For cryptographic use, elliptic curves over real numbers are not useful.
Curves with integer roots and coefficients are required.
Instead of continuous curves, these curves
are just a finite collection of fixed points,
whose coordinates are calculated using modular arithmetic with a prime modulus p.
The equation satisfied by these coordinates is y2 = x3 + ax + b (mod p).
Calculation results appear here.
Note that, except when y = 0 or the modulus is 2, each value of x is associated with two values of y, whose values sum to the modulus p. These correspond to the positive and negative values for y on the real-valued curve. (In modular arithmetic, two numbers which sum to the modulus effectively sum to zero, and so mutually negate each other.) Because the modulus is odd, one value of y is odd and the other is even. This allows a point to be represented in compressed form with the y value reduced to a single bit, since its value can otherwise be calculated from x and the curve's equation.