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Deletions are marked like this.  Additions are marked like this. 
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list st of Elliptic curves which for which ρE,2 is surjective mod 2 but not mod 4.  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. [[attachment:2not4or4not8v3.sage2not4or4not8v3.sage]] 
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List of Elliptic curves which for which ρE,2 is surjective mod 4 but not mod 8.  The results of the above program for 2not4 curves is [[attachment:li4.sobjli4.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 [[attachment:2not4or4not8galoisapproach.sage2not4or4not8galoisapproach.sage]] The results of the above program for 2not4 curves is [[attachment:li8.sobjli8.sobj]] 
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 2not4or4not8galoisapproach.sage
The results of the above program for 2not4 curves is li8.sobj