Calculator for for Elliptic Curves
Program for calculating elliptic curves of the form y² = x³ + ax + b
This equation algebraically links the four quantities a, b, x, and y. Given three of these values, the fourth can be calculated, provided a real solution exists. Depending on the input values, there may be one, two, or three solutions, especially when calculating the x-coordinate, which is based on a cubic equation.
Please enter three of the four values a, b, x, and y. The empty field will be calculated for each one. If all four fields are filled, y will be recalculated. If multiple solutions exist, a preferred solution will be entered in the input field, while other possible solutions will be displayed below. Additionally, a residual is output, indicating how well the calculated values numerically satisfy the equation y² = x³ + ax + b. A small residual indicates high computational accuracy.
This program is particularly suitable for investigating individual points on elliptic curves. It allows to verify whether a given point actually lies on a specific curve, as well as to construct new points from known values. Taking numerical rounding errors into account prevents computationally correct results from appearing distorted by unavoidable machine inaccuracies.
a:
b:
x:
y:
Elliptic curves possess a rich algebraic structure that extends far beyond their graphical representation. In cryptography, specific points on the curve are used for secure key procedures. In number theory, rationally computable points play a central role in addressing profound questions. Computer programs like this one therefore serve as tools for experimentally accessing elliptic curves and for verifying theoretical results.
An example of a rationally computable point on an elliptic curve is x=2, y=5, a=−3, b=23.
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