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2978 bytes, as of the version from 2026-10-25 17:54 (current). Recorded here, not run.
# Source and verification notes for T496
This table holds the full linear isometry-group orders of all 24 positive
definite even unimodular lattices in dimension 24. The parameter is the root
system, rather than a running row number. Its empty member is named `Leech`
and denotes the same lattice as `Lambda,24` in T495. Root norms are 2; the
Leech minimum norm is 4. The stored orders include the central involution.
The generator input is an unchanged download, on 2026-10-25, of Richard
Borcherds's file https://math.berkeley.edu/~reb/lattices/norm0 . The index
https://math.berkeley.edu/~reb/lattices/index.html identifies its 24 rows.
Field 5 names the root system and field 6 gives the group order modulo root
reflections. For the Leech lattice the reflection group is trivial. Multiplying
by the component Weyl orders gives the full lattice-group order. The first
field supplies the increasing Coxeter-number ordering used in the table.
The separate file `independent-quotients.json` records group factors from
Gaëtan Chenevier, The characteristic masses of Niemeier lattices, Journal de
théorie des nombres de Bordeaux 32 (2020), 545–583, Section 5.2; DOI
10.5802/jtnb.1134, arXiv:2002.03707. These factors were read from the paper,
not obtained by dividing the generator's answers. For the Leech lattice,
Example 1.3 identifies Co0 and Table C.5 gives the identity characteristic
polynomial mass 1/8315553613086720000. The publisher's PDF was available at
https://www.numdam.org/item/10.5802/jtnb.1134.pdf .
The rank-24 mass is checked using the Bernoulli-product formula in Section 2
of Oliver D. King, A mass formula for unimodular lattices with no roots,
Mathematics of Computation 72 (2003), 839–863, DOI
10.1090/S0025-5718-02-01455-2, arXiv:math/0012231. The rank-8 and rank-16
cases are positive controls. Two negative controls change one order by 1 and
omit the Leech row. The exact rank-24 result is
1027637932586061520960267/129477933340026851560636148613120000000.
`checks.py` builds its independent Weyl orders from Sage's positive-root
heights. It also enumerates all ADE root systems of total rank 24 with a
common Coxeter number, finding the same 23 nonempty systems. It does not
call the generator's order function for the independent answers.
Run `sage -python checks.py` for offline checks. With NUMBERDB_CHECK_STORED=1
and an API key it additionally reads T496, repeats all numerical checks on
its stored values, compares the Leech entry with T495, compares the Conway
and Mathieu relations with T77, evaluates the 18 applicable polynomials of
T257 at 1, and runs `verify(sample=None)`. T495 and T496 were drafts at build
time, so their authenticated reads need the owner's key.
All results are integers obtained with exact arithmetic. No decimal digit
count or approximation error applies. All 24 classes are included; the
largest written value has 31 digits. The attached `check.py` is the unchanged
shared NumberDB build-check helper.