Over the real numbers, **y² = x³ + 7** looks like a single smooth curve with **two branches** (upper and lower) that are mirror images across the x-axis.
- **One connected piece**: Unlike some elliptic curves that form a separate closed loop plus a tail, this one has just a single open curve because x³ + 7 has only one real root (at x = –∛7 ≈ –1.91).
- **Leftmost point**: The curve meets the x-axis at roughly (–1.91, 0) with a **vertical tangent**, creating a rounded “nose” or “toe” on the left.
- **Extending rightward**: From that point, the curve splits into an upper branch that curves upward and a lower branch that curves downward. Both stretch out to infinity toward the right, growing steeper as x increases.
- **Symmetry**: Because of the y² term, everything is perfectly symmetric across the x-axis.
**What “smooth with no singularities” means:**
- **No self-intersections** (unlike a figure-eight or lemniscate)
- **No cusps** or sharp corners
- **No isolated points**
The curve is continuously differentiable everywhere. If you zoom in at any point, it looks like a gently bending line. The “smooth” part is exactly what makes the geometric addition law work in elliptic curve cryptography: you can draw a line through any two points and it will intersect the curve cleanly in exactly one more point.
