Weight systems for 5 dimensions
 The database of all weight systems for 5d polyhedra with the IP property has a
searchable web frontend.
It is now also
available in Parquet format at the Hugging Face Hub.
 Older results:

All 5d weighted projective spaces with
CY 4fold hypersurfaces defined by quasihomogeneous polynomials
satisfying a transversality condidition:
1,100,055 (23MB);
subset of weight systems defining reflexive polytopes:
252,933 (6MB)
 All weight systems for reflexive 5d polyhedra (CY 4folds)
with degrees
6100 (2.7MB),
101150 (12MB),
151200 (31MB),
201250 (63MB),
251260 (18MB),
261270 (20MB),
271280 (22MB),
281290 (24MB),
291300 (26MB)
 All weight systems for reflexive 6d polyhedra with degrees
7100 (3.3 MB),
101150 (20 MB)
 All files
 The format of the files is
w1 ... w6 deg=d M:#p #v N:#p #v H:h11,h12,h13 [chi] . . . (reflexive)
w1 ... w6 deg=d M:#p #v N:#p #v V:h11,h12,h13 [chi] . . . (reflexive, transverse)
w1 ... w6 deg=d M:#p #v F:#f V:h11,h12,h13 [chi] . . . (transverse)
where
 #p (#v) = number of points (vertices) in the M or N lattice;
#f = number of facets (for nonreflexive polyhedra);
 H = toric Hodge data (nontransverse case, Batyrev's formula),
V = chiral ring data (Vafa's formula;
for all weights that define both reflexive polytopes and transverse
polynomials, i.e. ... N ... V ..., we checked that both formulas give the
same result).
For 6d polyhedra the Euler number is unknown: ignore the [0].
 History and references:
 In hepth/9701023
Klemm, Lian, Roan and Yau computed all Fermat weights
and the transverse weights up to degree 400.
 In hepth/9701175
Kreuzer and Skarke computed all reflexive weights up to degree 150 and all transverse
weights up to degree 4000.
 In hepth/9812195
Lynker, Schimmrigk and Wisskirchen constructed all 1,100,055
transverse configurations. We took their weights and added the
M and Nlattice data in our file format.
 Later Max Kreuzer also computed all
reflexive 4fold weight systems up to degree 300 and all
reflexive 5fold weight systems up to degree 150. These can
be used as starting points for complete intersections
of codimension 2 and 3, respectively (cf.
math.AG/0103214).
 In arXiv:1808.02422
Schöller and Skarke found all weight systems that determine 5d IP
polytopes.