1#############################################################################
2##
3##  This  file  tests  the large finite fields.
4##
5#@local fieldpairs,fieldsizes,iF,iPI,iW,izs,zs,iFI,x
6gap> START_TEST("ffeconway.tst");
7
8#
9# Disable various warnings which depend on state of prime number and
10# Conway Polynomial databases and on whether FactInt is loaded
11#
12gap> iPI := InfoLevel(InfoPrimeInt);;
13gap> iF := InfoLevel(InfoFactor);;
14gap> if IsBound(InfoFactInt) then iFI := InfoLevel(InfoFactInt); fi;
15gap> iW := InfoLevel(InfoWarning);;
16gap> SetInfoLevel(InfoPrimeInt,0);
17gap> SetInfoLevel(InfoFactor,0);
18gap> if IsBound(InfoFactInt) then SetInfoLevel(InfoFactInt,0); fi;
19gap> SetInfoLevel(InfoWarning,0);
20
21#
22# A range of field sizes to hit various cases for the underlying vector
23# arithmetic.
24#
25gap> fieldsizes := [ [2,17], [2,32], [2,60], [2,76], [2,87], [3,11],
26> [3,20], [3,60], [17,4], [257,2], [257,11], [65521,2], [65537,2],
27> [268435399,2], [4294967291,2], [1152921504606846883,3] ];;
28gap> Add( fieldsizes, [NextPrimeInt(2^64),2] );
29
30#
31# When we come to test cross-field, we need to make sure we don't try and
32# create overlarge fields
33# leave the output here so we trip here if the Conway polynomial database changes
34#
35gap> fieldpairs := Concatenation(List(fieldsizes, pd -> List(Filtered([1..pd[2]-1],
36> 	d2 -> IsCheapConwayPolynomial(pd[1],Lcm(pd[2],d2))), d2 ->
37> 	[pd[1],pd[2],d2])));
38[ [ 2, 17, 1 ], [ 2, 17, 2 ], [ 2, 17, 3 ], [ 2, 17, 4 ], [ 2, 17, 5 ],
39  [ 2, 17, 6 ], [ 2, 17, 7 ], [ 2, 32, 1 ], [ 2, 32, 2 ], [ 2, 32, 3 ],
40  [ 2, 32, 4 ], [ 2, 32, 6 ], [ 2, 32, 8 ], [ 2, 32, 12 ], [ 2, 32, 16 ],
41  [ 2, 32, 24 ], [ 2, 60, 1 ], [ 2, 60, 2 ], [ 2, 60, 3 ], [ 2, 60, 4 ],
42  [ 2, 60, 5 ], [ 2, 60, 6 ], [ 2, 60, 8 ], [ 2, 60, 10 ], [ 2, 60, 12 ],
43  [ 2, 60, 15 ], [ 2, 60, 20 ], [ 2, 60, 24 ], [ 2, 60, 30 ], [ 2, 60, 40 ],
44  [ 2, 76, 1 ], [ 2, 76, 2 ], [ 2, 76, 4 ], [ 2, 76, 19 ], [ 2, 76, 38 ],
45  [ 2, 87, 1 ], [ 2, 87, 3 ], [ 2, 87, 29 ], [ 3, 11, 1 ], [ 3, 11, 2 ],
46  [ 3, 11, 3 ], [ 3, 11, 4 ], [ 3, 11, 5 ], [ 3, 11, 6 ], [ 3, 11, 7 ],
47  [ 3, 20, 1 ], [ 3, 20, 2 ], [ 3, 20, 3 ], [ 3, 20, 4 ], [ 3, 20, 5 ],
48  [ 3, 20, 6 ], [ 3, 20, 8 ], [ 3, 20, 10 ], [ 3, 20, 12 ], [ 3, 20, 15 ],
49  [ 3, 60, 1 ], [ 3, 60, 2 ], [ 3, 60, 3 ], [ 3, 60, 4 ], [ 3, 60, 5 ],
50  [ 3, 60, 6 ], [ 3, 60, 10 ], [ 3, 60, 12 ], [ 3, 60, 15 ], [ 3, 60, 20 ],
51  [ 3, 60, 30 ], [ 17, 4, 1 ], [ 17, 4, 2 ], [ 17, 4, 3 ], [ 257, 2, 1 ],
52  [ 257, 11, 1 ], [ 65521, 2, 1 ], [ 65537, 2, 1 ], [ 268435399, 2, 1 ],
53  [ 4294967291, 2, 1 ], [ 1152921504606846883, 3, 1 ],
54  [ 18446744073709551629, 2, 1 ] ]
55
56#
57# construct generating elements
58#
59gap> zs := List(fieldsizes, pd -> Z(pd[1],pd[2]));
60[ z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z ]
61
62#
63# and another way
64#
65gap> zs = List(fieldsizes, pd -> Z(pd[1]^pd[2]));
66true
67
68#
69# Construct some more interesting elements and test View, Print and Display
70#
71gap> izs := List(zs, Inverse);
72[ z2+z16, z2+z3+z6+z8+z14+z31,
73  z+z2+z3+z4+z7+z11+z16+z18+z21+z24+z25+z29+z31+z32+z33+z35+z38+z40+z41+z43+z4\
744+z59,
75  1+z+z4+z13+z14+z18+z19+z22+z23+z24+z26+z28+z30+z32+z33+z34+z35+z36+z37+z75,
76  1+z2+z4+z6+z7+z9+z10+z11+z12+z13+z15+z19+z25+z27+z29+z86, z+2z10,
77  1+2z2+2z3+2z4+z7+z8+z9+z10+2z12+z19,
78  2z+2z2+2z3+z4+2z5+2z7+z9+2z10+z13+2z14+2z15+z19+z20+2z21+2z23+2z25+2z26+2z27\
79+z29+z31+2z32+z33+z34+2z35+2z37+2z38+z39+2z40+z43+z59, 8+9z+11z3, 2+171z,
80  99+86z10, 61667+23125z, 21846+43691z, 89478467+89478466z,
81  2147483646+2147483645z, 1+576460752303423442z2, 2+9223372036854775814z ]
82gap> Print(izs,"\n");
83[ Z(2,17)^2+Z(2,17)^16,
84  Z(2,32)^2+Z(2,32)^3+Z(2,32)^6+Z(2,32)^8+Z(2,32)^14+Z(2,32)^31,
85  Z(2,60)+Z(2,60)^2+Z(2,60)^3+Z(2,60)^4+Z(2,60)^7+Z(2,60)^11+Z(2,60)^16+Z(2,60\
86)^18+Z(2,60)^21+Z(2,60)^24+Z(2,60)^25+Z(2,60)^29+Z(2,60)^31+Z(2,60)^32+Z(2,60)\
87^33+Z(2,60)^35+Z(2,60)^38+Z(2,60)^40+Z(2,60)^41+Z(2,60)^43+Z(2,60)^44+Z(2,60)^\
8859,
89  Z(2)^0+Z(2,76)+Z(2,76)^4+Z(2,76)^13+Z(2,76)^14+Z(2,76)^18+Z(2,76)^19+Z(2,76)\
90^22+Z(2,76)^23+Z(2,76)^24+Z(2,76)^26+Z(2,76)^28+Z(2,76)^30+Z(2,76)^32+Z(2,76)^\
9133+Z(2,76)^34+Z(2,76)^35+Z(2,76)^36+Z(2,76)^37+Z(2,76)^75,
92  Z(2)^0+Z(2,87)^2+Z(2,87)^4+Z(2,87)^6+Z(2,87)^7+Z(2,87)^9+Z(2,87)^10+Z(2,87)^\
9311+Z(2,87)^12+Z(2,87)^13+Z(2,87)^15+Z(2,87)^19+Z(2,87)^25+Z(2,87)^27+Z(2,87)^2\
949+Z(2,87)^86, Z(3,11)+2*Z(3,11)^10,
95  Z(3)^0+2*Z(3,20)^2+2*Z(3,20)^3+2*Z(3,20)^4+Z(3,20)^7+Z(3,20)^8+Z(3,20)^9+Z(3\
96,20)^10+2*Z(3,20)^12+Z(3,20)^19,
97  2*Z(3,60)+2*Z(3,60)^2+2*Z(3,60)^3+Z(3,60)^4+2*Z(3,60)^5+2*Z(3,60)^7+Z(3,60)^\
989+2*Z(3,60)^10+Z(3,60)^13+2*Z(3,60)^14+2*Z(3,60)^15+Z(3,60)^19+Z(3,60)^20+2*Z(\
993,60)^21+2*Z(3,60)^23+2*Z(3,60)^25+2*Z(3,60)^26+2*Z(3,60)^27+Z(3,60)^29+Z(3,60\
100)^31+2*Z(3,60)^32+Z(3,60)^33+Z(3,60)^34+2*Z(3,60)^35+2*Z(3,60)^37+2*Z(3,60)^38\
101+Z(3,60)^39+2*Z(3,60)^40+Z(3,60)^43+Z(3,60)^59,
102  Z(17)^10+9*Z(17,4)+11*Z(17,4)^3, Z(257)^48+171*Z(257,2),
103  Z(257)^198+86*Z(257,11)^10, Z(65521)^1451+23125*Z(65521,2),
104  ZmodpZObj(21846,65537)+43691*Z(65537,2),
105  ZmodpZObj(89478467,268435399)+89478466*Z(268435399,2),
106  ZmodpZObj(2147483646,4294967291)+2147483645*Z(4294967291,2),
107  ZmodpZObj(1,1152921504606846883)+576460752303423442*Z(1152921504606846883,3)\
108^2,
109  ZmodpZObj(2,18446744073709551629)+9223372036854775814*Z(18446744073709551629\
110,2) ]
111gap> for x in izs do
112>    Display(x);
113> od;
114z2+z16
115z2+z3+z6+z8+z14+z31
116z+z2+z3+z4+z7+z11+z16+z18+z21+z24+z25+z29+z31+z32+z33+z35+z38+z40+z41+z43+z44+\
117z59
1181+z+z4+z13+z14+z18+z19+z22+z23+z24+z26+z28+z30+z32+z33+z34+z35+z36+z37+z75
1191+z2+z4+z6+z7+z9+z10+z11+z12+z13+z15+z19+z25+z27+z29+z86
120z+2z10
1211+2z2+2z3+2z4+z7+z8+z9+z10+2z12+z19
1222z+2z2+2z3+z4+2z5+2z7+z9+2z10+z13+2z14+2z15+z19+z20+2z21+2z23+2z25+2z26+2z27+z\
12329+z31+2z32+z33+z34+2z35+2z37+2z38+z39+2z40+z43+z59
1248+9z+11z3
1252+171z
12699+86z10
12761667+23125z
12821846+43691z
12989478467+89478466z
1302147483646+2147483645z
1311+576460752303423442z2
1322+9223372036854775814z
133
134#
135# Test arithmetic within the field
136#
137gap> zs + izs;
138[ z+z2+z16, z+z2+z3+z6+z8+z14+z31,
139  z2+z3+z4+z7+z11+z16+z18+z21+z24+z25+z29+z31+z32+z33+z35+z38+z40+z41+z43+z44+\
140z59, 1+z4+z13+z14+z18+z19+z22+z23+z24+z26+z28+z30+z32+z33+z34+z35+z36+z37+z75,
141  1+z+z2+z4+z6+z7+z9+z10+z11+z12+z13+z15+z19+z25+z27+z29+z86, 2z+2z10,
142  1+z+2z2+2z3+2z4+z7+z8+z9+z10+2z12+z19,
143  2z2+2z3+z4+2z5+2z7+z9+2z10+z13+2z14+2z15+z19+z20+2z21+2z23+2z25+2z26+2z27+z2\
1449+z31+2z32+z33+z34+2z35+2z37+2z38+z39+2z40+z43+z59, 8+10z+11z3, 2+172z,
145  99+z+86z10, 61667+23126z, 21846+43692z, 89478467+89478467z,
146  2147483646+2147483646z, 1+z+576460752303423442z2, 2+9223372036854775815z ]
147gap> zs - izs;
148[ z+z2+z16, z+z2+z3+z6+z8+z14+z31,
149  z2+z3+z4+z7+z11+z16+z18+z21+z24+z25+z29+z31+z32+z33+z35+z38+z40+z41+z43+z44+\
150z59, 1+z4+z13+z14+z18+z19+z22+z23+z24+z26+z28+z30+z32+z33+z34+z35+z36+z37+z75,
151  1+z+z2+z4+z6+z7+z9+z10+z11+z12+z13+z15+z19+z25+z27+z29+z86, z10,
152  2+z+z2+z3+z4+2z7+2z8+2z9+2z10+z12+2z19,
153  2z+z2+z3+2z4+z5+z7+2z9+z10+2z13+z14+z15+2z19+2z20+z21+z23+z25+z26+z27+2z29+2\
154z31+z32+2z33+2z34+z35+z37+z38+2z39+z40+2z43+2z59, 9+9z+6z3, 255+87z,
155  158+z+171z10, 3854+42397z, 43691+21847z, 178956932+178956934z,
156  2147483645+2147483647z, 1152921504606846882+z+576460752303423441z2,
157  18446744073709551627+9223372036854775816z ]
158gap> List(izs, x->x^2);
159[ z+z15, z+z2+z5+z7+z13+z30,
160  1+z+z2+z3+z6+z10+z15+z17+z20+z23+z24+z28+z30+z31+z32+z34+z37+z39+z40+z42+z43\
161+z58, z+z3+z4+z12+z14+z17+z19+z21+z24+z25+z26+z27+z28+z29+z30+z31+z37+z74+z75,
162  1+z+z2+z3+z4+z5+z7+z8+z13+z14+z15+z18+z19+z24+z25+z26+z27+z28+z29+z85+z86,
163  1+2z9, 1+2z+z2+z3+2z4+z6+2z7+2z8+2z9+z10+2z11+2z12+z18+z19,
164  2+2z+2z2+z3+2z4+2z6+z8+2z9+z12+2z13+2z14+z18+z19+2z20+2z22+2z24+2z25+2z26+z2\
1658+z30+2z31+z32+z33+2z34+2z36+2z37+z38+2z39+z42+z58, 5+4z+11z2+3z3, 175+85z,
166  35+86z9+33z10, 3174+50331z, 50973+58255z, 59652311+149130777z,
167  3221225468+3221225468z, 1+576460752303423442z+576460752303423442z2,
168  9223372036854775818+18446744073709551628z ]
169gap> List([1..Length(zs)] , i-> zs[i]/izs[i]);
170[ z2, z2, z2, z2, z2, z2, z2, z2, z2, 254+6z, z2, 65504+3z, 65534+z,
171  268435396+2z, 4294967289+z, z2, 18446744073709551627+4z ]
172gap> List(izs, AdditiveInverse);
173[ z2+z16, z2+z3+z6+z8+z14+z31,
174  z+z2+z3+z4+z7+z11+z16+z18+z21+z24+z25+z29+z31+z32+z33+z35+z38+z40+z41+z43+z4\
1754+z59,
176  1+z+z4+z13+z14+z18+z19+z22+z23+z24+z26+z28+z30+z32+z33+z34+z35+z36+z37+z75,
177  1+z2+z4+z6+z7+z9+z10+z11+z12+z13+z15+z19+z25+z27+z29+z86, 2z+z10,
178  2+z2+z3+z4+2z7+2z8+2z9+2z10+z12+2z19,
179  z+z2+z3+2z4+z5+z7+2z9+z10+2z13+z14+z15+2z19+2z20+z21+z23+z25+z26+z27+2z29+2z\
18031+z32+2z33+2z34+z35+z37+z38+2z39+z40+2z43+2z59, 9+8z+6z3, 255+86z,
181  158+171z10, 3854+42396z, 43691+21846z, 178956932+178956933z,
182  2147483645+2147483646z, 1152921504606846882+576460752303423441z2,
183  18446744073709551627+9223372036854775815z ]
184
185#
186# and across fields
187#
188gap> List(fieldpairs, pdd -> Z(pdd[1],pdd[2])+Z(pdd[1],pdd[3]));
189[ 1+z, z+z4+z9+z10+z15+z17+z19+z21+z23+z25+z26+z27+z29+z30+z33,
190  z2+z3+z6+z10+z13+z15+z16+z18+z19+z20+z24+z25+z35+z38+z39+z40+z41+z44+z46+z48\
191+z49,
192  1+z2+z3+z4+z5+z7+z10+z11+z12+z13+z14+z16+z17+z18+z19+z21+z22+z23+z24+z25+z26\
193+z30+z31+z32+z35+z37+z38+z40+z47+z48+z50+z51+z56+z58+z62+z63+z67,
194  z+z3+z7+z8+z9+z10+z13+z14+z17+z19+z20+z21+z22+z24+z25+z27+z28+z30+z32+z33+z3\
1956+z38+z41+z42+z45+z46+z47+z48+z50+z51+z52+z54+z58+z59+z60+z61+z63+z65+z68+z69+\
196z71+z73+z75+z76+z77+z82,
197  z2+z3+z5+z7+z10+z11+z13+z16+z18+z19+z20+z21+z22+z25+z26+z27+z28+z30+z31+z32+\
198z34+z35+z37+z38+z39+z40+z44+z45+z47+z54+z55+z56+z58+z60+z61+z63+z65+z67+z69+z7\
1993+z74+z79+z83+z87+z89+z96+z97+z99+z100, <<an element of GF(2, 119)>>, 1+z,
200  1+z3+z4+z5+z6+z8+z9+z14+z17+z18+z19+z20+z21+z24+z29+z30,
201  z+z5+z7+z9+z12+z14+z15+z16+z17+z20+z22+z23+z24+z25+z28+z31+z33+z36+z37+z38+z\
20239+z42+z43+z44+z52+z56+z61+z62+z65+z67+z68+z69+z70+z71+z72+z73+z76+z77+z79+z80\
203+z81+z82+z83+z85+z86+z89+z94,
204  z4+z5+z6+z8+z10+z12+z14+z15+z17+z18+z19+z20+z21+z22+z23+z24+z25+z29+z30,
205  1+z6+z11+z12+z13+z15+z18+z22+z23+z30+z31+z36+z37+z38+z39+z41+z44+z45+z47+z48\
206+z50+z51+z52+z55+z56+z57+z58+z59+z60+z63+z64+z66+z68+z69+z74+z76+z81+z83+z84+z\
20785+z86+z88+z89+z90+z91+z92+z93,
208  z+z2+z3+z6+z7+z10+z12+z14+z17+z19+z20+z21+z22+z29+z31,
209  z2+z9+z10+z11+z15+z16+z17+z18+z20+z23+z25+z27+z29+z33+z35+z36+z39+z40+z42+z4\
2105+z47+z48+z52+z54+z55+z56+z57+z59+z62+z67+z68+z71+z74+z75+z76+z80+z81+z83+z85+\
211z86+z89+z90+z92+z93,
212  z+z2+z4+z5+z6+z7+z10+z11+z14+z15+z16+z20+z23+z26+z28+z29+z30+z31,
213  1+z+z4+z6+z7+z9+z12+z14+z17+z19+z21+z22+z28+z30+z33+z34+z36+z38+z40+z41+z45+\
214z48+z49+z51+z52+z53+z59+z61+z63+z67+z69+z70+z75+z78+z81+z83+z85+z86+z87+z90+z9\
2153+z94, 1+z,
216  z2+z4+z5+z7+z9+z11+z14+z17+z19+z20+z21+z23+z24+z25+z27+z30+z35+z36+z38+z39+z\
21741+z43+z50+z51+z53+z54+z55+z57+z59,
218  1+z+z5+z7+z8+z11+z12+z15+z17+z22+z23+z24+z27+z28+z29+z31+z34+z35+z37+z38+z39\
219+z42+z47+z48+z51+z52+z53,
220  1+z2+z3+z4+z7+z8+z9+z10+z11+z16+z18+z19+z20+z21+z23+z26+z27+z31+z32+z34+z36+\
221z37+z39+z40+z41+z42+z45+z46+z47+z49+z50+z51+z53+z54+z56+z57+z58,
222  z+z2+z3+z6+z7+z9+z11+z13+z17+z19+z21+z22+z23+z25+z27+z29+z30+z31+z36+z38+z39\
223+z41+z45+z46+z48+z51+z53+z54+z55+z56+z59,
224  z2+z4+z6+z7+z9+z11+z12+z14+z16+z17+z19+z20+z21+z22+z25+z26+z31+z34+z35+z36+z\
22537+z38+z44+z46+z48+z49+z51+z52+z57+z58+z59, <<an element of GF(2, 120)>>,
226  z+z3+z4+z6+z8+z10+z11+z15+z18+z19+z22+z24+z26+z28+z30+z31+z32+z33+z34+z36+z3\
2277+z39+z40+z41+z42+z47+z48+z50+z53+z54+z56+z57+z59,
228  z4+z6+z7+z13+z14+z16+z17+z18+z19+z20+z21+z22+z24+z26+z28+z31+z33+z34+z35+z37\
229+z39+z43+z47+z48+z51+z52+z54+z58,
230  1+z4+z6+z8+z9+z14+z16+z24+z28+z29+z32+z34+z38+z39+z40+z42+z47+z48+z50+z51+z5\
2312+z54+z56+z57+z59,
232  1+z+z2+z3+z4+z7+z11+z14+z15+z16+z17+z18+z22+z25+z26+z29+z34+z35+z36+z37+z38+\
233z40+z44+z45+z46+z47+z48+z50+z52+z55+z56+z57+z58, <<an element of GF(2, 120)>>,
234  z2+z4+z7+z10+z13+z15+z18+z20+z23+z25+z26+z38+z39+z41+z43+z44+z46+z48+z50+z51\
235+z55+z56+z57+z59, <<an element of GF(2, 120)>>, 1+z,
236  z3+z6+z7+z8+z10+z14+z15+z18+z20+z22+z23+z25+z26+z27+z30+z31+z32+z33+z35+z36+\
237z38+z42+z45+z46+z47+z50+z51+z54+z55+z57+z58+z60+z62+z63+z64+z65+z66+z67+z69+z7\
2381+z73+z75,
239  1+z2+z3+z4+z7+z8+z9+z17+z18+z19+z21+z26+z29+z30+z31+z32+z34+z35+z36+z39+z43+\
240z44+z45+z46+z48+z49+z52+z53+z56+z57+z60+z63+z64+z65+z69+z70+z71+z73+z74+z75,
241  1+z2+z5+z6+z7+z8+z10+z12+z13+z15+z16+z18+z19+z20+z26+z28+z33+z35+z37+z40+z41\
242+z43+z44+z45+z46+z48+z49+z52+z53+z54+z55+z57+z60+z62+z65+z67+z72+z73+z74,
243  z2+z3+z5+z6+z8+z9+z11+z19+z23+z27+z31+z32+z34+z35+z36+z38+z42+z46+z48+z49+z5\
2440+z53+z58+z59+z60+z62+z63+z65+z66+z67+z68+z69+z70+z74, 1+z,
245  1+z2+z3+z5+z9+z15+z22+z25+z26+z27+z28+z29+z30+z31+z32+z34+z36+z40+z42+z43+z4\
2464+z48+z51+z52+z55+z56+z57+z60+z61+z62+z63+z64+z65+z67+z68+z69+z70+z71+z72+z75+\
247z77+z79+z81+z83+z84+z86,
248  z+z2+z6+z8+z9+z10+z14+z15+z16+z19+z20+z22+z23+z26+z28+z30+z31+z33+z34+z37+z4\
2491+z46+z50+z51+z54+z55+z56+z57+z58+z59+z60+z61+z63+z64+z66+z68+z71+z72+z76+z78+\
250z81+z82+z83+z84+z85, 2+z,
251  2+z+z2+2z3+z4+2z5+2z6+z7+2z9+2z11+z12+2z14+z16+z17+2z18+2z19+2z20+2z21,
252  1+2z+2z3+2z4+2z6+2z8+2z9+z11+z12+z14+2z15+2z17+z18+z19+2z21+2z23+z25+2z26+2z\
25327+2z28+2z29+2z30+z32,
254  2z+2z3+z8+z9+z12+z15+2z17+2z19+z21+z22+z23+z24+z25+z30+z34+z35+z37+2z38+2z40\
255+z41+2z42+z43,
256  2+z+z4+2z6+z7+2z9+z10+2z14+z16+2z19+z20+2z21+2z22+z23+2z25+z26+2z27+z28+2z29\
257+z30+z31+2z34+2z35+z36+z38+z40+z41+z42+z44+2z45+2z47+z51+z52,
258  1+z+z4+z6+2z8+z9+z11+2z13+z14+z17+z19+z20+2z21+z23+z25+2z26+z27+z28+z29+z31+\
259z32+z33+z34+z35+2z38+z40+z41+2z42+2z45+z46+2z47+z50+2z51+z52+2z57+z58+2z59+z60\
260+z61+2z62+2z63+2z64+z65,
261  1+z2+2z4+z5+2z6+2z7+z8+2z9+2z10+z11+2z12+2z13+z14+z16+2z17+2z18+z21+z22+z24+\
2622z27+2z28+2z29+2z32+z33+z35+z36+z40+2z41+2z42+z45+2z46+2z48+2z49+z51+z52+z53+2\
263z54+z55+z56+z57+2z58+z59+z60+z61+z63+z64+2z65+2z66+2z67+z68+2z73+2z76, 2+z,
264  z+z2+2z3+2z4+z7+2z8+z9+z10+2z12+2z13+2z14+z15+2z17+2z18+z19,
265  2+z2+z3+2z5+z7+2z11+z12+z14+z15+z16+z18+z19+2z20+2z22+z25+z27+z29+2z30+z32+z\
26633+2z34+z35+2z36+2z39+z40+z41+2z43+2z45+z46+2z48+2z49+2z50+z53+2z54+2z55+z56+2\
267z57, 1+2z+z2+2z3+2z4+2z5+z6+z7+2z11+2z12+z13+z14+z15+2z16+z17+z18+z19,
268  2z+2z2+z4+z5+z6+2z8+z11+2z12+z16+z19,
269  2z+z2+z3+2z4+z6+2z7+2z8+2z9+2z10+2z12+2z13+z15+z16+z17+2z18+z19+z20+2z21+z24\
270+z25+2z26+2z27+z28+2z29+z30+2z31+z32+z33+z35+z36+2z37+z39+2z40+2z42+z45+z46+z4\
2717+2z48+2z50+z51+2z52+z53+z54+z55+2z57+2z58+2z59,
272  2+z+2z3+2z4+2z5+z6+2z7+2z8+z9+2z11+2z12+z13+2z15+z16+z19+z23+2z24+2z25+2z26+\
273z27+2z28+z29+2z30+2z33+2z35+z39,
274  z+z3+2z4+2z7+z8+2z10+z11+2z12+z13+z14+z15+z17+z18+2z19,
275  2+z+2z2+2z5+2z6+z8+2z9+z10+z12+2z15+z18+z19+z20+z23+2z25+2z26+z27+2z28+2z29+\
2762z30+z31+2z32+2z33+2z34+2z38+2z39+2z40+z42+z44+z45+2z46+z48+z49+2z50+2z52+z53+\
277z54+2z55+2z56+z57+z58,
278  2+z+2z2+2z3+z4+z7+2z8+2z9+2z10+z11+z13+z14+z15+2z19+2z21+2z23+2z24+z25+2z29+\
2792z31+z32+z34+z35+z36+z37+z38+z39+2z40+2z42+2z44+z45+2z46+z49+z50+z51+2z53+z54+\
2802z55+2z57+2z58+2z59, 2+z,
281  2+2z+2z2+2z4+2z5+z6+2z7+z8+2z9+2z11+z13+2z14+2z15+z16+2z17+2z18+z19+2z20+z21\
282+z22+2z23+2z25+2z27+z28+2z29+2z31+z35+2z36+z37+2z38+2z39+z40+2z42+z43+2z45+2z4\
2836+z47+z49+z51+2z52+2z53+z55+2z58+z59,
284  1+z+2z2+2z3+z4+2z5+2z6+z7+z9+z10+z11+z12+2z16+z17+z18+z19+z21+z25+z26+2z27+z\
28528+2z29+2z30+z31+z34+z35+z36+2z37+z38+2z39+z40+2z42+2z43+2z46+z47+2z48+2z49+z5\
2862+2z53+2z55+z57+2z58+2z59,
287  1+z+z3+2z5+2z7+2z8+2z9+z10+z12+z13+2z14+2z19+z20+z22+2z23+z26+2z27+z28+2z29+\
288z30+z32+2z33+z34+2z35+z37+2z38+2z40+z41+2z42+z43+2z44+2z45+z46+2z47+z48+z49+2z\
28951+z53+2z56+z57+2z58+z59,
290  2+2z+2z3+z4+2z5+z7+z8+z9+z10+2z11+2z12+2z13+z14+z18+2z20+z24+z25+z28+2z29+2z\
29130+2z31+z32+z33+z34+2z35+z36+2z37+z38+z40+z44+z46+z47+z49+z50+2z52+2z53+2z57+2\
292z58+2z59,
293  2+2z2+2z3+2z7+2z8+2z11+2z12+2z13+2z14+2z16+2z17+2z18+z19+2z20+z22+z24+z25+2z\
29428+z30+2z34+z35+z37+z38+z39+2z40+2z41+z42+2z45+2z46+2z47+2z48+z51+2z53+2z54+z5\
2955+2z56+z57+z58+z59,
296  z+2z3+2z4+z5+z6+2z8+z10+z11+2z12+2z15+2z16+z19+z21+2z22+z25+2z29+2z30+2z31+2\
297z32+z33+2z34+z38+2z39+z41+z43+2z48+2z49+z50+z53+2z54+2z55+2z56+2z57+z58+z59,
298  1+2z+z3+z4+2z5+z6+z8+2z10+2z11+z12+2z14+z15+z16+z17+z18+z19+2z20+z21+z22+z23\
299+2z25+2z27+2z30+2z31+z32+z33+z34+2z36+2z37+2z39+2z40+2z41+z44+2z45+z47+z48+z49\
300+2z53+2z54+2z55+z56+2z59,
301  1+2z+2z4+2z6+z7+2z8+z13+z16+z17+2z19+z20+z22+2z23+2z24+z25+z26+z27+z28+2z33+\
302z35+2z38+z39+2z40+2z41+z42+2z44+2z45+z47+z49+2z50+z51+z52+2z54+2z55+2z56+z57+z\
30358+z59,
304  1+z+2z2+2z3+2z4+z6+2z9+2z10+z11+z14+z15+2z16+2z17+2z20+2z21+2z22+2z26+2z27+2\
305z28+2z29+2z31+z32+z33+z34+z36+z37+2z38+z41+z42+2z45+2z46+2z47+2z50+2z52+2z53+2\
306z54+z56+z57+z58+z59,
307  1+z+2z2+z3+z4+2z6+2z7+2z9+z10+z11+2z12+z14+2z15+z16+z19+2z21+2z24+z26+2z28+2\
308z29+2z30+z32+z33+z34+z35+2z37+2z40+z41+2z42+2z46+2z48+z49+2z50+2z53+z54+z55+z5\
3096+z58+2z59, 3+z, 12+z2+9z3, 10+9z+13z3+11z4+5z5+10z6+15z7+5z8+13z9+6z10+9z11,
310  3+z, 3+z, 17+z, 3+z, 3+z, 2+z, 2+z, 2+z ]
311gap> List(fieldpairs, pdd -> Z(pdd[1],pdd[2])-Z(pdd[1],pdd[3]));
312[ 1+z, z+z4+z9+z10+z15+z17+z19+z21+z23+z25+z26+z27+z29+z30+z33,
313  z2+z3+z6+z10+z13+z15+z16+z18+z19+z20+z24+z25+z35+z38+z39+z40+z41+z44+z46+z48\
314+z49,
315  1+z2+z3+z4+z5+z7+z10+z11+z12+z13+z14+z16+z17+z18+z19+z21+z22+z23+z24+z25+z26\
316+z30+z31+z32+z35+z37+z38+z40+z47+z48+z50+z51+z56+z58+z62+z63+z67,
317  z+z3+z7+z8+z9+z10+z13+z14+z17+z19+z20+z21+z22+z24+z25+z27+z28+z30+z32+z33+z3\
3186+z38+z41+z42+z45+z46+z47+z48+z50+z51+z52+z54+z58+z59+z60+z61+z63+z65+z68+z69+\
319z71+z73+z75+z76+z77+z82,
320  z2+z3+z5+z7+z10+z11+z13+z16+z18+z19+z20+z21+z22+z25+z26+z27+z28+z30+z31+z32+\
321z34+z35+z37+z38+z39+z40+z44+z45+z47+z54+z55+z56+z58+z60+z61+z63+z65+z67+z69+z7\
3223+z74+z79+z83+z87+z89+z96+z97+z99+z100, <<an element of GF(2, 119)>>, 1+z,
323  1+z3+z4+z5+z6+z8+z9+z14+z17+z18+z19+z20+z21+z24+z29+z30,
324  z+z5+z7+z9+z12+z14+z15+z16+z17+z20+z22+z23+z24+z25+z28+z31+z33+z36+z37+z38+z\
32539+z42+z43+z44+z52+z56+z61+z62+z65+z67+z68+z69+z70+z71+z72+z73+z76+z77+z79+z80\
326+z81+z82+z83+z85+z86+z89+z94,
327  z4+z5+z6+z8+z10+z12+z14+z15+z17+z18+z19+z20+z21+z22+z23+z24+z25+z29+z30,
328  1+z6+z11+z12+z13+z15+z18+z22+z23+z30+z31+z36+z37+z38+z39+z41+z44+z45+z47+z48\
329+z50+z51+z52+z55+z56+z57+z58+z59+z60+z63+z64+z66+z68+z69+z74+z76+z81+z83+z84+z\
33085+z86+z88+z89+z90+z91+z92+z93,
331  z+z2+z3+z6+z7+z10+z12+z14+z17+z19+z20+z21+z22+z29+z31,
332  z2+z9+z10+z11+z15+z16+z17+z18+z20+z23+z25+z27+z29+z33+z35+z36+z39+z40+z42+z4\
3335+z47+z48+z52+z54+z55+z56+z57+z59+z62+z67+z68+z71+z74+z75+z76+z80+z81+z83+z85+\
334z86+z89+z90+z92+z93,
335  z+z2+z4+z5+z6+z7+z10+z11+z14+z15+z16+z20+z23+z26+z28+z29+z30+z31,
336  1+z+z4+z6+z7+z9+z12+z14+z17+z19+z21+z22+z28+z30+z33+z34+z36+z38+z40+z41+z45+\
337z48+z49+z51+z52+z53+z59+z61+z63+z67+z69+z70+z75+z78+z81+z83+z85+z86+z87+z90+z9\
3383+z94, 1+z,
339  z2+z4+z5+z7+z9+z11+z14+z17+z19+z20+z21+z23+z24+z25+z27+z30+z35+z36+z38+z39+z\
34041+z43+z50+z51+z53+z54+z55+z57+z59,
341  1+z+z5+z7+z8+z11+z12+z15+z17+z22+z23+z24+z27+z28+z29+z31+z34+z35+z37+z38+z39\
342+z42+z47+z48+z51+z52+z53,
343  1+z2+z3+z4+z7+z8+z9+z10+z11+z16+z18+z19+z20+z21+z23+z26+z27+z31+z32+z34+z36+\
344z37+z39+z40+z41+z42+z45+z46+z47+z49+z50+z51+z53+z54+z56+z57+z58,
345  z+z2+z3+z6+z7+z9+z11+z13+z17+z19+z21+z22+z23+z25+z27+z29+z30+z31+z36+z38+z39\
346+z41+z45+z46+z48+z51+z53+z54+z55+z56+z59,
347  z2+z4+z6+z7+z9+z11+z12+z14+z16+z17+z19+z20+z21+z22+z25+z26+z31+z34+z35+z36+z\
34837+z38+z44+z46+z48+z49+z51+z52+z57+z58+z59, <<an element of GF(2, 120)>>,
349  z+z3+z4+z6+z8+z10+z11+z15+z18+z19+z22+z24+z26+z28+z30+z31+z32+z33+z34+z36+z3\
3507+z39+z40+z41+z42+z47+z48+z50+z53+z54+z56+z57+z59,
351  z4+z6+z7+z13+z14+z16+z17+z18+z19+z20+z21+z22+z24+z26+z28+z31+z33+z34+z35+z37\
352+z39+z43+z47+z48+z51+z52+z54+z58,
353  1+z4+z6+z8+z9+z14+z16+z24+z28+z29+z32+z34+z38+z39+z40+z42+z47+z48+z50+z51+z5\
3542+z54+z56+z57+z59,
355  1+z+z2+z3+z4+z7+z11+z14+z15+z16+z17+z18+z22+z25+z26+z29+z34+z35+z36+z37+z38+\
356z40+z44+z45+z46+z47+z48+z50+z52+z55+z56+z57+z58, <<an element of GF(2, 120)>>,
357  z2+z4+z7+z10+z13+z15+z18+z20+z23+z25+z26+z38+z39+z41+z43+z44+z46+z48+z50+z51\
358+z55+z56+z57+z59, <<an element of GF(2, 120)>>, 1+z,
359  z3+z6+z7+z8+z10+z14+z15+z18+z20+z22+z23+z25+z26+z27+z30+z31+z32+z33+z35+z36+\
360z38+z42+z45+z46+z47+z50+z51+z54+z55+z57+z58+z60+z62+z63+z64+z65+z66+z67+z69+z7\
3611+z73+z75,
362  1+z2+z3+z4+z7+z8+z9+z17+z18+z19+z21+z26+z29+z30+z31+z32+z34+z35+z36+z39+z43+\
363z44+z45+z46+z48+z49+z52+z53+z56+z57+z60+z63+z64+z65+z69+z70+z71+z73+z74+z75,
364  1+z2+z5+z6+z7+z8+z10+z12+z13+z15+z16+z18+z19+z20+z26+z28+z33+z35+z37+z40+z41\
365+z43+z44+z45+z46+z48+z49+z52+z53+z54+z55+z57+z60+z62+z65+z67+z72+z73+z74,
366  z2+z3+z5+z6+z8+z9+z11+z19+z23+z27+z31+z32+z34+z35+z36+z38+z42+z46+z48+z49+z5\
3670+z53+z58+z59+z60+z62+z63+z65+z66+z67+z68+z69+z70+z74, 1+z,
368  1+z2+z3+z5+z9+z15+z22+z25+z26+z27+z28+z29+z30+z31+z32+z34+z36+z40+z42+z43+z4\
3694+z48+z51+z52+z55+z56+z57+z60+z61+z62+z63+z64+z65+z67+z68+z69+z70+z71+z72+z75+\
370z77+z79+z81+z83+z84+z86,
371  z+z2+z6+z8+z9+z10+z14+z15+z16+z19+z20+z22+z23+z26+z28+z30+z31+z33+z34+z37+z4\
3721+z46+z50+z51+z54+z55+z56+z57+z58+z59+z60+z61+z63+z64+z66+z68+z71+z72+z76+z78+\
373z81+z82+z83+z84+z85, 1+z,
374  2+2z+2z2+2z3+z4+2z5+z8+z9+2z10+z11+z12+z15+z16+2z18+2z19+2z20+z21,
375  1+2z+2z2+2z3+z4+z6+z8+2z11+z12+2z13+2z14+2z15+2z16+2z17+z18+z20+2z21+z22+z23\
376+z24+2z25+z26+2z28+z30+2z31,
377  2z+z5+2z7+2z8+z9+z10+2z11+z12+z13+z14+z17+z18+2z19+2z20+2z23+2z26+2z28+2z31+\
3782z33+2z35+z36+2z38+z39+z40+2z41+2z42+2z43,
379  1+z+z3+2z4+2z5+z6+2z9+z11+z12+z15+2z18+2z19+z20+2z21+z22+2z24+z25+2z27+z28+z\
38029+2z34+z36+2z37+2z38+z40+2z41+2z43+2z45+z47+2z51+z52+2z54,
381  1+z+z3+z4+2z5+2z6+z9+z10+z11+z13+2z14+2z15+z16+2z17+2z18+z19+2z22+2z23+2z24+\
382z26+z28+z30+z31+2z33+z34+z35+z37+2z38+z40+2z41+z42+z44+2z45+z46+2z47+z49+2z50+\
383z51+2z52+z54+2z55+z56+z57+z58+z59+2z61+z62+z64,
384  2+z+z5+z6+2z9+2z11+2z13+z14+z16+2z20+z23+2z24+z27+z28+2z29+2z30+z31+2z32+2z3\
3854+2z35+z36+2z37+z38+z39+2z40+2z41+z42+z45+2z46+z47+2z49+2z51+2z52+2z55+z56+2z5\
3867+z58+z61+2z63+z64+2z65+2z67+2z69+2z70+z71+z72+z73+z75+2z76, 1+z,
387  z+2z2+z3+z4+2z7+z8+2z9+2z10+z12+z13+z14+2z15+z17+z18+2z19,
388  z4+z5+2z6+2z7+z9+z10+2z12+z14+z15+z17+2z18+2z19+2z20+z21+2z22+2z25+z26+z28+z\
38930+z31+z32+z33+2z35+2z37+z38+z39+2z40+z41+2z42+z43+2z45+z47+z48+z49+2z50+z52+2\
390z54+z55+z56+2z58+2z59,
391  2+2z2+z3+z4+z5+2z6+2z7+z11+z12+2z13+2z14+2z15+z16+2z17+2z18+2z19,
392  z2+2z4+2z5+2z6+z8+2z11+z12+2z16+2z19,
393  2+z+2z4+z6+z7+z8+2z9+2z10+2z11+z12+z13+2z14+z15+z18+2z19+2z21+z22+2z24+2z25+\
3942z26+2z27+2z29+2z30+2z31+z32+z33+2z34+2z35+z36+z38+2z39+z40+2z41+z48+2z50+2z51\
395+2z52+2z55+2z56,
396  2+z+2z7+2z8+2z9+2z10+2z11+2z12+z13+2z14+z15+2z16+2z17+z18+2z20+2z21+2z22+z24\
397+2z25+2z26+2z28+z29+2z30+2z32+z33+z34+z36+2z38+z39,
398  z+2z3+z4+z7+2z8+z10+2z11+z12+2z13+2z14+2z15+2z17+2z18+z19,
399  2z+2z2+z3+z4+z5+2z8+2z9+2z11+2z12+2z14+z16+z17+2z18+2z19+z21+z22+2z23+z25+2z\
40026+2z28+2z29+z30+2z36+2z37+2z38+z39+z40+2z41+z42+2z44+2z46+z47+2z48+2z49+2z50+\
4012z52+z55+z57+z58+2z59,
402  2z+2z2+2z3+2z6+2z7+z8+2z9+2z10+z11+2z13+z14+z15+z16+z17+z19+z20+2z21+z22+z23\
403+z24+2z25+z26+z27+z28+2z29+2z31+z32+2z33+z34+2z35+z36+z37+2z39+z40+2z41+z44+2z\
40446+z47+2z49+2z51+z52+2z53+z55+2z56, 1+z,
405  1+z2+z4+z5+2z6+z7+2z8+z9+z11+2z13+z14+z15+2z16+z17+z18+2z19+z20+2z21+2z22+z2\
4063+z25+z27+2z28+z29+z31+2z35+z36+2z37+z38+z39+2z40+z42+2z43+z45+z46+2z47+2z49+2\
407z51+z52+z53+2z55+z58+2z59,
408  2+z+z2+z3+2z4+z5+z6+2z7+2z9+2z10+2z11+2z12+z16+2z17+2z18+2z19+2z21+2z25+2z26\
409+z27+2z28+z29+z30+2z31+2z34+2z35+2z36+z37+2z38+z39+2z40+z42+z43+z46+2z47+z48+z\
41049+2z52+z53+z55+2z57+z58+z59,
411  2+z+2z3+z5+z7+z8+z9+2z10+2z12+2z13+z14+z19+2z20+2z22+z23+2z26+z27+2z28+z29+2\
412z30+2z32+z33+2z34+z35+2z37+z38+z40+2z41+z42+2z43+z44+z45+2z46+z47+2z48+2z49+z5\
4131+2z53+z56+2z57+z58+2z59,
414  1+z3+2z4+z5+2z7+2z8+2z9+2z10+z11+z12+z13+2z14+2z18+z20+2z24+2z25+2z28+z29+z3\
4150+z31+2z32+2z33+2z34+z35+2z36+z37+2z38+2z40+2z44+2z46+2z47+2z49+2z50+z52+z53+z\
41657+z58+z59,
417  1+2z+z2+z3+z7+z8+z11+z12+z13+z14+z16+z17+z18+2z19+z20+2z22+2z24+2z25+z28+2z3\
4180+z34+2z35+2z37+2z38+2z39+z40+z41+2z42+z45+z46+z47+z48+2z51+z53+z54+2z55+z56+2\
419z57+2z58+2z59,
420  z+z3+z4+2z5+2z6+z8+2z10+2z11+z12+z15+z16+2z19+2z21+z22+2z25+z29+z30+z31+z32+\
4212z33+z34+2z38+z39+2z41+2z43+z48+z49+2z50+2z53+z54+z55+z56+z57+2z58+2z59,
422  2+2z3+2z4+z5+2z6+2z8+z10+z11+2z12+z14+2z15+2z16+2z17+2z18+2z19+z20+2z21+2z22\
423+2z23+z25+z27+z30+z31+2z32+2z33+2z34+z36+z37+z39+z40+z41+2z44+z45+2z47+2z48+2z\
42449+z53+z54+z55+2z56+z59,
425  2+z4+z6+2z7+z8+2z13+2z16+2z17+z19+2z20+2z22+z23+z24+2z25+2z26+2z27+2z28+z33+\
4262z35+z38+2z39+z40+z41+2z42+z44+z45+2z47+2z49+z50+2z51+2z52+z54+z55+z56+2z57+2z\
42758+2z59,
428  2+z+z2+z3+z4+2z6+z9+z10+2z11+2z14+2z15+z16+z17+z20+z21+z22+z26+z27+z28+z29+z\
42931+2z32+2z33+2z34+2z36+2z37+z38+2z41+2z42+z45+z46+z47+z50+z52+z53+z54+2z56+2z5\
4307+2z58+2z59,
431  2+z+z2+2z3+2z4+z6+z7+z9+2z10+2z11+z12+2z14+z15+2z16+2z19+z21+z24+2z26+z28+z2\
4329+z30+2z32+2z33+2z34+2z35+z37+z40+2z41+z42+z46+z48+2z49+z50+z53+2z54+2z55+2z56\
433+2z58+z59, 14+z, 5+2z+16z2+8z3, 4z+13z2+8z3+4z4+8z6+16z7+9z8+14z9+16z10+11z11,
434  254+z, 254+z, 65504+z, 65534+z, 268435396+z, 4294967289+z,
435  1152921504606846881+z, 18446744073709551627+z ]
436gap> List(fieldpairs, pdd -> Z(pdd[1],pdd[2])*Z(pdd[1],pdd[3]));
437[ z, 1+z2+z4+z9+z10+z13+z14+z15+z16+z18+z20+z21+z24+z25+z29+z30+z31+z33,
438  1+z+z3+z9+z10+z11+z14+z19+z20+z21+z22+z23+z25+z26+z33+z35+z36+z37+z41+z43+z4\
4396+z47+z49,
440  z+z3+z4+z5+z7+z10+z14+z15+z17+z19+z26+z28+z31+z32+z37+z40+z41+z44+z45+z47+z4\
4418+z51+z53+z55+z57+z60+z62+z64+z65+z66+z67,
442  1+z3+z5+z6+z7+z9+z10+z11+z12+z13+z14+z16+z18+z19+z22+z26+z28+z29+z31+z34+z42\
443+z44+z47+z51+z55+z56+z58+z60+z64+z68+z71+z72+z73+z74+z75+z76+z77+z80+z81+z82+z\
44484,
445  1+z+z4+z6+z8+z10+z11+z13+z15+z16+z17+z19+z20+z23+z26+z31+z37+z38+z39+z42+z45\
446+z46+z50+z51+z52+z56+z57+z62+z63+z65+z66+z69+z71+z72+z73+z76+z81+z83+z84+z85+z\
44787+z88+z91+z93+z98+z100+z101,
448  1+z2+z4+z6+z8+z11+z13+z16+z17+z20+z21+z23+z24+z27+z28+z29+z32+z35+z36+z38+z4\
4490+z48+z49+z51+z52+z54+z55+z56+z58+z59+z61+z68+z69+z70+z72+z75+z77+z78+z79+z80+\
450z81+z87+z91+z92+z96+z101+z102+z107+z109+z111+z112+z113+z115+z118, z,
451  z+z2+z4+z5+z6+z7+z9+z10+z15+z18+z19+z20+z21+z22+z25+z30+z31,
452  z2+z3+z8+z13+z17+z19+z22+z23+z24+z27+z29+z31+z33+z37+z39+z40+z42+z45+z48+z51\
453+z52+z54+z55+z58+z59+z60+z63+z64+z68+z69+z70+z76+z78+z79+z81+z82+z83+z86+z87+z\
45488+z89+z93+z94+z95,
455  z2+z5+z6+z7+z9+z11+z13+z15+z16+z18+z19+z20+z21+z22+z23+z24+z25+z26+z30+z31,
456  z+z3+z6+z8+z10+z16+z18+z19+z25+z26+z28+z32+z35+z36+z40+z41+z42+z43+z44+z45+z\
45747+z48+z49+z50+z51+z52+z53+z54+z55+z56+z57+z58+z60+z65+z67+z69+z71+z72+z74+z77\
458+z83+z84+z88+z89+z91+z93+z94+z95, 1+z8+z9+z11+z13+z18+z20+z21+z22+z23+z30,
459  z+z2+z3+z4+z5+z6+z7+z8+z10+z11+z13+z15+z18+z20+z21+z24+z26+z27+z28+z29+z30+z\
46032+z37+z38+z39+z40+z43+z46+z47+z48+z49+z52+z53+z54+z57+z62+z66+z67+z68+z70+z74\
461+z77+z79+z80+z82+z83+z86+z87+z89+z90+z91+z93+z94+z95,
462  1+z4+z5+z6+z8+z9+z11+z12+z16+z17+z21+z24+z27+z29+z30+z31,
463  1+z+z3+z5+z6+z10+z11+z13+z14+z17+z18+z19+z20+z23+z25+z27+z29+z33+z34+z41+z42\
464+z43+z45+z46+z47+z50+z51+z53+z54+z55+z56+z57+z60+z61+z62+z64+z67+z68+z70+z73+z\
46575+z77+z81+z82+z84+z88+z89+z94, z,
466  1+z4+z6+z10+z15+z17+z18+z19+z20+z21+z24+z28+z30+z31+z32+z33+z34+z37+z40+z41+\
467z45+z51+z52+z54+z55+z56+z58,
468  z+z6+z8+z9+z12+z13+z16+z18+z23+z24+z25+z28+z29+z30+z32+z35+z36+z38+z39+z40+z\
46943+z48+z49+z52+z53+z54,
470  z+z2+z3+z4+z5+z8+z9+z10+z11+z12+z17+z19+z20+z21+z22+z24+z27+z28+z32+z33+z35+\
471z37+z38+z40+z41+z42+z43+z46+z47+z48+z50+z51+z52+z54+z55+z57+z58+z59,
472  1+z2+z5+z7+z10+z14+z17+z18+z19+z20+z23+z24+z25+z28+z31+z33+z34+z36+z37+z40+z\
47341+z44+z45+z46+z47+z49+z52+z54+z55+z56+z57,
474  1+z4+z7+z10+z13+z15+z18+z19+z20+z21+z23+z25+z27+z30+z33+z34+z35+z37+z38+z41+\
475z42+z44+z47+z49+z50+z52+z53+z58+z59,
476  1+z2+z3+z4+z5+z6+z7+z12+z16+z17+z19+z20+z22+z26+z27+z28+z29+z30+z33+z36+z37+\
477z38+z40+z41+z45+z46+z47+z48+z52+z53+z54+z56+z57+z59+z62+z64+z69+z73+z75+z85+z8\
4787+z89+z92+z95+z100+z101+z104+z105+z106+z108+z110+z113+z115+z118,
479  1+z2+z3+z7+z8+z9+z11+z16+z17+z20+z22+z23+z26+z27+z29+z30+z31+z35+z36+z37+z38\
480+z39+z40+z43+z44+z45+z48+z49+z51+z54+z55+z57+z58,
481  z2+z5+z7+z8+z14+z15+z17+z18+z19+z20+z21+z22+z23+z25+z27+z29+z32+z34+z35+z36+\
482z38+z40+z44+z48+z49+z52+z53+z55+z59,
483  1+z+z3+z4+z7+z8+z9+z10+z12+z15+z19+z22+z26+z29+z32+z34+z35+z36+z40+z42+z43+z\
48444+z45+z48+z49+z51+z52+z53+z55+z57+z58,
485  z+z3+z4+z5+z8+z12+z15+z16+z17+z18+z19+z23+z26+z27+z30+z35+z36+z37+z38+z39+z4\
4861+z45+z46+z47+z48+z49+z51+z53+z56+z57+z58+z59, <<an element of GF(2, 120)>>,
487  1+z4+z11+z12+z14+z16+z17+z21+z22+z24+z25+z27+z30+z32+z33+z34+z36+z40+z41+z47\
488+z49+z51+z52+z56+z57+z58, <<an element of GF(2, 120)>>, z,
489  1+z+z4+z5+z7+z8+z9+z11+z14+z16+z20+z21+z25+z26+z28+z29+z32+z35+z38+z39+z43+z\
49046+z47+z48+z51+z52+z55+z56+z58+z59+z61+z63+z64+z65+z66+z67+z68+z70+z72+z74,
491  1+z3+z4+z8+z9+z10+z14+z15+z18+z22+z23+z24+z25+z29+z30+z32+z34+z38+z40+z44+z4\
4925+z46+z47+z49+z50+z53+z54+z57+z58+z61+z64+z65+z66+z70+z71+z72+z74+z75,
493  z+z2+z3+z6+z7+z8+z9+z11+z13+z14+z16+z17+z19+z20+z21+z27+z29+z34+z36+z38+z41+\
494z42+z44+z45+z46+z47+z49+z50+z53+z54+z55+z56+z58+z61+z63+z66+z68+z73+z74+z75,
495  z2+z3+z4+z6+z7+z9+z10+z12+z20+z24+z28+z32+z33+z35+z36+z37+z39+z43+z47+z49+z5\
4960+z51+z54+z59+z60+z61+z63+z64+z66+z67+z68+z69+z70+z71+z75, z,
497  1+z2+z4+z5+z6+z7+z8+z11+z12+z13+z14+z20+z23+z27+z29+z31+z32+z33+z35+z37+z41+\
498z43+z44+z45+z49+z52+z53+z56+z57+z58+z61+z62+z63+z64+z65+z66+z68+z69+z70+z71+z7\
4992+z73+z76+z78+z80+z82+z84+z85,
500  z3+z7+z9+z10+z11+z15+z16+z17+z20+z21+z23+z24+z27+z29+z31+z32+z34+z35+z38+z42\
501+z47+z51+z52+z55+z56+z57+z58+z59+z60+z61+z62+z64+z65+z67+z69+z72+z73+z77+z79+z\
50282+z83+z84+z85+z86, 2z, 1+z+2z2+2z3+z4+z5+z9+2z12+2z14+z17+z18,
503  1+2z+2z2+z3+2z4+z6+z7+2z8+2z9+2z10+z11+2z13+2z14+2z16+z18+z20+z21+2z22+2z24+\
5042z25+z26+z27+2z31,
505  2z+2z2+z3+z6+2z7+z8+2z11+2z12+2z13+z16+2z17+z18+2z22+2z24+2z26+z27+z28+z29+2\
506z30+2z31+2z32+z33+z37+z38+2z41,
507  2z2+2z4+z5+z6+2z9+2z11+z12+z13+z16+z17+z18+2z19+z21+z23+z24+z27+z28+2z31+z32\
508+2z33+2z34+2z36+2z39+2z41+2z44+z45+2z46+2z47+2z48+2z51+2z52+2z53+2z54,
509  1+z+z2+2z3+z5+z6+2z8+z9+2z13+2z16+2z17+2z18+2z19+z20+z21+z22+2z23+z24+2z25+z\
51026+2z27+z28+z33+2z34+2z35+z36+2z38+2z39+z40+2z41+z42+z44+z45+z47+z48+z49+z50+2\
511z51+2z52+z53+2z54+2z55+z56+2z57+2z59+2z60+2z61+2z62+2z63,
512  1+2z+z5+2z8+2z9+z12+z13+2z16+2z17+z19+z20+z24+z25+z27+z29+z30+z31+z32+2z33+2\
513z34+z35+2z38+z39+2z42+z44+z45+2z46+z48+z49+z50+z52+2z54+2z55+2z57+z58+z59+2z62\
514+z64+2z66+z68+z69+z71+2z72+z73+z75, 2z, 1+2z+2z3+z9+2z14+2z15+z16+2z18+2z19,
515  2+2z+2z2+z6+z7+2z10+z17+z19+2z20+2z21+z24+2z25+2z27+z28+z29+2z30+z31+2z32+2z\
51635+z36+2z37+z38+2z40+2z41+2z42+z44+2z46+z47+z49+z51+z52+z54+z56+2z57+2z58+z59,
517  1+z2+2z3+2z6+z7+2z9+2z10+2z11+2z12+z14+z15+z16+2z17+z18+z19,
518  1+2z+z2+z4+2z5+z6+z7+2z8+z9+2z10+2z11+z12+z17,
519  1+z+2z2+2z3+z5+2z6+z7+2z10+2z11+2z12+2z13+z14+z15+z16+2z17+2z18+z20+2z21+2z2\
5203+2z24+2z25+z28+2z30+z31+2z32+z34+2z36+z37+z41+2z42+2z43+2z44+z45+z47+z48+2z49\
521+z53+z54+2z55+z56+2z59,
522  1+z+2z2+z3+2z6+z7+z10+2z11+z13+2z14+z15+z16+z17+2z18+z19+z21+z22+z23+z24+2z2\
5235+z28+2z29+2z31+2z33+z35+z36+z38+2z39,
524  2+z+2z3+z5+2z9+z10+z12+z13+z14+z15+z16+z18+z19,
525  2+z+2z4+2z6+2z7+z8+2z10+z11+z12+z13+z16+z18+2z19+z20+2z22+z23+z25+2z26+2z28+\
526z29+z30+2z31+2z32+2z33+z37+z38+z40+z41+z42+2z46+2z47+2z48+z50+z51+2z52+2z53+2z\
52754+2z55+z56+z57,
528  2+z+2z2+2z4+2z6+2z7+2z9+2z11+2z12+2z13+z14+z15+2z16+2z17+2z18+2z20+2z21+z22+\
529z23+2z24+2z25+z28+z29+z30+2z31+z32+2z33+2z34+2z35+z36+2z37+2z39+2z40+z41+2z42+\
5302z43+2z44+2z45+z46+z47+2z48+z49+z50+z51+z52+z53+2z54+z57+2z58, 2z,
531  1+2z+2z2+z4+z5+z7+z9+z10+z11+2z12+z17+2z18+2z19+z21+2z22+z23+z27+z29+z30+z32\
532+z33+2z34+2z35+2z36+2z37+2z38+z40+2z41+2z43+2z46+2z47+z48+z50+z52+2z53+2z54+z5\
5336+2z59,
534  2+z+2z2+z3+z4+2z5+z6+2z7+2z10+z12+z13+z14+2z15+2z16+2z17+z18+z19+2z20+z21+2z\
53524+z28+z29+2z31+2z32+2z33+z34+2z35+z37+z38+2z43+2z47+z48+2z49+2z50+z53+2z54+2z\
53656+z58+2z59,
537  1+z+z2+z3+2z4+2z5+2z9+z10+2z11+z13+z16+z20+z22+z23+z26+2z27+z29+z30+z31+2z32\
538+2z33+z34+2z38+2z40+z42+2z43+2z45+2z46+z47+2z48+z49+z50+2z52+z54+2z57+z58+2z59
539    ,
540  2+2z+2z3+z4+2z5+z6+z9+2z10+2z12+2z13+2z16+z19+z20+2z22+2z24+z25+2z27+2z28+z2\
5419+2z31+2z34+2z35+z36+z37+z38+z40+z44+z45+z47+z48+z50+z51+2z53+2z54+2z58+2z59,
542  1+2z+2z5+z6+2z9+2z10+z11+2z12+2z13+z14+z16+2z17+2z18+2z19+z21+z22+z23+z24+z2\
5435+2z26+z27+z28+2z29+2z30+z31+2z32+z33+2z34+z35+2z36+2z38+2z39+2z42+z43+2z44+2z\
54446+2z47+2z48+2z49+z52+2z54+2z55+z56+2z57+z58+z59,
545  1+z2+z3+z5+2z6+z7+z8+2z9+2z10+2z11+z12+2z13+2z14+z15+2z17+2z21+2z22+2z23+z24\
546+2z26+z27+z28+z30+2z31+z32+z35+z36+z38+2z39+z40+z41+z42+2z49+2z50+z51+z54+2z55\
547+2z56+2z57+2z58+z59,
548  2+z+2z3+2z5+z6+z7+2z8+z9+z10+z11+2z12+z13+z14+z15+z17+z18+z19+2z20+z23+z26+2\
549z27+z28+z30+2z31+2z34+2z35+2z36+2z37+z38+2z39+z41+2z42+z44+z45+2z46+z48+z49+z5\
5500+2z54+2z55+2z56+z57,
551  1+z+2z2+z3+z4+z5+z6+2z7+2z8+2z9+2z10+z11+z15+z16+z17+z18+z20+z22+z23+2z25+2z\
55226+2z27+2z28+z29+2z30+2z32+z33+z34+2z35+2z36+z38+2z42+z43+2z44+2z45+2z46+z48+z\
55350+2z51+z52+z53+2z55+2z56+2z57+z58+z59,
554  1+z+z2+z5+z6+z7+z8+z10+z12+2z14+2z15+2z16+2z17+2z18+2z20+z21+2z23+z24+z26+2z\
55529+z30+z32+2z33+z36+z37+2z38+2z40+z41+z42+z43+2z44+2z46+2z47+2z48+2z51+2z53+2z\
55654+2z55+z57+z58+z59,
557  2+z+2z2+z3+2z5+2z6+2z7+z8+z12+2z13+z14+z16+z17+2z20+z21+z22+2z24+2z25+2z26+2\
558z28+2z29+2z31+z32+2z34+2z35+z38+2z39+z40+z41+z42+2z43+z44+2z47+2z49+z50+2z51+2\
559z54+z55+z56+z57+z59, 3z, 7+7z+4z2+z3,
560  5+12z+z2+11z3+3z5+8z6+z7+11z8+10z9+16z10+7z11, 3z, 3z, 17z, 3z, 3z, 2z, 2z,
561  2z ]
562gap> List(fieldpairs, pdd -> Z(pdd[1],pdd[2])/Z(pdd[1],pdd[3]));
563[ z, z+z4+z7+z8+z9+z10+z13+z14+z17+z18+z19+z23+z24+z25+z26+z27+z28+z29+z32,
564  1+z+z9+z10+z12+z13+z20+z21+z26+z28+z30+z31+z32+z33+z34+z37+z39+z46+z48+z49+z\
56550,
566  z+z5+z6+z10+z11+z12+z15+z16+z20+z24+z25+z28+z34+z36+z37+z38+z40+z43+z44+z45+\
567z46+z47+z48+z49+z53+z55+z57+z58+z59+z63+z64,
568  z+z2+z5+z8+z9+z13+z14+z15+z16+z20+z21+z22+z24+z26+z27+z28+z29+z30+z31+z32+z3\
5693+z38+z41+z42+z44+z46+z48+z49+z50+z53+z55+z59+z62+z63+z65+z66+z68+z70+z72+z75+\
570z77+z78+z79+z80+z82+z83,
571  1+z2+z7+z8+z9+z10+z11+z12+z13+z16+z17+z19+z20+z22+z27+z28+z29+z30+z34+z36+z3\
5727+z39+z40+z41+z45+z47+z51+z52+z54+z56+z58+z59+z60+z63+z65+z67+z68+z69+z72+z74+\
573z81+z84+z88+z89+z90+z93+z95+z97+z101,
574  1+z2+z3+z4+z8+z10+z11+z12+z14+z15+z16+z21+z22+z24+z32+z33+z34+z36+z39+z40+z4\
5754+z50+z51+z52+z54+z62+z63+z65+z68+z70+z71+z74+z77+z80+z88+z89+z92+z94+z101+z10\
5762+z104+z107+z108+z109+z110+z111+z118, z,
577  z2+z4+z5+z6+z7+z9+z10+z15+z18+z19+z20+z21+z22+z25+z30+z31,
578  1+z3+z8+z9+z10+z12+z13+z16+z17+z18+z20+z23+z24+z28+z29+z31+z34+z36+z38+z40+z\
57942+z43+z46+z47+z48+z51+z55+z56+z57+z59+z63+z64+z65+z66+z69+z71+z72+z73+z77+z79\
580+z81+z82+z83+z85+z86+z87+z88+z90+z91+z92+z93+z94+z95,
581  z+z2+z4+z6+z8+z9+z12+z16+z17+z18+z20+z21+z23+z24+z26+z28+z29+z30,
582  1+z3+z6+z7+z8+z10+z11+z13+z15+z16+z17+z18+z19+z21+z22+z29+z31+z32+z36+z37+z3\
5838+z39+z40+z41+z42+z45+z46+z47+z48+z49+z50+z51+z53+z55+z58+z61+z62+z65+z67+z68+\
584z72+z73+z76+z78+z81+z84+z87+z88+z92+z93+z94+z95,
585  z2+z7+z8+z11+z14+z16+z17+z19+z21+z24+z25+z27+z28+z30+z31,
586  1+z2+z3+z5+z9+z11+z13+z15+z16+z17+z18+z19+z20+z23+z26+z27+z33+z39+z40+z41+z4\
5875+z46+z48+z49+z50+z51+z53+z55+z56+z59+z60+z61+z62+z63+z64+z65+z67+z68+z69+z70+\
588z73+z77+z82+z83+z84+z88+z89+z90+z91+z93+z94+z95,
589  1+z3+z4+z5+z7+z10+z11+z13+z14+z15+z16+z18+z19+z20+z22+z23+z24+z26+z29+z30+z3\
5901,
591  z+z2+z7+z10+z11+z13+z14+z18+z20+z21+z24+z25+z27+z30+z33+z34+z37+z38+z39+z41+\
592z43+z44+z46+z48+z50+z51+z52+z55+z56+z58+z59+z63+z65+z66+z67+z68+z69+z71+z76+z7\
5938+z80+z81+z83+z88+z89+z90+z91+z92+z94, z,
594  1+z+z4+z6+z10+z15+z17+z18+z19+z20+z21+z24+z28+z30+z31+z32+z33+z34+z37+z40+z4\
5951+z45+z51+z52+z54+z55+z56+z58,
596  z+z2+z5+z6+z7+z8+z10+z11+z12+z13+z15+z17+z19+z20+z22+z23+z24+z25+z26+z27+z29\
597+z32+z34+z38+z39+z41+z43+z50+z52+z56+z57,
598  1+z2+z3+z5+z6+z7+z8+z11+z14+z16+z19+z20+z22+z23+z26+z27+z30+z31+z32+z36+z42+\
599z43+z44+z45+z47+z48+z50+z52+z58+z59,
600  1+z4+z6+z7+z8+z9+z10+z13+z17+z20+z21+z25+z26+z35+z37+z38+z44+z45+z47+z48+z56\
601+z57+z58+z59,
602  1+z2+z4+z5+z6+z7+z9+z11+z16+z21+z22+z24+z26+z27+z31+z32+z33+z35+z36+z37+z45+\
603z48+z49+z50+z51+z52+z53+z55+z57+z58,
604  1+z3+z5+z8+z9+z11+z13+z16+z17+z20+z21+z22+z26+z27+z28+z29+z32+z33+z35+z42+z4\
6056+z47+z49+z50+z51+z52+z55+z60+z61+z62+z63+z66+z67+z69+z70+z72+z73+z74+z78+z80+\
606z81+z82+z83+z87+z88+z92+z93+z96+z99+z100+z102+z103+z105+z106+z107+z108+z115+z1\
60716,
608  z+z2+z3+z4+z6+z8+z9+z10+z11+z12+z13+z14+z15+z17+z20+z24+z25+z28+z30+z37+z40+\
609z41+z42+z45+z46+z48+z50+z51+z54+z57+z58,
610  z2+z3+z5+z6+z7+z10+z11+z13+z15+z16+z21+z22+z26+z29+z30+z32+z33+z35+z36+z37+z\
61138+z43+z45+z46+z47+z49+z50+z51+z53+z54+z56+z58,
612  z2+z3+z4+z6+z9+z12+z13+z14+z16+z24+z25+z26+z30+z31+z32+z34+z37+z40+z42+z43+z\
61344+z45+z46+z49+z50,
614  z3+z8+z9+z13+z14+z17+z19+z22+z23+z24+z25+z28+z29+z32+z33+z36+z38+z40+z41+z42\
615+z43+z44+z46+z47+z51+z53+z54+z57,
616  z2+z3+z4+z7+z12+z17+z19+z20+z21+z22+z23+z28+z29+z30+z31+z32+z34+z36+z37+z38+\
617z41+z50+z51+z57+z58+z60+z61+z63+z66+z67+z68+z69+z73+z74+z75+z76+z79+z81+z82+z8\
6183+z85+z87+z89+z93+z94+z97+z98+z99+z100+z101+z102+z104+z106+z107+z110+z112+z116\
619+z117,
620  z2+z4+z5+z6+z8+z10+z11+z12+z16+z17+z18+z20+z21+z23+z24+z25+z27+z30+z34+z36+z\
62138+z39+z41+z43+z44+z45+z48+z49+z51+z53+z54+z56+z59,
622  1+z+z2+z6+z11+z13+z17+z18+z22+z23+z24+z26+z31+z32+z33+z35+z40+z44+z45+z47+z4\
6238+z49+z51+z55+z57+z59+z62+z66+z67+z69+z71+z74+z76+z77+z79+z81+z82+z83+z88+z89+\
624z90+z93+z94+z96+z97+z98+z99+z102+z104+z105+z106+z108+z111+z112+z114+z118+z119,
625  z,
626  1+z4+z5+z7+z8+z9+z11+z14+z16+z20+z21+z25+z26+z28+z29+z32+z35+z38+z39+z43+z46\
627+z47+z48+z51+z52+z55+z56+z58+z59+z61+z63+z64+z65+z66+z67+z68+z70+z72+z74,
628  z2+z4+z7+z8+z9+z11+z12+z13+z15+z16+z17+z19+z22+z23+z25+z26+z31+z34+z35+z37+z\
62938+z41+z42+z44+z45+z47+z50+z53+z54+z55+z56+z57+z58+z61+z62+z64+z67+z68+z69+z70\
630+z71+z72+z73+z75,
631  1+z2+z3+z8+z9+z10+z11+z13+z17+z19+z21+z25+z26+z27+z28+z29+z31+z32+z35+z36+z4\
6321+z43+z47+z51+z52+z59+z60+z61+z63+z64+z67+z69+z75,
633  z2+z3+z8+z9+z11+z14+z15+z16+z20+z21+z22+z24+z27+z28+z30+z33+z34+z39+z40+z41+\
634z44+z46+z48+z50+z51+z53+z58+z61+z62+z63+z64+z65+z66+z67+z68+z69+z70+z71+z72+z7\
6353+z74+z75, z,
636  1+z2+z4+z5+z6+z8+z11+z13+z15+z16+z18+z20+z24+z25+z27+z29+z30+z31+z33+z37+z40\
637+z42+z44+z45+z48+z49+z50+z53+z54+z55+z58+z59+z62+z63+z65+z67+z68+z69+z70+z71+z\
63872+z73+z74+z75+z77+z79+z80+z84+z86,
639  z2+z4+z7+z9+z12+z13+z18+z19+z21+z22+z24+z25+z28+z29+z30+z33+z34+z36+z39+z43+\
640z45+z46+z50+z51+z52+z55+z57+z58+z60+z61+z62+z65+z66+z69+z71+z73+z74+z75+z77+z7\
6418+z79+z80+z81+z82, 2z,
642  2+z+2z2+2z5+2z6+z7+z8+z9+2z10+z12+z14+z15+2z16+2z17+2z18+z19+z20,
643  2+2z5+2z6+2z7+2z8+2z9+2z11+2z13+2z14+2z16+2z19+z21+z22+z23+2z24+z25+2z26+z29\
644+z30+2z31+2z32,
645  1+z+z2+2z3+z6+z7+z8+z9+2z10+2z11+z12+2z14+z15+z26+z27+2z29+z30+z31+2z33+2z34\
646+z35+z36+z41+2z42,
647  2+2z+z2+2z3+z4+z5+z7+z8+z11+2z13+2z14+2z20+2z21+2z22+2z26+z28+z29+z30+z31+2z\
64832+z33+2z34+2z35+2z38+z39+z40+2z41+2z42+2z43+z44+z46+z48+z49+2z52+2z53+2z54,
649  1+z+z2+z4+2z5+2z7+z8+z9+z10+2z11+z13+z14+2z16+2z18+2z19+z20+2z22+2z25+z26+2z\
65028+z29+z30+2z31+2z32+z33+2z36+z38+2z39+2z42+2z43+z44+2z45+2z46+2z48+2z49+2z50+\
6512z54+2z55+z57+z58+z60+2z65,
652  2+z+z3+2z4+2z5+2z6+2z7+z8+2z10+z11+z14+2z15+2z18+z19+z24+2z25+z26+z27+z30+z3\
6531+2z32+2z33+2z35+z36+z38+z39+2z40+z41+2z43+2z45+z46+2z48+2z50+z52+2z53+z55+z59\
654+2z60+z61+z63+z65+2z66+z67+2z68+z69+z70+z73+z75+z76, 2z,
655  1+z+2z3+z9+2z14+2z15+z16+2z18+2z19,
656  1+2z2+2z3+2z4+z5+2z6+2z8+2z9+z10+z13+2z16+2z17+z18+z19+2z20+z22+2z25+z27+2z2\
6578+z29+2z31+z32+z33+z35+z37+2z40+z41+2z48+2z50+z52+z53+z54+2z55+2z57+z58,
658  2z+z3+z4+z5+z8+2z9+2z10+z11+2z12+2z14+2z16+2z18+2z19,
659  2+z3+2z4+2z6+2z7+2z10+z11+2z13+2z14+2z17+2z18+z19,
660  2+z3+2z4+2z6+2z8+z9+z10+2z12+z13+2z14+2z16+z19+z21+z22+2z23+z24+2z25+2z26+2z\
66128+2z29+z30+z32+2z33+2z34+2z35+z37+z39+z41+2z42+z44+2z45+z46+2z47+z48+z49+z51+\
662z52+2z53+2z54+2z55+z57+2z58+z59,
663  2+2z2+z3+2z4+z6+z8+z10+z11+2z13+z15+2z16+z17+z19+z20+2z22+z23+2z24+z25+2z26+\
6642z27+z28+2z29+z30+z31+z32+2z33+z35+z36+2z37+2z39,
665  z+z2+2z3+2z5+2z7+2z8+2z9+z10+2z11+2z12+z13+z14+z15+z18,
666  z+2z3+2z4+z5+2z6+2z7+z8+2z12+2z15+z16+2z17+z18+2z19+2z20+z21+2z24+2z25+2z26+\
6672z28+2z29+2z31+z33+z34+z35+z37+z38+z40+2z41+z42+2z43+z45+z46+z47+2z48+2z51+2z5\
6683+z54+2z56+2z57+2z58+2z59,
669  1+z2+z3+2z5+z6+2z8+z11+2z12+z14+2z16+2z19+2z20+2z21+z22+z23+z25+z26+2z28+z29\
670+z30+2z32+2z33+z34+z35+z36+2z38+2z39+z40+2z41+2z45+z47+z55+2z56+2z57, 2z,
671  1+z+2z2+z4+z5+z7+z9+z10+z11+2z12+z17+2z18+2z19+z21+2z22+z23+z27+z29+z30+z32+\
672z33+2z34+2z35+2z36+2z37+2z38+z40+2z41+2z43+2z46+2z47+z48+z50+z52+2z53+2z54+z56\
673+2z59,
674  2+2z+2z2+z4+z5+2z6+z8+2z12+2z13+2z15+z16+z17+2z18+2z19+2z20+z22+z23+z24+2z25\
675+2z26+z27+2z29+2z30+2z32+z33+z34+2z35+z36+2z37+2z38+z39+2z40+2z41+z42+z43+z44+\
676z45+z46+z50+z51+z54+z55+2z56+2z58+2z59,
677  2+z3+2z4+2z5+z9+2z10+z11+2z12+2z13+z14+2z17+z19+2z21+z22+z24+z25+2z26+z27+z2\
6788+z29+z30+2z31+z32+z34+2z35+2z36+2z37+z39+z41+z42+2z43+z44+z45+z46+z48+z49+z50\
679+z51+z52+2z53+2z54+2z55+z56+z59,
680  1+2z2+2z5+z6+z7+2z10+z11+2z12+2z13+2z14+2z16+z18+z19+z20+2z22+2z23+2z24+2z25\
681+z26+2z27+2z28+2z30+z31+z32+2z33+2z34+2z35+2z36+2z37+2z38+z40+z41+2z43+z44+2z4\
6825+2z46+z47+z48+z50+2z52+2z53+z55+2z58,
683  2z2+z3+z4+z6+z7+z11+z14+z16+2z17+2z18+z19+z20+z21+z22+z23+z26+2z27+2z28+z29+\
684z30+2z31+2z32+2z34+2z35+2z36+z38+z39+z40+2z42+z45+2z46+2z49+2z51+z52+z53+2z54+\
6852z55+2z56+2z57+z58+2z59,
686  2+z+2z5+z6+z7+z8+z9+z14+z15+z16+2z17+2z19+z20+2z21+z23+z24+2z25+2z26+2z28+2z\
68730+2z32+z33+2z34+z36+2z37+z38+z40+z41+z42+2z43+z44+2z48+z50+2z52+2z53+2z54+z55\
688+z56+2z58,
689  2+2z+z2+2z3+2z4+2z6+z9+2z11+z14+2z15+2z16+z17+2z18+2z19+2z20+2z21+z23+z24+z2\
6905+z26+2z27+z29+2z30+2z31+z32+2z34+2z36+z37+2z38+2z39+z40+2z41+z42+z43+z44+2z45\
691+2z46+z47+2z48+2z49+z50+2z51+z52+z55+z56+z57+z59,
692  z3+2z5+2z6+2z7+z9+z10+2z11+z12+z15+2z17+z18+z19+2z20+z21+z22+z24+z25+z27+z28\
693+z30+z32+2z33+z34+2z35+2z36+2z37+z38+2z40+2z41+2z45+z46+z47+2z48+z49+2z50+z53+\
6942z54+2z55+z56+z57+z59,
695  2+z+2z4+2z5+z6+z7+z9+z10+2z11+z12+2z13+z15+z16+2z17+2z22+z25+2z26+z29+2z32+z\
69633+z35+2z37+2z38+z40+2z41+z44+z45+2z46+2z48+2z49+2z50+z52+z56+2z58+z59,
697  2z+2z2+2z4+z5+2z7+z8+2z9+2z10+2z12+z14+2z15+z16+z17+2z18+z19+z20+z21+2z22+z2\
6985+2z26+z28+2z29+z30+z32+z33+2z34+z35+2z36+z39+z40+2z41+z42+z44+2z46+z47+2z48+z\
69949+z51+2z52+z53+z54+z57+z58+2z59, 6z, 9+15z+10z2+11z3,
700  5+7z+14z2+13z3+10z4+10z5+4z6+13z7+5z8+12z9+3z10+7z11, 86z, 86z, 42396z,
701  21846z, 178956933z, 2147483646z, 576460752303423442z, 9223372036854775815z ]
702gap> ForAll(fieldsizes, pd -> ForAll(DivisorsInt(pd[2]), d2 ->
703>     Z(pd[1],pd[2])^((pd[1]^pd[2]-1)/(pd[1]^d2-1)) = Z(pd[1],d2) ));
704true
705gap> AsInternalFFE(0*Z(2,10));
7060*Z(2)
707gap> AsInternalFFE(Z(2,10)^0);
708Z(2)^0
709gap> AsInternalFFE(Z(2,10));
710Z(2^10)
711gap> AsInternalFFE(0*Z(7,6));
7120*Z(7)
713gap> AsInternalFFE(Z(7,6)^0);
714Z(7)^0
715gap> AsInternalFFE(Z(7,6));
716fail
717gap> AsInternalFFE(0*Z(65537,2));
718fail
719gap> AsInternalFFE(Z(65537,2)^0);
720fail
721gap> AsInternalFFE(Z(65537,2));
722fail
723gap> SetInfoLevel(InfoPrimeInt,iPI);
724gap> SetInfoLevel(InfoFactor,iF);
725gap> if IsBound(InfoFactInt) then SetInfoLevel(InfoFactInt,iFI); fi;
726gap> SetInfoLevel(InfoWarning,iW);
727gap> STOP_TEST( "ffeconway.tst", 1);
728