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list st of Elliptic curves which for which $\rho_{2,E}$ 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 $\rho_{2,E}$ is surjective mod 4 but not mod 8.  The results of the above program for 2not4 curves is [[attachment:li4.sobjli4.sobj]] [[attachment:2not4 output.txt2not4 output.txt]] 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:2not4galoisapproachv2.sage2not4galoisapproachv2.sage]] 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. {{{ .rcpagelink {width: 33%; } .rctime, .rceditor {fontsize: 0.88em; whitespace: nowrap; } /* admonition start */ div.caution, div.important, div.note, div.tip, div.warning {backgroundcolor: #F9F9FF; border: 1pt solid #E5E5E5; color: black; margin: 10pt 30pt; minheight: 64px; paddingright: 64px; } }}} 
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 2not4 output.txt
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 2not4galoisapproachv2.sage
The results of the above program for 4not8 curves is li8.sobj
I am working on verifying this result using the Galois approach.
.rcpagelink {width: 33%; } .rctime, .rceditor {fontsize: 0.88em; whitespace: nowrap; } /* admonition start */ div.caution, div.important, div.note, div.tip, div.warning {backgroundcolor: #F9F9FF; border: 1pt solid #E5E5E5; color: black; margin: 10pt 30pt; minheight: 64px; paddingright: 64px; }