(* This Mathematica file is a supplement to the paper Hurwitz-Belyi Maps.
All covers explicitly given in the paper are given here as well, 
including the degree 192 cover from Section 10 which is only abbreviated
in the paper.  The covers here were pasted in from the tex file
of the paper, as a means of avoiding typos in the paper.   

The earlier Arxiv version of this paper has a slightly different
section structure and six more covers.  The corresponding
Mathematica file is still available as HBM.mma
*)

(* There are 36 rational covers explicitly
given in the paper.  These covers are all in this file
as f[label] where label is on the list labels *)
labels = 
{64,7,24,"7alt","12aa","12ab",30,40,
{3,1},{4,3,2},{13,13,8,6,3,2,1},{9,9,9,8,8,4,4,2,1},
{5,3,1},{5,3,2},{5,4,3},{5,4,1},{5,4,3,3},{5,1},{5,5,4,2},{10,8,8,6,5,4,4,3},
{10,6,5,4,4,3},{5,4}, {9,6,5,4}, {5,5,5,5,4,3},{10,9,5,2,2},
{4,3},{7,4,3,3,1},{7,6,3,1,1}, 
{9,7},{9,7,2},{9,7,7,7,1,1,1},{9,7,7,4,3,1,1},{8,7,6,3},{12,8,8,7,3,2},
{15,15,15,9,9,9,5,3,3,3,3,3,3,1},
{15,15,15,12,12,12,12,12,9,9,9,6,6,6,6,6,6,6,6,5,4,3}}

(* There are also three irrational pairs with labels on the list irratlabels *)
irratlabels = {21 (* s=+-Sqrt[21]*), {6,5,4,1} (* s = +-Sqrt[6]*), 
{7,4,3,2} (* s = +- Sqrt[2] *)}

(* The command factdisc prints the discriminant of polynomials of the form f[variables,x] *)
factdisc[poly_] := Module[{fact,fnumb,fpoly},fact = Factor[Discriminant[poly,x]];
fnumb = fact[[1]];
fpoly = fact/fact[[1]];
FactoredForm[fnumb] fpoly]

(* Typing doit checks that 40 of the 42 covers have the right discriminantal forms,
with prettified discriminants being printed after about half a minute.
The covers of the two largest degrees, 96 and 192, are excluded because runtimes
would be too long.  Checks here can be made by substituting j=1 and factoring *)
doit := Map[factdisc,Map[f,Drop[labels,{39,40}]]]

(* Section one.  Introduction. *)

f[64] =  (x+2)^9 x^18  (x^2-2)^18 (x-2) - v (x+1)^16 (x^3-3 x+1)^16

(* Section two. Two Belyi maps unexpectedly defined over Q.  *)

f[7] =  x^3 (x^2+2 x-5)^2 + v 4 (2 x-1) (3 x-4)

modulifield1 = 
(48828125 x^23+283203125 x^22-4345703125
    x^21-21400390625 x^20+134842187500 x^19 +461968375000
    x^18-1670830050000 x^17-2095451850000 x^16+7249113240000
    x^15 +6576215456000 x^14-23053309281280 x^13-10284915779584
    x^12+50191042453504 x^11  +9449308979200 x^10-74715419574272
    x^9+5031544553472 x^8+71884253429760 x^7  -35243151065088 x^6-41613745192960
    x^5+29347637362688 x^4+14541349978112 x^3 +1765701320704 x^2+100126425088
    x+2684354560)

modulifield2 = (-4200 + 59640*x - 113232*x^2 - 132804*x^3 - 280392*x^4 +
 360780*x^5 + 324612*x^6 + 
 12348*x^7 - 104100*x^8 - 300936*x^9 + 87780*x^10 + 146316*x^11 - 52440*x^12 - 
 10638*x^13 + 4818*x^14 - 
3360*x^15 + 1812*x^16 - 102*x^17 + 70*x^18 - 31*x^19 - 
 40*x^20 + 26*x^21 - 8*x^22 + x^23)

f[24] := 5^5 x^6 (x-1)^4  (5 x+4)^2 - v 2^4 3^3 (2 x+1)^3 (5 x^2-6 x+2)^2

(* Section three.  Hurwitz maps, Belyi pencils, and Hurwitz-Belyi maps *)

belyipencil = {(t^6-8 t^3 v+9 t^2 v-2 v^2),
(t^6-3 t^5+10 t^3 v-15 t^2 v+9 t v-2 v^2) , 
(t^6-6 t^5 v+15 t^4 v-20 t^3 v+6 t^2 v^2+9 t^2 v-6 t v^2+v^2), 
(t^4-2 t^3+2 t v-v) ,
(t^4-4 t v+3 v),
(2 t^3-3 t^2+v) , 
(t)}

belyipencilcheck = Factor[Discriminant[Apply[Times,belyipencil],t]]

(* Section four. The two rational Belyi maps as Hurwitz-Belyi maps *)

f["7alt"] = (4 x^2+x) (4 x^3+25 x^2+18 x+3)^2 - v 4 x^2 (3 x+2)^5

f["12aa"] =  x^5 (9 x^2-21 x+16)^3 (x+3) - 2^8 w (x-1)^3 (9 x^2-12  x+8)^2

f["12ab"] = 5^5 (x-1)^4 x^6 (5 x+4)^2  -2^4 3^3  w(2 x+1)^3 (5 x^2-6 x+2)^2

(* Section five. The braid-triple method *)

(* Section six.  Hurwitz-Belyi maps exhibiting spin separation *)

f[30] = (2^2 3^3  (7 x^2+14 x+4)^7 x^5 (2 x+1)^3 (x^2+3 x+1)^2 (2 x^2+x+2)^2 + w (7 x^2+6 x+2)^5 (5 x+2)^4 
    (14 x^3+39 x^2+18 x+2)^3 (x+2))

f[40] =   (2^2 3^4 (5 x^2-12 x+3)^7  (5 x^2-15 x+12)^5  (x^2-3 x+6)^4   
 (4 x^2-15 x+15)^2 x (5 x-9)  + w (x^2-3)^5 (5 x^3-45 x^2+120 x-108)^4 *
(400 x^6-2700 x^5+7425 x^4-10530 x^3+7830 x^2-2430  x-27)^3)

(* s needs to be set equal to plus or minus Sqrt[21] for f21 *)
f[21] =  (2 5 (2700000 x^7+x^6 (630000 s-3780000)+x^5 (724500-829500
    s) +x^4 (2228100-474600 s)  +x^3 (1404725 s-5328225) +x^2 (7020216-1485456
    s)+x (856800 s-4384800)-252000 s+972000)^3  + 3 7^7 j (5239 s-21429) x^5 (10 x-9)^4
   (150 x^2+x (40 s-15)-8 s+88)^3)

(* Section seven. Hurwitz-Belyi maps with |G|=2^a 3^b p and nu = (3,1) *)  

f[{3,1}]  =  (x-4) x^3 + 4 j  (2 x+1)

f[{4,3,2}]   =   (4 x^3-3 x+2)^3 -27 j  x^3 (3 x-2)^2 

f[{13,13,8,6,3,2,1}] = ((16 x^4+40 x^3-3 x^2-116 x-8)(4096 x^14+20480 x^13-25856 x^12-
 196736 x^11+47189 x^10+680764 x^9-69384 x^8-1135104 x^7+7638144 x^6  -16337408
   x^5+9620480 x^4-2785280 x^3+741376 x^2-16384 x-32768)^3  -2^13 3^12 j (x-2)^3 x^6 (x+4)^2 (2 x-1) (3 x^2+2 x-4)^13)

f[{9,9,9,8,8,4,4,2,1}] = 
(
(x^3+12 x^2+12 x-8)(x^17-52 x^16+42136 x^15-593008
   x^14+10147846 x^13 +225862160 x^12+1467000268 x^11+6342760760 x^10+593082769
   x^9 -1815237116 x^8-5586407260 x^7-258348008 x^6+8975722736 x^5
    -8292246656
   x^4+3424464320 x^3-664160384 x^2+44883968 x-131072)^3
  -  2^4 3^9 j x (x^2-71
   x+32)^4 (x^2+2 x-1)^8 (x^3+18 x^2-48 x-8)^9)

(* Section eight. Hurwitz-Belyi maps with |G|=2^a 3^b 5 and nu = (3,1) *)  

f[{5,3,1}] = 5^2 (5 x^3-45 x^2+39 x+25)^3-2^14 3^3 j x^3 (3 x-25)   

f[{5,3,2}] = (9 x^3+3 x^2-53 x+81)^3 (x+9)-2^14 3^2 j x^3 (3 x-5)^2

f[{5,4,3}] = (4 x^4-24 x^3+24 x^2-48 x+27)^3 -2^2 3^3 j x^3 (3  x-4)^5  

f[{5,4,1}] =  (16 x^3-87 x^2+48 x+16)^3 (16 x+1) -2^2 3^12 j  x^4 (x-5)


f[{5,4,3,3}] = (4 (256 x^5+640 x^4-440 x^3-3325 x^2-6400 x-4096)^3  
- 3^12 j x^4 (32 x^2+95 x+80)^3)   

f[{5,1}] = (x^2-5)^3 -3^3 j  (2 x-5)

f[{5,5,4,2}] = (2^7 x (18 x^5-144 x^4+336 x^3-224 x^2+801 x-162)^3 - j (2 x-9)^2 (36 x^2-52 x-9)^5 )     

f[{10,8,8,6,5,4,4,3}] = 
((184528125 x^16-984150000 x^15+2263545000 x^14 -2768742000 x^13 +1616849100 x^12+181316880 x^11-
1023304104 x^10+721510416 x^9   -166620402 x^8-72763728 x^7+59318552 x^6-4952016 x^5
-12051828 x^4   +7406640 x^3-2117016 x^2+314928 x-19683)^3   + 2^20  3^8  j (9 x^2-10   x+3)^8 x^6  (5 x-3)^5 
(3 x^2-1)^4 (3 x-1)^3) 

f[{10,6,5,4,4,3}] = ((x^10-38 x^9+591 x^8-4920 x^7+24050
    x^6-71236 x^5+125638 x^4  
-124536 x^3+40365 x^2+85050 x-91125)^3 (x^2-14 x-5) 
+ 2^20 3^3 j x^6 (x-5)^5  (x^2-4    x+5)^4 (x-9)^3)

(* s is plus or minus Sqrt[6] *)
f[{6,5,4,1}] =  ((3 x+s-3) (-225 x^5+1305 x^4 s+4005 x^4-8932 x^3 s-22662 x^3+6594 x^2 s 
+16254 x^2-28476 x s-69741 x+11673 s+28593)^3  - 12288 j x^5 (5 x-9)^4 (53236 s+130401) (-15 x+76 s+186)
)

f[{5,4}] = 5^2 (10 x^3+15 x^2+48 x-100)^3 + 3^15 j  x^4


f[{9,6,5,4}] = ((9 x^8-72 x^7+180 x^6-104 x^5-26 x^4-568 x^3 +1620 x^2-1944 x+729)^3  
+ 2^16 j (x-3)^4 x^6 (2 x-3)^5)


f[{5,5,5,5,4,3}] = 
( (1024 x^9-13824 x^8+81360 x^7-272928 x^6+585144 x^5-879336
    x^4 +1012365 x^3 -896832 x^2+516096 x-131072)^3  
  - 54 j (72 x^4-508 x^3+1350 x^2-1629 x+768)^5 x^3 )


f[{10,9,5,2,2}] = 
 ((3125 x^9-9375 x^8+7500 x^7-6500 x^6+9150 x^5-4410 x^4-2484 x^3  -2916 x^2-2187 x+6561)^3 (x-3) 
+ 2^22 3^3 j x^9 (5 x-6)^5 (3 x^2+2 x+3)^2)

indices5 = {{5,3,1},{5,3,2},{5,4,3},{5,4,1},{5,4,3,3},{5,1},{5,5,4,2},
{10,8,8,6,5,4,4,3},{10,6,5,4,4,3},{5,4},
(* {6,5,4,1}, *) {9,6,5,4},
{5,5,5,5,4,3},{10,9,5,2,2}}

(* Section nine. Hurwitz-Belyi maps with |G|=2^a 3^b 7 and nu = (3,1) *)  

f[{4,3}]  =  4 (x-12) (9 x^2-20 x-27)^3 + 3 7^7 j  x^3

f[{7,4,3,3,1}] = (9 x^6-126 x^4+252 x^3-63 x^2-252 x+196)^3 + 2^6 j (3 x-2)^4 (3 x^2-9 x+7)^3 (3 x + 14) 

f[{7,6,3,1,1}] =  (9 x^6-102 x^5+295 x^4-212 x^3+39 x^2+90 x+9)^3 - 2^14 j x^6 (2 x-3)^3 (9 x^2-66 x-7)
  
 
(* s needs to be set equal to plus or minus Sqrt[2] *)
f[{7,4,3,2}] = 
((-7 x+19 s+27) (-49 x^5+x^4 (217 s-63)+x^3 (332 s-478)+  x^2 (154 s-658)+  x (196 s+147)+441   s+637)^3 
-216 j x^4 (35123 s+49688) (-2 x+s-4)^2 (7 s-4 x)^3)

f[{9,7}]  =  ((441 x^4+1764 x^3+702 x^2-140 x+49)^3 (343 x^4+2940 x^3+6594 x^2-468 x+63)
-2^42 j x^7)


f[{9,7,2}]  =  (7^4 x^5-441 x^4-3366 x^3+2430 x^2-3^7 x+3^7)^3   (49 x^2+6 x+9) (x+3) -2^30 3^9 j x^9 (x-1)^2

f[{9,7,7,7,1,1,1}] = ((16 x^11+256 x^10+1312 x^9+2208 x^8 -1248 x^7-6720 x^6 -1512
    x^5+5652 x^4   -6147 x^3-3912 x^2+11712 x-1536)^3 
+ 108 j (x-1) (x+2) (x+8) (8 x^3+15 x^2-9 x-8)^7  )

f[{9,7,7,4,3,1,1}] = 
((11664 x^10+31104 x^9-38880 x^8-276960 x^7-458528 x^6-245952
    x^5 +244440 x^4+549396 x^3+475389 x^2+225504 x+46656)^3  
-2^2 3^2 7^7 j  (8 x^2+15 x+9)^7 x^4 (x-3) (3 x^2+6 x+4) )

f[{8,7,6,3}] = ( 4  (4 x^7+22 x^6-60 x^5-166 x^4  +236 x^3+858 x^2-3626 x+2401)^3   
(2 x-1) (2 x^2+16 x-49) + 3^18 j x^7 (x-2)^6   (x+4)^3)

f[{12,8,8,7,3,2}] = 
((64 x^12-576 x^11+2400
    x^10-5696 x^9  +7344 x^8-3168 x^7  -4080 x^6  +8640 x^5-7380 x^4 -1508 x^3+8982
    x^2-7644 x+2401)^3 (4 x^4-20 x^3+78 x^2-92 x+49) 
-2^8 3^12  j  (2 x^2-4 x+3)^8 x^7 (x-2)^3  (x+1)^2)

indices7 = {{4,3},{7,4,3,3,1},{7,6,3,1,1},(* {7,4,3,2}, *) {9,7},{9,7,2},{9,7,7,7,1,1,1},{9,7,7,4,3,1,1},{8,7,6,3},{12,8,8,7,3,2}}

(* Section ten.  Some 5-point Hurwitz-Belyi maps *)

f[{15,15,15,9,9,9,5,3,3,3,3,3,3,1}] = 
((3 x^8-6 x^7-60 x^6+202 x^5-110 x^4-74 x^3-52 x^2-10 x-1)^3   
     (729 x^24-10206 x^23+15552 x^22-2045790 x^21+52397442 x^20-543319218
    x^19 +3209261832 x^18-12210163074 x^17+31525143435 x^16-55955395164
    x^15 +66094935696 x^14-43882703964 x^13-2654708692 x^12+42096515820
    x^11 -51857004992 x^10+37353393228 x^9-17942013057 x^8+5711207034
    x^7 -1071984720 x^6+65222394 x^5+12734514 x^4-1277306 x^3-182088 x^2  
-3850 x-3)^3  + 2^10 j   (3 x^3-7 x^2+11 x-1)^15   (3 x^3-9 x^2+3 x+1)^9 (x-3)^5 
  (x^4+8 x^3-36 x^2+17 x+1)^3 (x-1)^3 x)

f[{15,15,15,12,12,12,12,12,9,9,9,6,6,6,6,6,6,6,6,5,4,3}] = 
11664*j*(1 - x)^3*(3 - x)^5*x^4*(1 - 11*x + 7*x^2 - 3*x^3)^15*(-1 - 3*x + 9*x^2 - 3*x^3)^9*(4 - 23*x + 64*x^2 - 72*x^3 + 36*x^4 - 6*x^5)^12*
  (1 - 12*x + 72*x^2 - 280*x^3 + 474*x^4 - 444*x^5 + 240*x^6 - 72*x^7 + 9*x^8)^6 - 
 (-64 + 11008*x - 569088*x^2 + 28200960*x^3 - 1245316608*x^4 + 40857490944*x^5 - 997176929040*x^6 + 18922957479000*x^7 - 290321160347007*x^8 + 
   3706186924919424*x^9 - 40192947783080832*x^10 + 376007011347296808*x^11 - 3069973819587968460*x^12 + 22077817199230072320*x^13 - 140890280188035072384*x^14 + 
   802698234842482438896*x^15 - 4103537425394473089972*x^16 + 18902495071545300975024*x^17 - 78732238221467866201440*x^18 + 297394502747557537424640*x^19 - 
   1021276371591640228279608*x^20 + 3195391643626531307093136*x^21 - 9126717513052781696513472*x^22 + 23838784429830319278894960*x^23 - 
   57036149439469873029136170*x^24 + 125193945303764687170548720*x^25 - 252463808193125135683667040*x^26 + 468319525651588816677101968*x^27 - 
   799955987202674508453230080*x^28 + 1259250700419983003671188432*x^29 - 1827647223992145955112791968*x^30 + 2446096789721832686977775904*x^31 - 
   3018318328866179900013947748*x^32 + 3431647825750099504993141008*x^33 - 3591103756833435892538268576*x^34 + 3453428522304898347310346304*x^35 - 
   3044980857568898774208717864*x^36 + 2453788753600241796658089264*x^37 - 1798898333120926710588200880*x^38 + 1191523388340334602011467896*x^39 - 
   705289279715200559904946047*x^40 + 365991749065400208421066704*x^41 - 160127684590245633626763168*x^42 + 53246652206593248680390280*x^43 - 
   7752070897301693127088980*x^44 - 6148917099653542869110544*x^45 + 7180354816545311859606048*x^46 - 4756885046910502161106896*x^47 + 
   2461784589173488297225680*x^48 - 1072959766950478916828160*x^49 + 405034066683678758023296*x^50 - 134004621531329746509312*x^51 + 
   39029007660450767747136*x^52 - 10008498803226766001280*x^53 + 2253447469492691799168*x^54 - 443110379854617924480*x^55 + 75500809670622110832*x^56 - 
   11027832725742833664*x^57 + 1360928059671174912*x^58 - 139138539422454144*x^59 + 11465766788840640*x^60 - 731468794028544*x^61 + 33879848424192*x^62 - 
   1012884030720*x^63 + 14659268544*x^64)^3

indices10 = {{15,15,15,9,9,9,5,3,3,3,3,3,3,1},
{15,15,15,12,12,12,12,12,9,9,9,6,6,6,6,6,6,6,6,5,4,3}}

(*  Section eleven.  Expectations in large degree *)


