic curves are not ellipses — the name comes from their historical connection to elliptic integrals.

An elliptic curve over a field $K$ is a non-singular projective algebraic curve of genus 1 with a specified rational point (serving as the identity element). In Weierstrass normal form over $K$ , it is given by $y^2 = x^3 + ax + b$ where $a, b \in K$ and the discriminant $\Delta = -16(4a^3 + 27b^2) \neq 0$ , ensuring no cusps or self-intersections.

Key Formula

y^2 = x^3 + ax + b, \quad 4a^3 + 27b^2 \neq 0

Where:

$a$ = Coefficient controlling the linear term in $x$
$b$ = Constant term
$4a^3 + 27b^2$ = Must be nonzero to ensure the curve is non-singular (no cusps or self-intersections)

How It Works

The discriminant condition

4a^3 + 27b^2 \neq 0

guarantees the curve is smooth. Points on an elliptic curve form an abelian group under a geometric addition law: given two points

P

and

Q

on the curve, draw the line through them, find the third intersection with the curve, and reflect it across the

x

-axis to obtain

P + Q

. A special "point at infinity"

\mathcal{O}

acts as the identity element. This group structure is what makes elliptic curves so powerful in both pure mathematics and applications like cryptography.

Worked Example

Problem: Verify that

y^2 = x^3 - x + 1

defines an elliptic curve, and check whether the point

(1, 1)

lies on it.

Identify coefficients: Here

a = -1

and

b = 1

y^2 = x^3 + (-1)x + 1

Check the discriminant condition: Compute

4a^3 + 27b^2 = 4(-1)^3 + 27(1)^2 = -4 + 27 = 23

. Since

23 \neq 0

, the curve is non-singular and therefore a valid elliptic curve.

4(-1)^3 + 27(1)^2 = 23 \neq 0

Test the point: Substitute

x = 1

y = 1

: the left side gives

1^2 = 1

, and the right side gives

1^3 - 1 + 1 = 1

. Both sides are equal.

1 = 1 \checkmark

Answer: The equation defines an elliptic curve, and the point

(1, 1)

lies on it.

Why It Matters

Elliptic curves are central to modern cryptography — the TLS protocol securing most web traffic relies on elliptic curve Diffie–Hellman key exchange. In pure mathematics, Andrew Wiles's proof of Fermat's Last Theorem hinged on showing that every semistable elliptic curve over

\mathbb{Q}

is modular, connecting them to deep areas of number theory.

Common Mistakes

Mistake: Assuming an elliptic curve is a type of ellipse.

Correction: The two are unrelated shapes. The name "elliptic" comes from elliptic integrals, which historically arose when computing arc lengths of ellipses. An elliptic curve is a cubic curve, not a conic section.