Question on ternary quadratic form

[hermann@stamm-wilbrandt.de]

After two modifications I got this ternary quadratic form:

? Qt

[1  0   0]

[0  5  28]

[0 28 157]

?

I know it can be transformed to ternary identity form.
In order to determine the transformation I define:

? X

[1 0 0]

[0 a b]

[0 c d]

?

And get this:

? X~*Qt*X

[1                                   0                                   
0]

[0            5*a^2 + 56*c*a + 157*c^2 (5*b + 28*d)*a + (28*c*b + 
157*d*c)]

[0 (5*b + 28*d)*a + (28*c*b + 157*d*c)            5*b^2 + 56*d*b + 
157*d^2]

?

Now I switch to wolframscript and determine a solution for the three 
equations:

In[4]:= FindInstance[5*a^2 + 56*c*a + 157*c^2==1&&5*b^2 + 56*d*b + 
157*d^2==1&&(
5*b + 28*d)*a + (28*c*b + 157*d*c)==0,{a,b,c,d},Integers]

Out[4]= {{a -> -6, b -> -11, c -> 1, d -> 2}}

In[5]:=

With that I define matrix Y:

? Y

[1  0   0]

[0 -6 -11]

[0  1   2]

?

And really Y is transformation matrix to ternary quadratic form:

? Y~*Qt*Y

[1 0 0]

[0 1 0]

[0 0 1]

?


How can I determine integer {a,b,c,d} solutions in PARI/GP without using 
wolframscript?


Regards,

Hermann.