I have figured out some things; it is much quicker, as far as computing, to find a way for the Hessian matrix of the ternary quadratic form, is to have it **represent** the (two by two) Hessian of the form $xy;$ this form, or its quadratic space, is often called The Hyperbolic Plane; see page 15 in Cassels.

Once this is done, there is just the business of adding an appropriate third row to "rows" to get a nice result. The final quadratic form is
(y - 1250*z)*x + (-797*z*y - 5751*z^2)
$$ xy -797yz - 1250 zx - 5751 z^2, $$
which is universal because we can take $z = 0, y = 1,$ and $x$ equal to the target number. Oh, the beginning form was your
$$ 77yz + 91 zx + 143xy $$

$$
\left(
\begin{array}{ccc}
830 &-3486 &-2145 \\
616& -2587 & -1592 \\
-3& -5& 12 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0 &143 &91 \\
143& 0&77 \\
91&77 &0 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
830 &616 &-3 \\
-3486& -2587 &-5 \\
-2145&-1592 &12 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
0&1 &-1250 \\
1&0 & -797 \\
-1250&-797 &-11502 \\
\end{array}
\right)
$$
**Note**: it turns out to be quite easy, with explicit matrices, to take the form with the visible hyperbolic plane to the form $xy - (abc) z^2,$ that GH has already proved equivalent to the original $ayz+bzx+cxy.$

$$
\left(
\begin{array}{ccc}
830 &-3486 &-2145 \\
616& -2587 & -1592 \\
1431507& -6012097& -3699553 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
0 &143 &91 \\
143& 0&77 \\
91&77 &0 \\
\end{array}
\right)
\left(
\begin{array}{ccc}
830 &616 & 1431507 \\
-3486& -2587 &-6012097 \\
-2145&-1592 & -3699553 \\
\end{array}
\right) =
\left(
\begin{array}{ccc}
0&1 &0 \\
1&0 & 0 \\
0& 0 &-2004002 \\
\end{array}
\right)
$$

========================================================

```
parisize = 4000000, primelimit = 500000
? h = [ 0,143,91; 143,0,77; 91,77,0]
%1 =
[ 0 143 91]
[143 0 77]
[ 91 77 0]
? rows = [ 830, -3486, -2145; 616, -2587, -1592 ]
%2 =
[830 -3486 -2145]
[616 -2587 -1592]
? columns = mattranspose(rows)
%3 =
[ 830 616]
[-3486 -2587]
[-2145 -1592]
? rows * h * columns
%4 =
[0 1]
[1 0]
?
?
?
? rows = [ 830, -3486, -2145; 616, -2587, -1592; -3,-5,12 ]
%5 =
[830 -3486 -2145]
[616 -2587 -1592]
[ -3 -5 12]
? matdet(rows)
%6 = 1
? columns = mattranspose(rows)
%7 =
[ 830 616 -3]
[-3486 -2587 -5]
[-2145 -1592 12]
? rows * h * columns
%8 =
[ 0 1 -1250]
[ 1 0 -797]
[-1250 -797 -11502]
? x
%9 = x
? y
%10 = y
? z
%11 = z
? g = rows * h * columns
%12 =
[ 0 1 -1250]
[ 1 0 -797]
[-1250 -797 -11502]
? vec = [ x,y,z]
%13 = [x, y, z]
? vect = mattranspose(vec)
%14 = [x, y, z]~
? vec * g * vect / 2
%15 = (y - 1250*z)*x + (-797*z*y - 5751*z^2)
?
```

======================================================

allintegers are to be represented including negative integers. $\endgroup$ – Somos Sep 17 at 20:11