Belyi maps have S_3 applied #6796
Description
In the new equations for Belyi maps, there was an S_3-action applied to get the nicest coordinates but this is not recorded, found by Alexandre Guillemot.
For example, even in
https://www.lmfdb.org/Belyi/2T1/2/2/1.1/a/
which is the map t = -x^2, the ramification points are at 0,oo so the permutation triple should be
(1 2), triv, (1 2)
As far as I can tell, everything is OK for the old maps; in the example above, it's
t = -1/(x^2-1)
which becomes
x^2 = 1-1/t
which is indeed ramified at t = 0,1.
The first quick patch would be to indicate this in the equation (just write "up to S_3-action, as a lax Belyi map").
It'll kinda suck to go back and reconstruct this S_3-action in some symmetric cases, so probably the simplest thing is just to rerun the computation of the simpler models but keep track of the S_3-action. In the example above, we would still write
x^2 + t = 0
but the Belyi map would be
phi(t,x) = 1/(1-t).
But I wonder if we shouldn't follow the decision we made for Hilbert modular forms. Initially we thought that it would be better to keep only one form up to Gal(F | QQ)-action because it took up less space, eventually we decided it was too confusing and causing and bugs so we just added them all. Does the same principle apply here, and we really should add these additional maps? Strictly speaking, they are not isomorphic as Belyi maps (so the S_3-action is like the Galois action). We needn't explode the size of the database as they would all have the same model and only the map would change. In the example above, it means we could have the simpler t = x^2 as well as the rest.