File:BSD data plot for elliptic curve 800h1.svg

English: A plot of the type of data used by Birch and Swinnerton-Dyer to support their conjecture. The curve in question is y² = x³ − 5x (curve 800h1 of the Cremona database). This is a curve of rank 1 (and one of the curves originally looked at by Birch and Swinnerton-Dyer). The horizontal axis is a bound X and the vertical axis is ${\displaystyle Y(X)=\prod _{p\leq X}{\frac {N_{p}}{p}}}$ where N_p is the number of points on the curve modulo p. The scale of the horizontal axis is log(log(X)) while the scale of the vertical axis is log(Y). The data should thus resemble a line of slope 1 (equal to the rank of the curve). The data points in blue are the points (X, Y(X)) where X runs through the first 100000 primes. The red line is a line of slope 1 for comparison purposes (the Y-intercept was simply chosen so that the line went through the data points). The data was computed and plotted using Sage.