In R.Miranda-U.Persson, Configurations of In fibers on elliptic K3-surfaces,
Math. Z. 201 (1989), 339-361 the authors study semi-stable elliptic
fibrations over P^1 of K3-surfaces with 6 singular fibres.
In their paper the authors give a list of possible
fiber types for such fibrations. It turns out that there are 112 such cases.
The corresponding J-invariant is a so-called Belyi-function.
More particularly, J is a rational function of degree 24,
it ramifies of order 3 in every point above 0, it ramifies of order
2 in every point above 1, and the only other ramification occurs
above infinity. To every such map we can associate a so-called
'dessin d'enfant' (coined by Grothendieck) which in its turn uniquely
determines the Belyi map. If f: C -> P^1 is a Belyi map, the dessin
is the inverse image under f of the real segment [0,1].
Recently (2004) H.Montanus and F.Beukers computed
all J-invariants corresponding to the Miranda-Persson list. Below
is the collection of dessins d'enfant. An entry like
14-3-2-2-2-1 means that one finds there all dessins of J-maps
with ramification orders
14,3,2,2,2,1 above infinity. Alternative one can say that the special
elliptic fibers are of type In with n=14,3,2,2,2,1. If to a partition
there corresponds only one picture, this means that J is a rational function
with coefficients in Q. If there are several pictures, the corresponding
fields of definition are indicated.
Furthermore, we compiled the
list of all J-maps with coefficients in Q. The other J-maps
can be defined over quadratic or cubic fields except one which
is defined over a quartic field. An explicit list is under preparation.
