There are several files that aid in checking 
the computations in section 2.3 of the paper
"Nontrivial elements of Sha through K3 surfaces,"
and several files that aid in using the results
and the explicit elliptic fibrations found. 

The files manual*.jpg contain 9 pages of manual
computations that verify all statements made in 
section 2.3.

The script section2.3.mg checks all the computations
using MAGMA. To verify that everything is checked we
recommend running it with the command 
   iload "section2.3.mg";
It saves the galois invariant polynomials that define
the elliptic fibration in the file "section2.3.state",
which will be fairly big, but uses the relatively 
efficient phi_j as coordinates. Running this script
takes only about 60 megabytes of memory.

The file phi2a.mg loads the polynomials from 
"section2.3.state" and expresses them in terms of 
the coordinates a_i (hence the name of the file). 
The coefficients are then expressed in terms of 
the symmetric polynomials c_i. This could be done 
more efficiently, but to keep the programming more
transparent, it is kept easy. It takes around 800
megabytes of memory and 90 hours of processor 
time (AMD Opteron(tm) Processor 246, 2000 Mhz).
The resulting polynomials that express the fibration
and the quadric Q from the paper are saved in 
"pre(c)coord(i)" where c runs through x,y,z, and i 
is 1 or 2, and "prequadricQ". These files are then 
slightly changed to contain functions that return 
these expressions for any given explicit c_i, d_i,
and N. Here the c_i are such that 
   f_4 = x^4 - c1 x^3 + c2 x^2 - c3 x + c4
is a factor of f, and the image of delta in k[x]/f_4
equals delta4 = d_3 x^3 + d_2 x^2 + d_1 x + d_0, 
while N is a square root of the norm of delta4 from 
k[x]/f_4 to k. The resulting files are quadricQ, 
and (c)coord(i).

The file "makegeneqs.mg" expresses the generic 
norm of delta4 from k[x]/f_4 to k in terms of the 
c_i and the d_i. It also gives the equation for 
the conic C_3 with coefficients in terms of the 
c_i and d_i. These are saved in the files "pregeneqs".
Running this file only takes around 11 megabytes
of memory. 

The file "geneqs" comes from the output "pregeneqs"
of "makegeneqs.mg". It contains functions that for 
any c_i, d_i, and N return the norm of delta4 from 
k[x]/f_4 to k, the conic C_3, equations for the 
fibration, and a function to find linear equations 
for the fibration over a P^1 that works (obviously)
only in the case that the conic C_3 contains a
rational point. These functions do not work in case
the generic equations for the fibration specialize
to zero, or the generic conic specializes to become
absolutely reducible. 

The file "geneqs.example" contains an example of 
how to use the functions in "geneqs".