Some observations to speed the polynomial selection in the number field sieve

If the yield of a polynomial pair is closely correlated with the coefficients of the polynomial pair, we can select polynomials by checking the coefficients first. This can speed the selection of good polynomials. In this paper, we aim to study the correlation between the polynomial coefficients and the yield of the polynomials. By heuristic analysis and some experiments, we find that the yield of polynomial with the ending coefficient containing many small primes is usually better than the one whose ending coefficient does not contain. The ending coefficient has closer correlation with the yield than the leading coefficient has. The number of real roots can be determined only by partial coefficients of the polynomial if it is skewed. All these observations can be used to speed the search of good polynomials for the number filed sieve.