How many rational points does an elliptic curve have?

How many rational points does an elliptic curve have?

one rational point
Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

What is a rational point on a curve?

A -rational point is a point on an algebraic curve , where and are in a field . For example, rational point in the field of ordinary rational numbers is a point satisfying the given equation such that both and. are rational numbers. The rational point may also be a point at infinity.

Why are elliptic curves called elliptic?

So elliptic curves are the set of points that are obtained as a result of solving elliptic functions over a predefined space. I guess they didn’t want to come up with a whole new name for this, so they named them elliptic curves.

What is the order of an elliptic curve?

The order of is linked to the order of the elliptic curve by Lagrange’s theorem, which states that the order of a subgroup is a divisor of the order of the parent group. In other words, if an elliptic curve contains points and one of its subgroups contains points, then is a divisor of .

What is the order of elliptic curve?

The order of a point on an elliptic curve is the order of that point as an element of the group defined on the curve. = O, and m P = O for all integers 1 ≤ m < m. If such m exists, P is said to have finite order, otherwise it has infinite order.

How do you find the order of an elliptic curved group?

How to determine the order of an elliptic curve group from its parameters? Let E:y2=x3+ax+b be an elliptic curve defined over a finite field Fq where q=pn, a,b∈Fq and p≠2,3. By Hasse’s theorem we know that the order of E(Fq) is in the range [q+1−2√q,q+1+2√q].

What is a rational view?

1 using reason or logic in thinking out a problem. 2 in accordance with the principles of logic or reason; reasonable. 3 of sound mind; sane.

What is rational or irrational?

Rational numbers are numbers that can be expressed as a fraction or part of a whole number. (examples: -7, 2/3, 3.75) Irrational numbers are numbers that cannot be expressed as a fraction or ratio of two integers. There is no finite way to express them. ( examples: √2, π, e)

What are curves in geometry?

A curve is a shape or a line which is smoothly drawn in a plane having a bent or turns in it. For example, a circle is an example of curved-shape. Plane or Two Dimensional Geometry. Solid or Three Dimensional Geometry.