Before you start marking or cutting, bleach the inside of the egg. It's very difficult to remove membrane after the cuts for this type of egg. As the whole egg is on view, it's essential that it's clean both inside and out.
Next, draw your 12 lines vertically, and leave in the egg marker. I am using a marker which I normally would use for larger eggs, but I have a Mark-It as well, which does the job just as well. This marker made it easier to take pictures for you. Remember to leave the egg in the marker at this stage.
This has now been marked with just 3 horizontal lines. (See on left in picture below)
The fewer the lines, the less curve you will have. It may be advisable to start with just a few until you build up confidence.
The next egg on the right, I have marked 5 lines instead of 3. This will give a more accentuated curve. This picture shows how to draw the spiral itself, which means using the eye more than following a 'point to point' line. The lines at the top and bottom *must* curve.