These Thurston maps are NET maps for every choice of translation term. They have degree 4. They are imprimitive, each factoring as a Thurston map with degree 2 followed by a Euclidean NET map with degree 2. The non-Euclidean factor is a NET map if it has four postcritical points. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0,lambda1} {lambda2} {lambda1+lambda2} These pure modular group Hurwitz classes each contain infinitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/4, 2/2, 2/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,infinity) The half-space computation determines rationality. The supplemental half-space computation is not needed. These NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points because every nonzero multiplier is at least 1 and the map is rational. Similarly, there are not even any slope function cycles. The slope function maps some slope to the nonslope. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 ends in the nonslope. If the slope function maps slope p/q to slope p'/q', then |q'| <= |q| for every slope p/q with |p| <= 50 and |q| <= 50. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,3)", "b=<1,1,1,c>(2,3)", "c=<1,1,1,1>(1,2)(3,4)", "d=(1,2)(3,4)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,b*c,d*a,b*c*b^-1>(2,3)", "b=(2,3)", "c=<1,1,1,1>(1,2)(3,4)", "d=(1,2)(3,4)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,2)(3,4)", "b=<1,1,1,1>(1,2)(3,4)", "c=<1,1,1,c>(2,3)", "d=(2,3)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,2)(3,4)", "b=<1,1,1,1>(1,2)(3,4)", "c=(2,3)", "d=<1,a,a^-1,c>(2,3)", "a*b*c*d");