Toric Fano polytopes: There are

  866 Fano polytopes  in 5 dimensions and their polar polytopes (facet equations),
  124 Fano polytopes  in 4 dimensions,
   18 Fano polytopes  in 3 dimensions,
    5 Fano polytopes  in 2 dimensions.

The polar polytopes (and other data) can be computed with the command poly.x -Dv of the software package PALP. (Vertices a of the polar polytope correspond to coefficient vectors for the bounding halfspace equations a.x+1≥0.)

NEF partitions and Complete Intersection Calabi-Yau 3-folds:

The 5d-Fano polytopes have between 3 and 159 nef partitions per polytope, which can be used to construct 36233 toric CICY 3-folds with 421 different pairs of Hodge numbers.

HOWTO compute these results with PALP:

Making reflexive polytopes:	
echo -e "3 1 1 1\n4 1 1 2\n6 1 2 3\n2 1 1 0 0 2 0 0 1 1" > 2d.cws
class.x -po 2d.bin 2d.cws                                               # 2d reflexive (binary)
class.x -b -pi 2d.bin > 2d.ref                                          # 2d reflexive (asci)

echo "4 1 1 1 1 /Z2: 0 1 0 1" > 3d.cws ; cws.x -w3 >> 3d.cws ; cws.x -c3 >> 3d.cws
class.x -po 3d.bin 3d.cws ; class.x -b -pi 3d.bin > 3d.ref              # 3d reflexive (asci)

http://quark.itp.tuwien.ac.at/~kreuzer/d4/                              # 4d reflexive database (4.4 GB)

The following is still under construction / verification:

Making Fanos: This requires palp-1.1+fano.tar.gz   (code will be contained in releases > palp-1.1) 

poly.x -U1e 2d.ref| poly.x -fN| poly.x -fv> ToricFano.2d        	# Result 2d: #Fano polytopes = 5

poly.x -U5 2d.ref 3d.fanocand			 	 		# redundant list of lifts
poly.x -DU1e 3d.fanocand | class.x -f -a -po 3d.fano.bin		# Result 3d: #Fano polytopes = 18
class.x -b -pi 3d.fano.bin > ToricFano.3d		     	        # conversion to asci    
poly.x -U1v 3d.ref | grep -c P             		     	        # test     

poly.x -U5 3d.ref 4d.fanocand			 	 		# redundant list of lifts
poly.x -DU1e 4d.fanocand | class.x -f -a -po 4d.fano.bin		# Result 4d: #Fano polytopes = 124
class.x -b -pi 4d.fano.bin > ToricFano.4d		     	        # conversion to asci    

class.x -b -di d4/zzdb | poly.x -fU5 5d.fanocand	 		# redundant list of lifts
poly.x -DU1e 5d.fanocand | class.x -f -a -po 5d.fano.bin		# Result 5d: #Fano polytopes = 866
class.x -b -pi 5d.fano.bin > ToricFano.5d		     	        # conversion to asci    



Computing NEF partitions
	
nef.x -N -D -P ToricFanos.5d FanoCICY.cd2              		        # NEF partitions and CICYs