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%; }
.rceditor {font-size: 0.88em; white-space: nowrap; }

/* admonition start */
div.warning {background-color: #F9F9FF; border: 1pt solid #E5E5E5; color: black; 
    margin: 10pt 30pt; min-height: 64px; padding-right: 64px; }