# This file lists 9 + 266 non-equivalent Belyi maps, in 9 + 99 Gal(Qbar/Q)-orbits, # that are contained (9 indirectly and 266 directly) in the Belyi-(1) families from this file: # read B1_inf_2_3: # 9 Belyi-maps contained in Belyi-(1) families after reparametrization: # BranchPattern (4), (4), (1^4) Belyi_inf23_256 := eval( eval(N8, {x=x/s, s=-s^4}), s=0); # 1/(1-x^4); # BranchPattern (1^4), (4), (4) Belyi_inf23_257 := eval( eval(N12, {x=x/s, s=-s^4}), s=0); # 1-x^4; # BranchPattern (1^2,4), (1^2,2^2), (6) Belyi_inf23_258 := limit(eval(N17, {x=s*x,s=s^2}), s=infinity); # (3*x^2+2)/x^6 # BranchPattern (2^3), (1^2,2^2), (6) Belyi_inf23_259 := limit( eval(N20, {x=s*x, s=s^2}), s=infinity); # (3*x^2-4)^2/x^6 # BranchPattern (1^2,6), (2^4), (1^2,6) Belyi_inf23_260 := limit( eval(N31, {x=x*s, s=s^2}), s=infinity); # 64*(x^2-1)/((x^2+8)*x^6) # BranchPattern (2,3^2), (2^4), (1^2,6) Belyi_inf23_261 := limit( eval(N36, {x=s*x,s=s^2}), s=infinity); # 64*(x^2-1)^3/((9*x^2-8)*x^6) # BranchPattern (1^4,8), (2^4,4), (3^4) Belyi_inf23_262 := eval(normal(eval(N55, {x=x/s, s=s^4})), s=0); # 27*(x^4-8)/(x^4-6)^3 # BranchPattern (2^4,4), (2^4,4), (3^4) Belyi_inf23_263 := limit( eval(N65, {x=s*x, s=-s^4}), s=infinity); # (x^4-1)^2/((1/3)*x^4+1)^3 # BranchPattern (1^4,2), (6), (3^2) Belyi_inf23_264 := limit( eval(N67, {x=s*x, s=3*s^2}), s=infinity); # 1-x^6/(x^2+1)^3 # 266 Belyi-maps (in 99 Galois orbits) that are directly contained in Belyi-(1) families (and # are thus not needed for completeness) if Read_Only_Necessary_Maps <> true then # BranchPattern (1,3), (4), (1^2,2) Belyi_inf23_265 := eval(N9, s=-9/16); # BranchPattern (2^2), (4), (1^2,2) Belyi_inf23_266 := eval(N10, s=-1/2); # BranchPattern (1^2,2), (4), (2^2) Belyi_inf23_267 := eval(N11, s=-1/2); # BranchPattern (1,4), (1,4), (1^2,3) Belyi_inf23_268 := eval(N13, s=2); # BranchPattern (2,3), (1,4), (1^2,3) Belyi_inf23_269 := eval(N14, s=RootOf(Z^2+28*Z-20)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^2,3), (1,4), (2,3) Belyi_inf23_270 := eval(N15, s=RootOf(Z^2+28*Z-20)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,2^2), (1,4), (2,3) Belyi_inf23_271 := eval(N16, s=-4/15); # BranchPattern (1^2,4), (1^2,4), (3^2) Belyi_inf23_272 := eval(N17, s=1); # BranchPattern (1^2,4), (1^2,2^2), (6) Belyi_inf23_273 := eval(N18, s=1); # BranchPattern (1^2,4), (1^2,4), (3^2) Belyi_inf23_274 := eval(N18, s=1/16); # BranchPattern (1,2,3), (1^2,4), (3^2) Belyi_inf23_275 := eval(N19, s=4/5); # BranchPattern (1,2,3), (1^2,2^2), (6) Belyi_inf23_276 := eval(N19, s=RootOf(5*Z^3-6*Z^2+2)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (2^3), (1^2,4), (3^2) Belyi_inf23_277 := eval(N20, s=-1/4); # BranchPattern (1,5), (2,4), (1^3,3) Belyi_inf23_278 := eval(N21, s=RootOf(3*Z^2+26*Z-27)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (2,4), (2,4), (1^3,3) Belyi_inf23_279 := eval(N22, s=-1); # BranchPattern (3^2), (2,4), (1^3,3) Belyi_inf23_280 := eval(N23, s=-1/2); # BranchPattern (1^2,4), (2,4), (1,2,3) Belyi_inf23_281 := eval(N24, s=RootOf(Z^4+14*Z^3-54*Z^2+70*Z-25)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,2,3), (2,4), (1,2,3) Belyi_inf23_282 := eval(N25, s=RootOf(100*Z^4+220*Z^3+171*Z^2+52*Z+4)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (2^3), (2,4), (1,2,3) Belyi_inf23_283 := eval(N26, s=-1/2); # BranchPattern (1^2,5), (1,2^3), (1,6) Belyi_inf23_284 := eval(N27, s=RootOf(24*Z^3+12*Z^2-1)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,5), (1,2,4), (1,3^2) Belyi_inf23_285 := eval(N27, s=RootOf(432*Z^5+513*Z^4+252*Z^3+40*Z^2-8*Z-4)); # Gal(Qbar/Q)-orbit with 5 Belyi maps # BranchPattern (1,2,4), (1,2,4), (1,3^2) Belyi_inf23_286 := eval(N28, s=RootOf(63*Z^2+14*Z-1)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,2,4), (1,2^3), (1,6) Belyi_inf23_287 := eval(N28, s=RootOf(945*Z^4+84*Z^3+150*Z^2-12*Z+1)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,2,4), (1,2,4), (1,3^2) Belyi_inf23_288 := eval(N28, s=RootOf(945*Z^4+756*Z^3+342*Z^2+20*Z+1)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,3^2), (1,2^3), (1,6) Belyi_inf23_289 := eval(N29, s=RootOf(4*Z^2+6*Z+3)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,3^2), (1,2,4), (1,3^2) Belyi_inf23_290 := eval(N29, s=RootOf(16*Z^4-24*Z^3-33*Z^2-18*Z-9)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (2^2,3), (1,2^3), (1,6) Belyi_inf23_291 := eval(N30, s=7/2); # BranchPattern (2^2,3), (1,2,4), (1,3^2) Belyi_inf23_292 := eval(N30, s=RootOf(2*Z^3-75*Z^2+840*Z-2352)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,6), (2^2,4), (1^2,3^2) Belyi_inf23_293 := eval(N31, s=RootOf(8*Z^2-12*Z+3)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^2,6), (2^4), (1^2,6) Belyi_inf23_294 := eval(N32, s=0); # BranchPattern (1^2,6), (2^2,4), (1^2,3^2) Belyi_inf23_295 := eval(N32, s=RootOf(2*Z^5-3*Z^4-360*Z^3-42*Z^2-882*Z+1029)); # Gal(Qbar/Q)-orbit with 5 Belyi maps # BranchPattern (1,2,5), (2^4), (1^2,6) Belyi_inf23_296 := eval(N33, s=0); # BranchPattern (1,2,5), (2^2,4), (1^2,3^2) Belyi_inf23_297 := eval(N33, s=RootOf(729*Z^4-6318*Z^3+26649*Z^2-65660*Z+68600)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,3,4), (2^4), (1^2,6) Belyi_inf23_298 := eval(N34, s=RootOf(4*Z^2-18*Z+21)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,3,4), (2^2,4), (1^2,3^2) Belyi_inf23_299 := eval(N34, s=RootOf(4*Z^7-140*Z^6+1440*Z^5-7173*Z^4+19872*Z^3-31374*Z^2+26460*Z-9261)); # Gal(Qbar/Q)-orbit with 7 Belyi maps # BranchPattern (2^2,4), (2^2,4), (1^2,3^2) Belyi_inf23_300 := eval(N35, s=RootOf(Z^3+6*Z^2+9*Z-4)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (2,3^2), (2^2,4), (1^2,3^2) Belyi_inf23_301 := eval(N36, s=RootOf(8*Z^2+12*Z+3)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^3,5), (2^2,4), (2,3^2) Belyi_inf23_302 := eval(N37, s=RootOf(128*Z^3+336*Z^2-81*Z-8)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,2,4), (2^4), (2,6) Belyi_inf23_303 := eval(N38, s=-2); # BranchPattern (1^2,2,4), (2^2,4), (2,3^2) Belyi_inf23_304 := eval(N38, s=RootOf(27*Z^4+108*Z^3+216*Z^2+176*Z+16)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1^2,3^2), (2^4), (2,6) Belyi_inf23_305 := eval(N39, s=-1); # BranchPattern (1^2,3^2), (2^2,4), (2,3^2) Belyi_inf23_306 := eval(N39, s=RootOf(8*Z^2+12*Z+3)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,2^2,3), (2^4), (2,6) Belyi_inf23_307 := eval(N40, s=0); # BranchPattern (1,2^2,3), (2^2,4), (2,3^2) Belyi_inf23_308 := eval(N40, s=RootOf(Z^4+80*Z^3-384*Z^2+1024*Z-4096)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1^3,6), (1,2^4), (3,6) Belyi_inf23_309 := eval(N41, s=27/4); # BranchPattern (1^3,6), (1,2^2,4), (3^3) Belyi_inf23_310 := eval(N41, s=81/8); # BranchPattern (1^3,6), (1,2^4), (3,6) Belyi_inf23_311 := eval(N42, s=RootOf(32*Z^2-13*Z+4)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^3,6), (1,2^2,4), (3^3) Belyi_inf23_312 := eval(N42, s=RootOf(128*Z^3-144*Z^2+63*Z-8)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,2,5), (1,2^4), (3,6) Belyi_inf23_313 := eval(N43, s=RootOf(8*Z^5-44*Z^4+424*Z^3-809*Z^2+644*Z-196)); # Gal(Qbar/Q)-orbit with 5 Belyi maps # BranchPattern (1^2,2,5), (1,2^2,4), (3^3) Belyi_inf23_314 := eval(N43, s=RootOf(16*Z^5-91*Z^4-1480*Z^3+3656*Z^2-2912*Z+784)); # Gal(Qbar/Q)-orbit with 5 Belyi maps # BranchPattern (1^2,3,4), (1,2^4), (3,6) Belyi_inf23_315 := eval(N44, s=RootOf(Z^4-8*Z^3+45*Z^2-70*Z-49)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1^2,3,4), (1,2^2,4), (3^3) Belyi_inf23_316 := eval(N44, s=RootOf(Z^6-18*Z^5+96*Z^4-180*Z^3-6*Z^2+252*Z+98)); # Gal(Qbar/Q)-orbit with 6 Belyi maps # BranchPattern (1,2^2,4), (1,2^4), (3,6) Belyi_inf23_317 := eval(N45, s=RootOf(108*Z^4+216*Z^3+144*Z^2+34*Z+1)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,2^2,4), (1,2^2,4), (3^3) Belyi_inf23_318 := eval(N45, s=RootOf(216*Z^4+432*Z^3+288*Z^2+62*Z-1)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,2,3^2), (1,2^4), (3,6) Belyi_inf23_319 := eval(N46, s=RootOf(32*Z^3+21*Z^2+6*Z+1)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1,2,3^2), (1,2^2,4), (3^3) Belyi_inf23_320 := eval(N46, s=RootOf(384*Z^4+368*Z^3+129*Z^2+18*Z+1)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (2^3,3), (1,2^4), (3,6) Belyi_inf23_321 := eval(N47, s=-81/4); # BranchPattern (2^3,3), (1,2^2,4), (3^3) Belyi_inf23_322 := eval(N47, s=243/4); # BranchPattern (1^3,7), (2^5), (1,3,6) Belyi_inf23_323 := eval(N48, s=RootOf(2*Z^3+3*Z^2-54*Z-153)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^3,7), (2^3,4), (1,3^3) Belyi_inf23_324 := eval(N48, s=RootOf(16*Z^7+104*Z^6-420*Z^5-4869*Z^4-9108*Z^3+25866*Z^2+117072*Z+124875)); # Gal(Qbar/Q)-orbit with 7 Belyi maps # BranchPattern (1^2,2,6), (2^5), (1,3,6) Belyi_inf23_325 := eval(N49, s=RootOf(16*Z^3-27*Z^2+54*Z-27)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,2,6), (2^3,4), (1,3^3) Belyi_inf23_326 := eval(N49, s=RootOf(32*Z^6-1464*Z^5+7863*Z^4-23292*Z^3+30618*Z^2-21060*Z+7047)); # Gal(Qbar/Q)-orbit with 6 Belyi maps # BranchPattern (1^2,3,5), (2^3,4), (1,3^3) Belyi_inf23_327 := eval(N50, s=-1/2); # BranchPattern (1^2,3,5), (2^5), (1,3,6) Belyi_inf23_328 := eval(N50, s=RootOf(Z^3+6*Z^2+309*Z-100)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,3,5), (2^3,4), (1,3^3) Belyi_inf23_329 := eval(N50, s=RootOf(Z^6+12*Z^5-666*Z^4-3972*Z^3-19359*Z^2+11760*Z-1600)); # Gal(Qbar/Q)-orbit with 6 Belyi maps # BranchPattern (1^2,4^2), (2^5), (1,3,6) Belyi_inf23_330 := eval(N51, s=RootOf(125*Z^3-147*Z^2+63*Z-9)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,4^2), (2^3,4), (1,3^3) Belyi_inf23_331 := eval(N51, s=RootOf(3125*Z^6-7500*Z^5+9900*Z^4-7524*Z^3+3105*Z^2-648*Z+54)); # Gal(Qbar/Q)-orbit with 6 Belyi maps # BranchPattern (1,2^2,5), (2^5), (1,3,6) Belyi_inf23_332 := eval(N52, s=RootOf(Z^3-18*Z^2+321*Z-560)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1,2^2,5), (2^3,4), (1,3^3) Belyi_inf23_333 := eval(N52, s=RootOf(Z^7+30*Z^6+135*Z^5-17740*Z^4-377745*Z^3-755874*Z^2+10350025*Z-129654000)); # Gal(Qbar/Q)-orbit with 7 Belyi maps # BranchPattern (1,2,3,4), (2^5), (1,3,6) Belyi_inf23_334 := eval(N53, s=RootOf(3*Z^4+256*Z^3+1692*Z^2+3888*Z+3024)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1,2,3,4), (2^3,4), (1,3^3) Belyi_inf23_335 := eval(N53, s=RootOf(12*Z^8-1844*Z^7-47175*Z^6-445044*Z^5-2187200*Z^4-6207648*Z^3-10299744*Z^2-9313920*Z-3556224)); # Gal(Qbar/Q)-orbit with 8 Belyi maps # BranchPattern (2^2,3^2), (2^5), (1,3,6) Belyi_inf23_336 := eval(N54, s=-128/125); # BranchPattern (2^2,3^2), (2^3,4), (1,3^3) Belyi_inf23_337 := eval(N54, s=RootOf(6250*Z^3+13125*Z^2+4800*Z-2048)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^4,8), (2^6), (3^2,6) Belyi_inf23_338 := eval(N55, s=6); # BranchPattern (1^4,8), (2^4,4), (3^4) Belyi_inf23_339 := eval(N55, s=9); # BranchPattern (1^4,8), (2^6), (3^2,6) Belyi_inf23_340 := eval(N56, s=-27/2); # BranchPattern (1^4,8), (2^4,4), (3^4) Belyi_inf23_341 := eval(N56, s=RootOf(4*Z^2-108*Z-243)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^3,2,7), (2^6), (3^2,6) Belyi_inf23_342 := eval(N57, s=RootOf(Z^2-20*Z-243)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^3,2,7), (2^4,4), (3^4) Belyi_inf23_343 := eval(N57, s=RootOf(2*Z^3-172*Z^2+243*Z-39366)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^3,3,6), (2^6), (3^2,6) Belyi_inf23_344 := eval(N58, s=-32/9); # BranchPattern (1^3,3,6), (2^4,4), (3^4) Belyi_inf23_345 := eval(N58, s=RootOf(27*Z^2+144*Z+128)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^3,4,5), (2^6), (3^2,6) Belyi_inf23_346 := eval(N59, s=RootOf(Z^3-48*Z^2+288*Z-3456)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^3,4,5), (2^4,4), (3^4) Belyi_inf23_347 := eval(N59, s=RootOf(Z^5-120*Z^4+3600*Z^3-43200*Z^2-259200*Z+1990656)); # Gal(Qbar/Q)-orbit with 5 Belyi maps # BranchPattern (1^2,2^2,6), (2^4,4), (3^4) Belyi_inf23_348 := eval(N60, s=-8); # BranchPattern (1^2,2^2,6), (2^6), (3^2,6) Belyi_inf23_349 := eval(N60, s=RootOf(4*Z^3-21*Z^2+192*Z-256)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,2^2,6), (2^4,4), (3^4) Belyi_inf23_350 := eval(N60, s=RootOf(8*Z^3-123*Z^2+384*Z-512)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1^2,2,3,5), (2^6), (3^2,6) Belyi_inf23_351 := eval(N61, s=RootOf(125*Z^4+1050*Z^3+3165*Z^2+4384*Z+2400)); # Gal(Qbar/Q)-orbit with 4 Belyi maps # BranchPattern (1^2,2,3,5), (2^4,4), (3^4) Belyi_inf23_352 := eval(N61, s=RootOf(3125*Z^7+34375*Z^6+159375*Z^5+398125*Z^4+584000*Z^3+525456*Z^2+293760*Z+86400)); # Gal(Qbar/Q)-orbit with 7 Belyi maps # BranchPattern (1^2,2,4^2), (2^4,4), (3^4) Belyi_inf23_353 := eval(N62, s=1/2); # BranchPattern (1^2,2,4^2), (2^6), (3^2,6) Belyi_inf23_354 := eval(N62, s=RootOf(16*Z^2-16*Z+1)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1^2,2,4^2), (2^4,4), (3^4) Belyi_inf23_355 := eval(N62, s=RootOf(32*Z^2-32*Z-1)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (1,2^2,3,4), (2^4,4), (3^4) Belyi_inf23_356 := eval(N63, s=-2); # BranchPattern (1,2^2,3,4), (2^4,4), (3^4) Belyi_inf23_357 := eval(N63, s=RootOf(Z^3-57*Z^2+3*Z-1)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1,2^2,3,4), (2^6), (3^2,6) Belyi_inf23_358 := eval(N63, s=RootOf(Z^3+24*Z^2+3*Z-1)); # Gal(Qbar/Q)-orbit with 3 Belyi maps # BranchPattern (1,2,3^3), (2^6), (3^2,6) Belyi_inf23_359 := eval(N64, s=-27/8); # BranchPattern (1,2,3^3), (2^4,4), (3^4) Belyi_inf23_360 := eval(N64, s=RootOf(8*Z^2+540*Z-729)); # Gal(Qbar/Q)-orbit with 2 Belyi maps # BranchPattern (2^4,4), (2^4,4), (3^4) Belyi_inf23_361 := eval(N65, s=-1/9); # BranchPattern (2^4,4), (2^6), (3^2,6) Belyi_inf23_362 := eval(N65, s=1/3); # BranchPattern (1^4,2), (2,4), (6) # Up to Mobius-equivalence, the following Belyi-map appears in both N67 and N68 # but we list only one to prevent Mobius-duplicates: Belyi_inf23_363 := eval(N68, s=-27/4); # equals eval(N67, {s=1/3, x=-1/3-2*x/9}) end if;