# Representing a point

We call a `Point` an element of the elliptic curve group, which we can represent with its coordinates. Let's define a `Point` class to encapsulate the $$x$$ and $$y$$ coordinates in the generalised curve equation of $$y^2 = x^3 + ax + b$$**.**

```python
from typing import Optional

@dataclass
class Point:
    x: Optional[int]
    y: Optional[int]

    curve: EllipticCurve

    def __post_init__(self):
        # Ignore validation for I
        if self.x is None and self.y is None:
            return

        # Encapsulate int coordinates in FieldElement
        self.x = FieldElement(self.x, self.curve.field)
        self.y = FieldElement(self.y, self.curve.field)

        # Verify if the point satisfies the curve equation
        if self not in self.curve:
            raise ValueError
```

{% hint style="info" %}
**Point-compression:** To reduce the storage size for a curve point, one can also store a sign and the`x` coordinate. It is then possible to  reconstruct the `y` by calculating `sign * sqrt(x^3+a*x+b).`

Ref: <https://bitcoin.stackexchange.com/a/29905>
{% endhint %}