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I have verified the results of 2not4 list, using a galois approach, which involves compute the order of the Gal(Q(E[4])/Q). The program I used is [[attachment:2not4galoisapproach.sage2not4galoisapproach.sage]]  I have verified the results of 2not4 list, using a Galois approach, which involves compute the order of the Gal(Q(E[4])/Q). The program I used is [[attachment:2not4galoisapproach.sage2not4galoisapproach.sage]] 
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The results of the above program for 2not4 curves is [[attachment:li8.sobjli8.sobj]]  The results of the above program for 4not8 curves is [[attachment:li8.sobjli8.sobj]] I am working on verifying this result using the Galois approach. 
I used a Heuristic approach to narrow down the list of elliptic curves for which \rho_{2,E} is surjective mod 2 but not mod 4 or \rho_{2,E} is surjective mod 4 but not mod. 2not4or4not8v3.sage
The results of the above program for 2not4 curves is li4.sobj
I have verified the results of 2not4 list, using a Galois approach, which involves compute the order of the Gal(Q(E[4])/Q). The program I used is 2not4galoisapproach.sage
The results of the above program for 4not8 curves is li8.sobj
I am working on verifying this result using the Galois approach.