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1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 339 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 340 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 341 1 {CSTYLE "" -1 -1 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 342 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 343 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 344 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 345 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 334 "" 0 "" {TEXT 400 15 "elliptic.mws " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 6 "Author" }{TEXT 257 27 ": Abderrahmane Ni taj" }}{PARA 0 "" 0 "" {TEXT 258 5 "email" }{TEXT 259 2 ": " }{TEXT -1 13 " " }{TEXT 260 21 "nitaj@math.unicaen.fr" }}{PARA 0 "" 0 "" {TEXT 261 3 "Web" }{TEXT -1 17 ": " }{TEXT 262 34 "http://www.math.unicaen.fr/~nitaj/" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 263 54 "Le t E be an elliptic curve in the Tate-Wierstrass form" }}{PARA 338 "" 0 "" {TEXT 403 43 "E: Y^2+a1XY+a3Y=x^3+a2X^2+a4X+a6," }} {PARA 339 "" 0 "" {TEXT 404 38 "with rational coefficients such that \+ " }{TEXT 412 5 "(0,0)" }{TEXT 413 41 " is a rational point (i.e. a4=a6 =0). Let " }{TEXT 405 1 "O" }{TEXT 406 26 " be the point at infinity. " }}{PARA 341 "" 0 "" {TEXT 407 80 "\nThis Maple worksheet gives a par ametrisation of all elliptic curves for which " }{TEXT 408 8 "m(0,0)= O" }{TEXT 409 27 " for some rational integer " }{TEXT 410 23 "m=2,3,4, 5,6,7,8,9,10,12" }{TEXT 411 111 ". It also gives all the isogenous cu rves associated to isogenies with kernels consisting of rational point s. \n" }}{PARA 335 "" 0 "" {TEXT -1 24 "For more details, see " }} {PARA 336 "" 0 "" {TEXT -1 3 "1. " }{TEXT 422 9 "A. Nitaj," }{TEXT -1 68 " Determination de courbes elliptiques pour la conjecture de Szpiro , " }{TEXT 401 36 "Acta Arithmetica 85 (1998), 351-376." }}{PARA 337 " " 0 "" {TEXT -1 4 "2. " }{TEXT 423 9 "A. Nitaj," }{TEXT -1 63 " Isoge nes des courbes elliptiques definies sur les rationnels, " }{TEXT 402 28 "J. Computational Mathematics" }{TEXT -1 13 " (to appear)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 264 95 "############### ###################################################################### ##########" }}{PARA 259 "" 0 "" {TEXT 265 8 "elliptic" }}{PARA 0 "" 0 "" {TEXT -1 99 "This procedure gives the quantities b2, b4, b6, b8, c4 , c6 and Delta related to the elliptic curve " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 289 "" 0 "" {TEXT 266 39 "E: Y^2+a1XY+a3Y=x ^3+a2X^2+a4X+a6, " }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{TEXT 307 1 " \+ " }{TEXT 305 10 "a1=E[1], " }}{PARA 278 "" 0 "" {TEXT -1 10 " a2=E[2] , " }}{PARA 279 "" 0 "" {TEXT -1 9 "a3=E[3], " }}{PARA 280 "" 0 "" {TEXT -1 9 "a4=E[4], " }}{PARA 281 "" 0 "" {TEXT -1 9 "a5=E[5], " }} {PARA 282 "" 0 "" {TEXT -1 9 "b2=E[6], " }}{PARA 283 "" 0 "" {TEXT -1 9 "b4=E[7], " }}{PARA 284 "" 0 "" {TEXT -1 9 "b6=E[8], " }}{PARA 285 " " 0 "" {TEXT -1 9 "b8=E[9], " }}{PARA 286 "" 0 "" {TEXT -1 10 "c4=E[10 ], " }}{PARA 287 "" 0 "" {TEXT -1 10 "c6=E[11], " }}{PARA 288 "" 0 "" {TEXT 306 12 "Delta=E[12]." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 32 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 " elliptic:=proc(a1,a2,a3,a4,a6)\nlocal b2,b4,b6,b8,c 4,c6,Dis,E:\nb2:=a1^2+4*a2:\nb4:=a1*a3+2*a4:\nb6:=a3^2+4*a6:\nb8:=a1^2 *a6-a1*a3*a4+4*a2*a6+a2*a3^2-a4^2:\nc4:=b2^2-24*b4:\nc6:=-b2^3+36*b2*b 4-216*b6:\nDis:=-b2^2*b8-8*b4^3-27*b6^2+9*b2*b4*b6:\nE:=[a1,a2,a3,a4,a 6,b2,b4,b6,b8,c4,c6,Dis]:\nend:\n" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 267 95 "############################################################## #################################" }}{PARA 261 "" 0 "" {TEXT 268 21 "E 1, 2(0,0)=O" }}{PARA 340 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "a1:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a2:=s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a3:=0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a4:=t:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6 "a6:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E1:=elliptic(a1,a2,a3,a4,a6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Delta1:=factor(E1[12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 264 "" 0 "" {TEXT 269 96 "############################### ################################################################\n" } {TEXT 270 9 "E2=E1/<2>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a1:=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "a2:=-2*s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a3:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a 4:=s^2-4*t:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a6:=0:" 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96 "###################################### #########################################################\n" }{TEXT 275 14 "E4=E3/<2> " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a1:=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "a2:=2*(s+t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a3:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a 4:=(s-t)^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a6:=0:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E4:=elliptic(a1,a2,a3,a4,a6) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Delta4:=factor(E4[12]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 267 "" 0 "" {TEXT 276 96 "### ###################################################################### ######################\n" }{TEXT 277 10 "E5=E3/<2>\n" }{TEXT 278 1 "\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 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"a2:=s^3*t*(s+t)*(2*s+t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a3:=s^3*t^2*(s+t)*(2*s+t)*(s^2+3*s*t+t^2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a4:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a6:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E48:=elliptic(a1,a2,a3,a4,a6):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "Delta48:=factor(E48[12]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 323 "" 0 "" {TEXT 378 96 "######################################################## #######################################\n" }{TEXT 379 11 "E49=E48/<2> " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a1:=-(2*s^3+2*s^2*t-2*s*t^2-t^3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a2:=s^3*t*(s+t)*(2*s+t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 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0 "" {MPLTEXT 1 0 42 "a3:=s ^3*t^2*(s+t)*(2*s+t)*(s^2+3*s*t+t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 249 "a4:=5*s^2*t*(s+t)^2*(2*s+t)*(4*s^6-10*s^4*t^2-12*s^3 *t^3-8*s^2*t^4-4*s*t^5-t^6):a6:=s^2*t*(s+t)^2*(2*s+t)*(16*s^12-48*s^11 *t-316*s^10*t^2-562*s^9*t^3-570*s^8*t^4-578*s^7*t^5-683*s^6*t^6-626*s^ 5*t^7-370*s^4*t^8-141*s^3*t^9-38*s^2*t^10-8*s*t^11-t^12):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E50:=elliptic(a1,a2,a3,a4,a6):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta50:=factor(E50[12]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 325 "" 0 "" {TEXT 382 96 "###################################### #########################################################\n" }{TEXT 383 12 "E51=E48/<10>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a1:=-(2*s^3+2*s^2*t-2*s*t^2-t^3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a2:=s^3*t*(s+t)*(2*s+t): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a3:=s^3*t^2*(s+t)*(2*s+ t)*(s^2+3*s*t+t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "a4: =-5*s*(s+t)*(s^10+7*s^9*t+83*s^8*t^2+346*s^7*t^3+739*s^6*t^4+935*s^5*t ^5+729*s^4*t^6+352*s^3*t^7+102*s^2*t^8+16*s*t^9+t^10): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "a6:=s*(s+t)*(3*s^16+33*s^15*t-543* s^14*t^2-5875*s^13*t^3-27930*s^12*t^4-83206*s^11*t^5-170156*s^10*t^6-2 48030*s^9*t^7-263424*s^8*t^8-206431*s^7*t^9-119674*s^6*t^10-50891*s^5* t^11-15541*s^4*t^12-3279*s^3*t^13-446*s^2*t^14-34*t^15*s-t^16):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E51:=elliptic(a1,a2,a3,a4,a6 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta51:=factor(E51[1 2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 326 "" 0 "" {TEXT 384 96 "############################### ################################################################\n" } {TEXT 385 23 "E52, 12(0,0)=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s': t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2* t^2+8*s*t^3-27*t^4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2: =-4*t^2*(s^2-t^2)*(s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a3:=32*t^5*(s-3*t)*(s^2-t^2)*(s^2+3*t^2)*(s^2 -2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a4:=0:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a6:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E52:=elliptic(a1,a2,a3,a4,a6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta52:=factor(E52[12]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 327 "" 0 "" {TEXT 386 96 "###################################### #########################################################\n" }{TEXT 387 11 "E53=E52/<2>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2*t^2+8*s*t^3-27*t^4) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2:=-4*t^2*(s^2-t^2)*( s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a3:=32*t^5*(s-3*t)*(s^2-t^2)*(s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a4:=-320*t^6*(s-t)^6*(s^2+3* t^2)^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "a6:=-64*t^6*(s- t)^6*(s^2+3*t^2)^2*(297*t^8+144*s*t^7-380*s^2*t^6+320*s^3*t^5-162*s^4* t^4+48*s^5*t^3-12*s^6*t^2+s^8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E53:=elliptic(a1,a2,a3,a4,a6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta53:=factor(E53[12]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 328 "" 0 "" {TEXT 388 96 "############################################################## #################################\n" }{TEXT 389 11 "E54=E52/<3>" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2*t^2+8*s*t^3-27*t^4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2:=-4*t^2*(s^2-t^2)*(s^2+3*t^2)*(s ^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a3:=32* t^5*(s-3*t)*(s^2-t^2)*(s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "a4:=80*t^4*(s^2-2*s*t+5*t^2)*(-s^4+6*s^2* t^2-24*s*t^3+3*t^4)*(t+s)^2*(-s+t)^4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "a6:=16*t^4*(s^2-2*s*t+5*t^2)*(-s^12+18*s^10*t^2-72*s ^9*t^3-131*s^8*t^4+1152*s^7*t^5-2676*s^6*t^6+1328*s^5*t^7+5833*s^4*t^8 -11456*s^3*t^9+10626*s^2*t^10-9384*t^11*s+667*t^12)*(t+s)^2*(-s+t)^4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "E54:=elliptic(a1,a2,a3, a4,a6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta54:=factor( E54[12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 329 "" 0 "" {TEXT 390 96 "################## ###################################################################### #######\n" }{TEXT 391 11 "E55=E52/<4>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s': t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2* t^2+8*s*t^3-27*t^4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2: =-4*t^2*(s^2-t^2)*(s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a3:=32*t^5*(s-3*t)*(s^2-t^2)*(s^2+3*t^2)*(s^2 -2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "a4:=-40* t^3*(s^2+3*t^2)*(s^8-12*s^6*t^2-24*s^5*t^3+54*s^4*t^4+48*s^3*t^5+180*s ^2*t^6-24*s*t^7+33*t^8)*(-s+t)^3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "a6:=-8*t^3*(s^2+3*t^2)*(s^16-24*s^14*t^2-88*s^13*t^3 +372*s^12*t^4+752*s^11*t^5-1176*s^10*t^6+504*s^9*t^7-14418*s^8*t^8+208 96*s^7*t^9-33480*s^6*t^10+65304*s^5*t^11-62396*s^4*t^12+39024*s^3*t^13 +32760*s^2*t^14+4680*s*t^15+12825*t^16)*(-s+t)^3:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 30 "E55:=elliptic(a1,a2,a3,a4,a6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Delta55:=factor(E55[12]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 330 "" 0 "" {TEXT 392 96 "###################################### #########################################################\n" }{TEXT 393 11 "E56=E52/<6>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2*t^2+8*s*t^3-27*t^4) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2:=-4*t^2*(s^2-t^2)*( 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331 "" 0 "" {TEXT 394 96 "############################# ##################################################################\n" }{TEXT 395 12 "E57=E52/<12>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "a1:=-(s^4+2*s^2*t^2+8*s*t^3- 27*t^4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a2:=-4*t^2*(s^2 -t^2)*(s^2+3*t^2)*(s^2-2*s*t+5*t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a3:=32*t^5*(s-3*t)*(s^2-t^2)*(s^2+3*t^2)*(s^2-2*s*t+5 *t^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "a4:=-10*t*(-s+t) *(s^14-18*s^13*t+111*s^12*t^2-476*s^11*t^3+1257*s^10*t^4-2854*s^9*t^5+ 4111*s^8*t^6-4456*s^7*t^7+2011*s^6*t^8+706*s^5*t^9-7067*s^4*t^10+7428* s^3*t^11-5061*s^2*t^12-330*t^13*s+21021*t^14):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 381 "a6:=-2*t*(-s+t)*(s^22-46*s^21*t+631*s^20*t^2- 5084*s^19*t^3+26731*s^18*t^4-107886*s^17*t^5+332909*s^16*t^6-830736*s^ 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