Table of all hyperbolic 4-to-3 rational Belyi maps and their dessins.
Mark van Hoeij and Raimundas Vidunas.
- The phrase "4-to-3" refers to a reduction of a 4-singularity equation (a Heun equation)
to a 3-singularity equation (a hypergeometric equation). For a precise description of the content of the 4-to-3 table,
see Remark 1 in the file NamingConvention, or the paper.
- The table of hyperbolic 4-to-3 rational Belyi maps contains 366 non-parametric cases (which we named: A1, ..., J28)
and 48 parametric cases (named: S1 .. S48) (see: NamingConvention).
- BelyiMaps: The maps A1-J28 and S1-S48 in this text-file
form a complete list of all hyperbolic 4-to-3 rational Belyi maps (up to conjugacy, and up to Mobius transformations).
The algebraic numbers are in Maple-syntax, but one can copy/edit/paste them to the syntax of
other computer algebra systems. The entire file can be read into Maple with the command "read BelyiMaps:"
(make sure to use a colon, not a semi-colon).
- 2F1_Heun_Sorted_by_Degree: This is the main file. It contains the main information
regarding these Belyi maps (ModuliField, Obstruction (if any),
Branching type, Decompositions (if any) and the Galois group (i.e. cartographic group).
It is sorted by type and degree (here is a version that is Sorted_by_Name).
- Dessins_and_ModuliFields: List of dessins (given as pairs of permutations g0, g1). This file contains
874 nonparametric 4-to-3 dessins, 50 parametric 4-to-3 dessins, and 4 dessins (from S49,S50,S51) that are not part of the 4-to-3 table.
- Some more information about
the fields Q(j),
the fields Q(j), Q(t), and Q(t,r)
- Remark: The files given here contain three more Belyi maps (S49, S50, S51) that are not
part of the 4-to-3 table (they are 3-to-3). These three maps were included because they occur in
the description of the decomposition structure.
- Click here for a complete list of files.
Related work parametric cases:
- Maximal degree for parametric d-to-3 cases is 6*(d-2).
- The 48 parametric cases S1..S48 for 4-to-3 Belyi are given [Vidunas and Filipuk, 2014] (max degree = 12).
- This is extended to 5-to-3 in FiveSing, which considers non-Belyi maps as well.
416 + 122 +
dessins and Belyi maps, with max degree 18, and (65+3) + 20 + 12 + 2 near-Belyi maps)
Related work non-parametric cases:
- Maximal degree for hyperbolic d-to-3 cases is 36*(d-7/3).
- Hyperbolic 3-to-3 transformations are given in [Vidunas 2005] (max degree = 24).
- The data on this page generalizes this to 4-to-3 (max degree = 60) (this required code specifically designed for 4-to-3, see below).
Computing 4-to-3 Belyi maps up to degree 60.
- Deterministic ComputeBelyi program for hyperbolic 4-to-3, and accompanying paper.
Syntax: The command ComputeBelyi(20, [4,2,5], [[1,1,1,1],,], x); returns all Belyi maps (up to birational
equivalence) with branching type 0:[1^4,4^4] 1:[2^10] infty:[5^4].
Our table was initially computed with a probabilistic method, which has now
become obsolete since we can now compute all Belyi maps in our table (and prove its completeness) with the
- This Maple file ComputeAll
first computes all branching patterns for our table, and then uses the deterministic algorithm ComputeBelyi
(download ComputeBelyi) to compute all 366 non-parametric Belyi maps in our table
(up to Mobius-equivalence) (the file ComputeAll produces 367 maps because it finds A21 twice).
Turning these Belyi maps into Heun equations and their hypergeometric type solutions.