Record Mahler Measures by Height

This table lists the polynomial with the smallest known Mahler measure among all irreducible, noncyclotomic polynomials with integer coefficients having a fixed height, H. Here the height of a polynomial is simply the maximum of the absolute values of its coefficients.

 H   D   Measure        Out  Coefficients
 1   10  1.176280818260   1  1 1 0 -1 -1 -1
 2   20  1.226993758166   2  1 1 -1 -2 0 2 1 -1 -1 0 1
 3   32  1.236227922245   4  1 2 2 1 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 2 3
 4   94  1.249426432422  11  1 1 1 1 0 0 -1 -2 -2 -2 -1 0 1 2 3 3 2 1 -1 -2 -3 -4 -3 -3 -1 1 2 4 4 3 2 0 -2 -3 -4 -4 -2 -1 1 3 3 4 3 1 0 -2 -3 -3
 5   72  1.243210037398   6  1 1 1 1 1 1 0 -1 -2 -2 -2 -2 -1 0 2 3 3 3 2 1 -1 -3 -4 -4 -3 -2 0 2 4 5 4 3 1 -1 -3 -5 -5
 6  108  1.253946827521  12  1 1 1 1 0 0 -1 -2 -2 -2 -1 0 1 2 3 3 3 2 0 -1 -3 -4 -4 -4 -2 0 2 4 5 5 4 2 -1 -3 -5 -6 -5 -4 -1 2 4 6 6 4 2 -1 -4 -5 -6 -5 -2 0 3 5 5
 7  118  1.254112341653  13  1 1 1 1 0 0 -1 -2 -2 -2 -1 0 1 2 3 3 2 0 -2 -3 -4 -4 -3 -1 2 4 5 5 4 2 -1 -4 -6 -6 -5 -3 0 3 6 7 6 4 1 -2 -5 -7 -7 -6 -3 0 3 6 7 7 5 2 -1 -4 -6 -7
 8  166  1.253310410960  13  1 1 1 1 0 -1 -2 -3 -3 -2 -1 1 3 4 5 4 2 0 -3 -5 -6 -6 -4 -1 2 5 7 7 6 3 -1 -4 -7 -8 -7 -5 -1 3 6 8 8 6 3 -1 -5 -7 -8 -7 -4 -1 3 6 7 7 5 2 -1 -4 -6 -6 -5 -3 0 2 4 5 4 3 1 -1 -2 -3 -3 -2 -1 0 1 1 1 1 0 0 0 -1
 9  132  1.250511624066  14  1 1 1 1 0 0 -1 -2 -2 -2 -1 0 1 2 3 3 2 1 -1 -3 -4 -5 -4 -2 0 3 5 6 6 4 1 -2 -5 -7 -7 -6 -3 1 4 7 8 7 5 1 -3 -6 -8 -8 -6 -3 1 5 8 9 8 5 1 -3 -7 -9 -9 -7 -3 1 5 8 9
11  132  1.252615987381  16  1 2 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 4 4 3 1 -1 -3 -5 -6 -5 -3 -1 1 3 5 6 6 5 3 0 -3 -6 -7 -7 -6 -4 -1 2 5 7 8 8 6 3 -1 -5 -8 -9 -9 -7 -4 0 4 7 9 10 9 6 2 -3 -7 -10 -11
13  118  1.257762678077  15  1 2 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 4 4 3 1 -1 -3 -5 -6 -6 -5 -3 0 3 6 8 9 8 5 1 -3 -7 -10 -11 -10 -7 -3 2 7 10 11 11 9 5 0 -5 -9 -12 -13 -11 -7 -2 4 9 12 13
17  142  1.255611215829  17  1 2 2 1 0 -1 -2 -3 -3 -2 -1 0 1 2 3 4 4 3 1 -2 -5 -7 -7 -5 -2 1 4 6 7 7 6 4 1 -3 -7 -10 -11 -9 -5 0 5 9 11 11 9 6 2 -3 -8 -12 -14 -13 -9 -3 4 10 14 15 13 8 2 -4 -9 -13 -15 -14 -10 -4 3 10 15 17

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Last modified June 5, 2009.