2 added 91 characters in body

For your sample problem, I get two flavors of identity, principal and non-principal.

For discriminant 229, I take the identity form as $$f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$f(x^3 + 3 x y^2 + 5 y^3, \; 3 x^2 y + 45 x y^2 + 226 y^3 ) \; = \; f^3(x,y).$$ As an automorph of $f$ is $$\left( \begin{array}{rr} 1 & 15 \\ 15 & 226 \end{array} \right) ,$$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.

For the other two classes, take $$g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$ A second family is $$f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \; = \; g^3(x,y).$$
  The following cycles of reduced forms are as in Buell's book, pages 21-30.

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 1 15 -1 0 form 1 15 -1 delta -15 1 form -1 15 1 delta 15 2 form 1 15 -1 minimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229 Automorph, written on right of Gram matrix: -1 -15 -15 -226 Trace: -227 gcd(a21, a22 - a11, a12) : 15 ========================================= ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
3 13 -5

0  form   3 13 -5   delta  -2
1  form   -5 7 9   delta  1
2  form   9 11 -3   delta  -4
3  form   -3 13 5   delta  2
4  form   5 7 -9   delta  -1
5  form   -9 11 3   delta  4
6  form   3 13 -5
minimum was   3rep 1 0 disc   229 dSqrt 15.13274595  M_Ratio  25.44444
Automorph, written on right of Gram matrix:
-16  -75
-45  -211
Trace:  -227   gcd(a21, a22 - a11, a12) : 15
=========================================

1

For your sample problem, I get two flavors of identity, principal and non-principal.

For discriminant 229, I take the identity form as $$f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$f(x^3 + 3 x y^2 + 5 y^3, \; 3 x^2 y + 45 x y^2 + 226 y^3 ) \; = \; f^3(x,y).$$ As an automorph of $f$ is $$\left( \begin{array}{rr} 1 & 15 \\ 15 & 226 \end{array} \right) ,$$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.

For the other two classes, take $$g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$ A second family is $$f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \; = \; g^3(x,y).$$

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$./indefCycle Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 1 15 -1 0 form 1 15 -1 delta -15 1 form -1 15 1 delta 15 2 form 1 15 -1 minimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229 Automorph, written on right of Gram matrix: -1 -15 -15 -226 Trace: -227 gcd(a21, a22 - a11, a12) : 15 ========================================= ========================================= jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
3 13 -5

0  form   3 13 -5   delta  -2
1  form   -5 7 9   delta  1
2  form   9 11 -3   delta  -4
3  form   -3 13 5   delta  2
4  form   5 7 -9   delta  -1
5  form   -9 11 3   delta  4
6  form   3 13 -5
minimum was   3rep 1 0 disc   229 dSqrt 15.13274595  M_Ratio  25.44444
Automorph, written on right of Gram matrix:
-16  -75
-45  -211
Trace:  -227   gcd(a21, a22 - a11, a12) : 15
=========================================