<< NumberTheory`Recognize`

(Subscript[k, 1]) = 1/(Sqrt[2]);
(Subscript[k, 2]) = (Sqrt[2]) - 1;
(Subscript[k, 3]) = ((Sqrt[3]) - 1)/(2 (Sqrt[ 2]));
(Subscript[k, 4]) = 3 - 2 (Sqrt[ 2]);
(Subscript[k, 5]) = ((Sqrt[(Sqrt[5]) - 1]) - (Sqrt[3 - (Sqrt[5])]))/
          2;
(Subscript[k, 6]) = (2 - (Sqrt[3])) ((Sqrt[3]) - (Sqrt[2]));
(Subscript[k, 7]) = (3 - (Sqrt[7]))/(4 (Sqrt[ 2]));
(Subscript[k, 8]) = ((((Sqrt[2]) + 
                  1))^2) (((1 - (Sqrt[2 (Sqrt[ 2]) - 2])))^2);
(Subscript[k, 9]) = ((Sqrt[2]) - (3^(1/4))) ((Sqrt[3]) - 1)/
            2;
(Subscript[k, 10]) = ((Sqrt[10]) - 3) ((((Sqrt[2]) - 1))^2);
(Subscript[k, 100]) = (((3 - 2 (Sqrt[ 2])))^2) (((2 + (Sqrt[5])))^2) ((((Sqrt[10]) - 
                  3))^2) ((((5^(1/4)) - (Sqrt[2])))^4);
(Subscript[k, 210]) = ((((Sqrt[2]) - 
                  1))^2) (2 - (Sqrt[3])) ((((Sqrt[7]) - (Sqrt[6])))^2) (8 - 
              3 (Sqrt[ 7])) ((((Sqrt[10]) - 
                  3))^2) (((4 - (Sqrt[15])))^2) ((Sqrt[15]) - (Sqrt[14])) (6 - (Sqrt[35]));

indx = Join[Range[10], {100, 210}];
\(\((minPolynomials = \(Recognize[
                  N[\@ModularLambda[I\ \@#], 500], 16, t] &\) /@ 
              indx // Factor)\) // TableForm\)
\(MapThread[\((Expand@\((#1 /. t -> #2)\) == 
            0)\) &, {minPolynomials, \(k\_# &\) /@ indx}]\)