Drawing Program for Elliptic Curves
Program for visualizing elliptic curves of the form y² = x³ + ax + b
This formula doesn't describe a function, but it resembles one. Solving for y actually yields two functions: plus the square root of x³ + ax + b and minus the squared root of x³ + ax + b. These two functions are their reflections across the x-axis. The elliptic curves have either no result, exactly one point (if y = 0), or two results for each x-value. They have either one, two (in the case of a double position) or three results for each y-value.
Please specify the values for parameters a and b, and a value for the range. The x- and y-axes will be plotted at this distance from the origin. With the default value of 6, this ranges from -6 to 6. The four colors used for the graph can be changed as desired. In addition to the graph, the discriminant is displayed, calculated as -16 * ( 4 a³ + 27 b² ). If this discriminant is not equal to 0, the curve is elliptic. With a positive discriminant, the graph of the elliptic curve shows an oval component to the left of the non-compact component. With a negative discriminant, only a non-compact component exists. An oval component can be quickly obtained by decreasing the value of parameter a, starting with the initial values.
Value a:Value b:
Range:
Elliptic curves play a role in several areas of mathematics. They are used in cryptography for encryption methods. In number theory, the rational values on the curve are important. With their help, Andrew Wiles was able to prove Fermat's Last Theorem in 1994, after more than 350 years. Furthermore, elliptic curves appear in algebraic geometry and are the subject of intensive research. They are also used in algorithmic mathematics to investigate more efficient computational methods.
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