\subsection{Level $165$: An $A_f$ not isomorphic to
its dual, though there are solutions to norm equation}

This has to be DELETED!!! It turns out that there is a solution that works -
it comes from the norm equation $\Norm (x) = -d$.

$[165, 3, 3, 0, 6, x^6 + 2*x^5 - 9*x^4 - 12*x^3 + 23*x^2 + 10*x +
1]$,

Has solutions to norm equation, but none of them work.


\begin{verbatim}
35 2 2
------

1/6*(K.1 - 20)
Homomorphism from image(35B) to image(35B) given on integral homology by:
[ 1 -2 -3  1]
[ 2 -4 -2  2]
[ 4 -2 -7  0]
[ 0  2 -1 -4]


1/3*(-K.1 - 16)
Homomorphism from image(35B) to image(35B) given on integral homology by:
[-14   4   6  -2]
[ -4  -4   4  -4]
[ -8   4   2   0]
[  0  -4   2  -4]
Good solution

***************************************************************************
165 1 2
-------

1/3*(4*K.1 - 24)
Homomorphism from image(165A) to image(165A) given on integral homology by:
[ -8  -8 -24   8]
[  0  -8  -8   0]
[  0  -8 -24   0]
[  8  -8  -8 -24]
Good solution


165 3 3
-------

1/2*(2059*K.1^2 + 47678*K.1 + 135555)
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
455*K.1^2 + 15210*K.1 + 122675
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
1/2*(3805*K.1^2 + 111770*K.1 + 719885)
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
-227*K.1^2 - 11210*K.1 - 133047
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
1/2*(-4535*K.1^2 - 131950*K.1 - 832935)
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
-5905*K.1^2 - 175170*K.1 - 1151185
Homomorphism from image(165C) to image(165C) (not printing 6x6 matrix)
None of the solution worked
\end{verbatim}