Magma V2.11-1 Wed Aug 4 2004 14:35:14 on fats [Seed = 1664316627] Type ? for help. Type -D to quit. > Attach("routines.m"); > > P:=PolynomialRing(Rationals()); > f:=x*(x-1)*(x-2)*(x+1)*(x+3); > F:=HyperellipticCurve(f); > J:=Jacobian(F); > Km:=KummerSurface(J); > KJ:=Scheme(Km); > R:=Parent(DefiningPolynomial(Km)); > > A:=PolynomialRing(Rationals(),3); > L:=[]; > > //We compute the plane section k1+k2+t*k3+k4 for some > //rational values of t and determine the corresponding Q1,Q2,Q3 > //from that. We store them with the corresponding value of t > for t in [t: t in [-20..20]| t notin [-2,0,2,3]] do for> print "t=",t; for> V:=k1+k2+t*k3+k4; for> for> //this routine does the hard work: It computes a model of the form for> //Q1*Q3=Q2^2 for V intersect Km. for> for> mp,Q1,Q2,Q3:=KummerSection(Km,V); for> c:=MonomialCoefficient(A!Q1,u^2); for> L cat:=[]; for> end for; t= -20 {@ (0 : 0 : 1 : -20 : 1 : 0), (-1/399 : 0 : -20/399 : 1 : 0 : 0) @} Q1A: 20*Ak2^2 - 806*Ak2*Ak3 + 6*Ak2 + 8238*Ak3^2 - 356*Ak3 + 118 -2351 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 145/18*k1^2*k2*k3 + 103/12*k1^2*k3^2 + 1/18*k1*k2^3 - 41/18*k1*k2^2*k3 + 25/2*k1*k2*k3^2 - 833/18*k1*k3^3 + 1/36*k2^4 - 10/9*k2^3*k3 + 67/6*k2^2*k3^2 - 10/9*k2*k3^3 + 1/36*k3^4 118*k1^2 + 6*k1*k2 - 356*k1*k3 + 20*k2^2 - 806*k2*k3 + 8238*k3^2 898/3249*k1^2 + 2560/159201*k1*k2 - 14758/53067*k1*k3 + 2666/53067*k2^2 - 105358/53067*k2*k3 + 1103780/53067*k3^2 9392859/2553800*k1^2 + 477603/2553800*k1*k2 - 14168889/1276900*k1*k3 + 159201/255380*k2^2 - 64158003/2553800*k2*k3 + 655748919/2553800*k3^2 k1^2 + 119/2260*k1*k2 - 4661/2260*k1*k3 + 397/2260*k2^2 - 7981/1130*k2*k3 + 165163/2260*k3^2 898/3249*k1^2 + 2560/159201*k1*k2 - 14758/53067*k1*k3 + 2666/53067*k2^2 - 105358/53067*k2*k3 + 1103780/53067*k3^2 [ [ 0, 1, 2260/159201, 45200/159201 ] ] t= -19 {@ (0 : 0 : 1 : -19 : 1 : 0), (-1/360 : 0 : -19/360 : 1 : 0 : 0) @} Q1A: 19*Ak2^2 - 727*Ak2*Ak3 + 5*Ak2 + 7066*Ak3^2 - 319*Ak3 + 112 -8487/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 139/18*k1^2*k2*k3 + 33/4*k1^2*k3^2 + 1/18*k1*k2^3 - 20/9*k1*k2^2*k3 + 107/9*k1*k2*k3^2 - 251/6*k1*k3^3 + 1/36*k2^4 - 19/18*k2^3*k3 + 121/12*k2^2*k3^2 - 19/18*k2*k3^3 + 1/36*k3^4 112*k1^2 + 5*k1*k2 - 319*k1*k3 + 19*k2^2 - 727*k2*k3 + 7066*k3^2 521/1800*k1^2 + 11/720*k1*k2 - 3449/14400*k1*k3 + 191/3600*k2^2 - 28567/14400*k2*k3 + 142481/7200*k3^2 1612800/458329*k1^2 + 72000/458329*k1*k2 - 4593600/458329*k1*k3 + 273600/458329*k2^2 - 10468800/458329*k2*k3 + 101750400/458329*k3^2 k1^2 + 63/1354*k1*k2 - 2561/1354*k1*k3 + 239/1354*k2^2 - 9115/1354*k2*k3 + 89745/1354*k3^2 521/1800*k1^2 + 11/720*k1*k2 - 3449/14400*k1*k3 + 191/3600*k2^2 - 28567/14400*k2*k3 + 142481/7200*k3^2 [ [ 0, 1, 677/43200, 12863/43200 ] ] t= -18 {@ (0 : 0 : 1 : -18 : 1 : 0), (-1/323 : 0 : -18/323 : 1 : 0 : 0) @} Q1A: 18*Ak2^2 - 652*Ak2*Ak3 + 4*Ak2 + 6010*Ak3^2 - 284*Ak3 + 106 -1904 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 133/18*k1^2*k2*k3 + 95/12*k1^2*k3^2 + 1/18*k1*k2^3 - 13/6*k1*k2^2*k3 + 203/18*k1*k2*k3^2 - 677/18*k1*k3^3 + 1/36*k2^4 - k2^3*k3 + 163/18*k2^2*k3^2 - k2*k3^3 + 1/36*k3^4 106*k1^2 + 4*k1*k2 - 284*k1*k3 + 18*k2^2 - 652*k2*k3 + 6010*k3^2 31690/104329*k1^2 + 1484/104329*k1*k2 - 20516/104329*k1*k3 + 5866/104329*k2^2 - 206780/104329*k2*k3 + 1958994/104329*k3^2 5529437/1645298*k1^2 + 104329/822649*k1*k2 - 7407359/822649*k1*k3 + 938961/1645298*k2^2 - 17005627/822649*k2*k3 + 313508645/1645298*k3^2 k1^2 + 36/907*k1*k2 - 1560/907*k1*k3 + 161/907*k2^2 - 5806/907*k2*k3 + 54253/907*k3^2 31690/104329*k1^2 + 1484/104329*k1*k2 - 20516/104329*k1*k3 + 5866/104329*k2^2 - 206780/104329*k2*k3 + 1958994/104329*k3^2 [ [ 0, 1, 1814/104329, 32652/104329 ] ] t= -17 {@ (0 : 0 : 1 : -17 : 1 : 0), (-1/288 : 0 : -17/288 : 1 : 0 : 0) @} Q1A: 17*Ak2^2 - 581*Ak2*Ak3 + 3*Ak2 + 5064*Ak3^2 - 251*Ak3 + 100 -6791/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 127/18*k1^2*k2*k3 + 91/12*k1^2*k3^2 + 1/18*k1*k2^3 - 19/9*k1*k2^2*k3 + 32/3*k1*k2*k3^2 - 605/18*k1*k3^3 + 1/36*k2^4 - 17/18*k2^3*k3 + 97/12*k2^2*k3^2 - 17/18*k2*k3^3 + 1/36*k3^4 100*k1^2 + 3*k1*k2 - 251*k1*k3 + 17*k2^2 - 581*k2*k3 + 5064*k3^2 6625/20736*k1^2 + 533/41472*k1*k2 - 457/3072*k1*k3 + 827/13824*k2^2 - 54739/27648*k2*k3 + 9095/512*k3^2 8294400/2588881*k1^2 + 248832/2588881*k1*k2 - 20818944/2588881*k1*k3 + 1410048/2588881*k2^2 - 48190464/2588881*k2*k3 + 420028416/2588881*k3^2 k1^2 + 103/3218*k1*k2 - 4981/3218*k1*k3 + 575/3218*k2^2 - 19537/3218*k2*k3 + 172757/3218*k3^2 6625/20736*k1^2 + 533/41472*k1*k2 - 457/3072*k1*k3 + 827/13824*k2^2 - 54739/27648*k2*k3 + 9095/512*k3^2 [ [ 0, 1, 1609/82944, 27353/82944 ] ] t= -16 {@ (0 : 0 : 1 : -16 : 1 : 0), (-1/255 : 0 : -16/255 : 1 : 0 : 0) @} Q1A: 16*Ak2^2 - 514*Ak2*Ak3 + 2*Ak2 + 4222*Ak3^2 - 220*Ak3 + 94 -1503 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 121/18*k1^2*k2*k3 + 29/4*k1^2*k3^2 + 1/18*k1*k2^3 - 37/18*k1*k2^2*k3 + 181/18*k1*k2*k3^2 - 179/6*k1*k3^3 + 1/36*k2^4 - 8/9*k2^3*k3 + 43/6*k2^2*k3^2 - 8/9*k2*k3^3 + 1/36*k3^4 94*k1^2 + 2*k1*k2 - 220*k1*k3 + 16*k2^2 - 514*k2*k3 + 4222*k3^2 2434/7225*k1^2 + 16/1445*k1*k2 - 686/7225*k1*k3 + 462/7225*k2^2 - 14286/7225*k2*k3 + 121008/7225*k3^2 339575/111392*k1^2 + 7225/111392*k1*k2 - 397375/55696*k1*k3 + 7225/13924*k2^2 - 1856825/111392*k2*k3 + 15251975/111392*k3^2 k1^2 + 11/472*k1*k2 - 11/8*k1*k3 + 85/472*k2^2 - 1355/236*k2*k3 + 22603/472*k3^2 2434/7225*k1^2 + 16/1445*k1*k2 - 686/7225*k1*k3 + 462/7225*k2^2 - 14286/7225*k2*k3 + 121008/7225*k3^2 [ [ 0, 1, 472/21675, 7552/21675 ] ] t= -15 {@ (0 : 0 : 1 : -15 : 1 : 0), (-1/224 : 0 : -15/224 : 1 : 0 : 0) @} Q1A: 15*Ak2^2 - 451*Ak2*Ak3 + Ak2 + 3478*Ak3^2 - 191*Ak3 + 88 -5279/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 115/18*k1^2*k2*k3 + 83/12*k1^2*k3^2 + 1/18*k1*k2^3 - 2*k1*k2^2*k3 + 85/9*k1*k2*k3^2 - 473/18*k1*k3^3 + 1/36*k2^4 - 5/6*k2^3*k3 + 227/36*k2^2*k3^2 - 5/6*k2*k3^3 + 1/36*k3^4 88*k1^2 + k1*k2 - 191*k1*k3 + 15*k2^2 - 451*k2*k3 + 3478*k3^2 1117/3136*k1^2 + 55/6272*k1*k2 - 1709/50176*k1*k3 + 431/6272*k2^2 - 99059/50176*k2*k3 + 394665/25088*k3^2 4415488/1525225*k1^2 + 50176/1525225*k1*k2 - 9583616/1525225*k1*k3 + 150528/305045*k2^2 - 22629376/1525225*k2*k3 + 174512128/1525225*k3^2 k1^2 + 33/2470*k1*k2 - 2967/2470*k1*k3 + 449/2470*k2^2 - 13369/2470*k2*k3 + 104791/2470*k3^2 1117/3136*k1^2 + 55/6272*k1*k2 - 1709/50176*k1*k3 + 431/6272*k2^2 - 99059/50176*k2*k3 + 394665/25088*k3^2 [ [ 0, 1, 1235/50176, 18525/50176 ] ] t= -14 {@ (0 : 0 : 1 : -14 : 1 : 0), (-1/195 : 0 : -14/195 : 1 : 0 : 0) @} Q1A: 14*Ak2^2 - 392*Ak2*Ak3 + 2826*Ak3^2 - 164*Ak3 + 82 -1148 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 109/18*k1^2*k2*k3 + 79/12*k1^2*k3^2 + 1/18*k1*k2^3 - 35/18*k1*k2^2*k3 + 53/6*k1*k2*k3^2 - 413/18*k1*k3^3 + 1/36*k2^4 - 7/9*k2^3*k3 + 11/2*k2^2*k3^2 - 7/9*k2*k3^3 + 1/36*k3^4 82*k1^2 - 164*k1*k3 + 14*k2^2 - 392*k2*k3 + 2826*k3^2 14362/38025*k1^2 + 44/7605*k1*k2 + 448/12675*k1*k3 + 314/4225*k2^2 - 24976/12675*k2*k3 + 186466/12675*k3^2 225/82*k1^2 - 225/41*k1*k3 + 1575/3362*k2^2 - 22050/1681*k2*k3 + 317925/3362*k3^2 k1^2 + 1/533*k1*k2 - 547/533*k1*k3 + 98/533*k2^2 - 2710/533*k2*k3 + 19880/533*k3^2 14362/38025*k1^2 + 44/7605*k1*k2 + 448/12675*k1*k3 + 314/4225*k2^2 - 24976/12675*k2*k3 + 186466/12675*k3^2 [ [ 0, 1, 82/2925, 1148/2925 ] ] t= -13 {@ (0 : 0 : 1 : -13 : 1 : 0), (-1/168 : 0 : -13/168 : 1 : 0 : 0) @} Q1A: 13*Ak2^2 - 337*Ak2*Ak3 - Ak2 + 2260*Ak3^2 - 139*Ak3 + 76 -3951/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 103/18*k1^2*k2*k3 + 25/4*k1^2*k3^2 + 1/18*k1*k2^3 - 17/9*k1*k2^2*k3 + 74/9*k1*k2*k3^2 - 119/6*k1*k3^3 + 1/36*k2^4 - 13/18*k2^3*k3 + 19/4*k2^2*k3^2 - 13/18*k2*k3^3 + 1/36*k3^4 76*k1^2 - k1*k2 - 139*k1*k3 + 13*k2^2 - 337*k2*k3 + 2260*k3^2 45/112*k1^2 + 3/1568*k1*k2 + 361/3136*k1*k3 + 127/1568*k2^2 - 6165/3136*k2*k3 + 21463/1568*k3^2 238336/91809*k1^2 - 3136/91809*k1*k2 - 435904/91809*k1*k3 + 40768/91809*k2^2 - 1056832/91809*k2*k3 + 7087360/91809*k3^2 k1^2 - 7/606*k1*k2 - 515/606*k1*k3 + 113/606*k2^2 - 961/202*k2*k3 + 19699/606*k3^2 45/112*k1^2 + 3/1568*k1*k2 + 361/3136*k1*k3 + 127/1568*k2^2 - 6165/3136*k2*k3 + 21463/1568*k3^2 [ [ 0, 1, 101/3136, 1313/3136 ] ] t= -12 {@ (0 : 0 : 1 : -12 : 1 : 0), (-1/143 : 0 : -12/143 : 1 : 0 : 0) @} Q1A: 12*Ak2^2 - 286*Ak2*Ak3 - 2*Ak2 + 1774*Ak3^2 - 116*Ak3 + 70 -839 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 97/18*k1^2*k2*k3 + 71/12*k1^2*k3^2 + 1/18*k1*k2^3 - 11/6*k1*k2^2*k3 + 137/18*k1*k2*k3^2 - 305/18*k1*k3^3 + 1/36*k2^4 - 2/3*k2^3*k3 + 73/18*k2^2*k3^2 - 2/3*k2*k3^3 + 1/36*k3^4 70*k1^2 - 2*k1*k2 - 116*k1*k3 + 12*k2^2 - 286*k2*k3 + 1774*k3^2 8770/20449*k1^2 - 64/20449*k1*k2 + 386/1859*k1*k3 + 1822/20449*k2^2 - 40082/20449*k2*k3 + 258900/20449*k3^2 715715/291848*k1^2 - 20449/291848*k1*k2 - 593021/145924*k1*k3 + 61347/145924*k2^2 - 2924207/291848*k2*k3 + 18138263/291848*k3^2 k1^2 - 21/764*k1*k2 - 513/764*k1*k3 + 145/764*k2^2 - 1693/382*k2*k3 + 21431/764*k3^2 8770/20449*k1^2 - 64/20449*k1*k2 + 386/1859*k1*k3 + 1822/20449*k2^2 - 40082/20449*k2*k3 + 258900/20449*k3^2 [ [ 0, 1, 764/20449, 9168/20449 ] ] t= -11 {@ (0 : 0 : 1 : -11 : 1 : 0), (-1/120 : 0 : -11/120 : 1 : 0 : 0) @} Q1A: 11*Ak2^2 - 239*Ak2*Ak3 - 3*Ak2 + 1362*Ak3^2 - 95*Ak3 + 64 -2807/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 91/18*k1^2*k2*k3 + 67/12*k1^2*k3^2 + 1/18*k1*k2^3 - 16/9*k1*k2^2*k3 + 7*k1*k2*k3^2 - 257/18*k1*k3^3 + 1/36*k2^4 - 11/18*k2^3*k3 + 41/12*k2^2*k3^2 - 11/18*k2*k3^3 + 1/36*k3^4 64*k1^2 - 3*k1*k2 - 95*k1*k3 + 11*k2^2 - 239*k2*k3 + 1362*k3^2 827/1800*k1^2 - 7/720*k1*k2 + 1517/4800*k1*k3 + 119/1200*k2^2 - 9373/4800*k2*k3 + 27907/2400*k3^2 921600/398161*k1^2 - 43200/398161*k1*k2 - 1368000/398161*k1*k3 + 158400/398161*k2^2 - 3441600/398161*k2*k3 + 19612800/398161*k3^2 k1^2 - 59/1262*k1*k2 - 619/1262*k1*k3 + 245/1262*k2^2 - 5185/1262*k2*k3 + 30203/1262*k3^2 827/1800*k1^2 - 7/720*k1*k2 + 1517/4800*k1*k3 + 119/1200*k2^2 - 9373/4800*k2*k3 + 27907/2400*k3^2 [ [ 0, 1, 631/14400, 6941/14400 ] ] t= -10 {@ (0 : 0 : 1 : -10 : 1 : 0), (-1/99 : 0 : -10/99 : 1 : 0 : 0) @} Q1A: 10*Ak2^2 - 196*Ak2*Ak3 - 4*Ak2 + 1018*Ak3^2 - 76*Ak3 + 58 -576 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 85/18*k1^2*k2*k3 + 21/4*k1^2*k3^2 + 1/18*k1*k2^3 - 31/18*k1*k2^2*k3 + 115/18*k1*k2*k3^2 - 71/6*k1*k3^3 + 1/36*k2^4 - 5/9*k2^3*k3 + 17/6*k2^2*k3^2 - 5/9*k2*k3^3 + 1/36*k3^4 58*k1^2 - 4*k1*k2 - 76*k1*k3 + 10*k2^2 - 196*k2*k3 + 1018*k3^2 538/1089*k1^2 - 20/1089*k1*k2 + 4/9*k1*k3 + 122/1089*k2^2 - 2116/1089*k2*k3 + 11530/1089*k3^2 31581/14450*k1^2 - 1089/7225*k1*k2 - 20691/7225*k1*k3 + 1089/2890*k2^2 - 53361/7225*k2*k3 + 554301/14450*k3^2 k1^2 - 6/85*k1*k2 - 26/85*k1*k3 + 1/5*k2^2 - 322/85*k2*k3 + 1713/85*k3^2 538/1089*k1^2 - 20/1089*k1*k2 + 4/9*k1*k3 + 122/1089*k2^2 - 2116/1089*k2*k3 + 11530/1089*k3^2 [ [ 0, 1, 170/3267, 1700/3267 ] ] t= -9 {@ (0 : 0 : 1 : -9 : 1 : 0), (-1/80 : 0 : -9/80 : 1 : 0 : 0) @} Q1A: 9*Ak2^2 - 157*Ak2*Ak3 - 5*Ak2 + 736*Ak3^2 - 59*Ak3 + 52 -1847/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 79/18*k1^2*k2*k3 + 59/12*k1^2*k3^2 + 1/18*k1*k2^3 - 5/3*k1*k2^2*k3 + 52/9*k1*k2*k3^2 - 173/18*k1*k3^3 + 1/36*k2^4 - 1/2*k2^3*k3 + 83/36*k2^2*k3^2 - 1/2*k2*k3^3 + 1/36*k3^4 52*k1^2 - 5*k1*k2 - 59*k1*k3 + 9*k2^2 - 157*k2*k3 + 736*k3^2 853/1600*k1^2 - 19/640*k1*k2 + 3829/6400*k1*k3 + 413/3200*k2^2 - 12353/6400*k2*k3 + 30519/3200*k3^2 332800/160801*k1^2 - 32000/160801*k1*k2 - 377600/160801*k1*k3 + 57600/160801*k2^2 - 1004800/160801*k2*k3 + 4710400/160801*k3^2 k1^2 - 81/802*k1*k2 - 93/802*k1*k3 + 167/802*k2^2 - 2785/802*k2*k3 + 13405/802*k3^2 853/1600*k1^2 - 19/640*k1*k2 + 3829/6400*k1*k3 + 413/3200*k2^2 - 12353/6400*k2*k3 + 30519/3200*k3^2 [ [ 0, 1, 401/6400, 3609/6400 ] ] t= -8 {@ (0 : 0 : 1 : -8 : 1 : 0), (-1/63 : 0 : -8/63 : 1 : 0 : 0) @} Q1A: 8*Ak2^2 - 122*Ak2*Ak3 - 6*Ak2 + 510*Ak3^2 - 44*Ak3 + 46 -359 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 73/18*k1^2*k2*k3 + 55/12*k1^2*k3^2 + 1/18*k1*k2^3 - 29/18*k1*k2^2*k3 + 31/6*k1*k2*k3^2 - 137/18*k1*k3^3 + 1/36*k2^4 - 4/9*k2^3*k3 + 11/6*k2^2*k3^2 - 4/9*k2*k3^3 + 1/36*k3^4 46*k1^2 - 6*k1*k2 - 44*k1*k3 + 8*k2^2 - 122*k2*k3 + 510*k3^2 2290/3969*k1^2 - 176/3969*k1*k2 + 346/441*k1*k3 + 202/1323*k2^2 - 2530/1323*k2*k3 + 3736/441*k3^2 91287/46208*k1^2 - 11907/46208*k1*k2 - 43659/23104*k1*k3 + 3969/11552*k2^2 - 242109/46208*k2*k3 + 1012095/46208*k3^2 k1^2 - 43/304*k1*k2 + 25/304*k1*k3 + 67/304*k2^2 - 481/152*k2*k3 + 4141/304*k3^2 2290/3969*k1^2 - 176/3969*k1*k2 + 346/441*k1*k3 + 202/1323*k2^2 - 2530/1323*k2*k3 + 3736/441*k3^2 [ [ 0, 1, 304/3969, 2432/3969 ] ] t= -7 {@ (0 : 0 : 1 : -7 : 1 : 0), (-1/48 : 0 : -7/48 : 1 : 0 : 0) @} Q1A: 7*Ak2^2 - 91*Ak2*Ak3 - 7*Ak2 + 334*Ak3^2 - 31*Ak3 + 40 -1071/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 67/18*k1^2*k2*k3 + 17/4*k1^2*k3^2 + 1/18*k1*k2^3 - 14/9*k1*k2^2*k3 + 41/9*k1*k2*k3^2 - 35/6*k1*k3^3 + 1/36*k2^4 - 7/18*k2^3*k3 + 17/12*k2^2*k3^2 - 7/18*k2*k3^3 + 1/36*k3^4 40*k1^2 - 7*k1*k2 - 31*k1*k3 + 7*k2^2 - 91*k2*k3 + 334*k3^2 5/8*k1^2 - 1/16*k1*k2 + 259/256*k1*k3 + 3/16*k2^2 - 483/256*k2*k3 + 945/128*k3^2 10240/5329*k1^2 - 1792/5329*k1*k2 - 7936/5329*k1*k3 + 1792/5329*k2^2 - 23296/5329*k2*k3 + 85504/5329*k3^2 k1^2 - 29/146*k1*k2 + 43/146*k1*k3 + 35/146*k2^2 - 419/146*k2*k3 + 1589/146*k3^2 5/8*k1^2 - 1/16*k1*k2 + 259/256*k1*k3 + 3/16*k2^2 - 483/256*k2*k3 + 945/128*k3^2 [ [ 0, 1, 73/768, 511/768 ] ] t= -6 {@ (0 : 0 : 1 : -6 : 1 : 0), (-1/35 : 0 : -6/35 : 1 : 0 : 0) @} Q1A: 6*Ak2^2 - 64*Ak2*Ak3 - 8*Ak2 + 202*Ak3^2 - 20*Ak3 + 34 -188 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 61/18*k1^2*k2*k3 + 47/12*k1^2*k3^2 + 1/18*k1*k2^3 - 3/2*k1*k2^2*k3 + 71/18*k1*k2*k3^2 - 77/18*k1*k3^3 + 1/36*k2^4 - 1/3*k2^3*k3 + 19/18*k2^2*k3^2 - 1/3*k2*k3^3 + 1/36*k3^4 34*k1^2 - 8*k1*k2 - 20*k1*k3 + 6*k2^2 - 64*k2*k3 + 202*k3^2 118/175*k1^2 - 4/49*k1*k2 + 1576/1225*k1*k3 + 298/1225*k2^2 - 2264/1225*k2*k3 + 7662/1225*k3^2 20825/10658*k1^2 - 2450/5329*k1*k2 - 6125/5329*k1*k3 + 3675/10658*k2^2 - 19600/5329*k2*k3 + 123725/10658*k3^2 k1^2 - 21/73*k1*k2 + 39/73*k1*k3 + 20/73*k2^2 - 190/73*k2*k3 + 622/73*k3^2 118/175*k1^2 - 4/49*k1*k2 + 1576/1225*k1*k3 + 298/1225*k2^2 - 2264/1225*k2*k3 + 7662/1225*k3^2 [ [ 0, 1, 146/1225, 876/1225 ] ] t= -5 {@ (0 : 0 : 1 : -5 : 1 : 0), (-1/24 : 0 : -5/24 : 1 : 0 : 0) @} Q1A: 5*Ak2^2 - 41*Ak2*Ak3 - 9*Ak2 + 108*Ak3^2 - 11*Ak3 + 28 -479/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 55/18*k1^2*k2*k3 + 43/12*k1^2*k3^2 + 1/18*k1*k2^3 - 13/9*k1*k2^2*k3 + 10/3*k1*k2*k3^2 - 53/18*k1*k3^3 + 1/36*k2^4 - 5/18*k2^3*k3 + 3/4*k2^2*k3^2 - 5/18*k2*k3^3 + 1/36*k3^4 28*k1^2 - 9*k1*k2 - 11*k1*k3 + 5*k2^2 - 41*k2*k3 + 108*k3^2 103/144*k1^2 - 25/288*k1*k2 + 307/192*k1*k3 + 11/32*k2^2 - 343/192*k2*k3 + 485/96*k3^2 16128/7225*k1^2 - 5184/7225*k1*k2 - 6336/7225*k1*k3 + 576/1445*k2^2 - 23616/7225*k2*k3 + 62208/7225*k3^2 k1^2 - 77/170*k1*k2 + 143/170*k1*k3 + 59/170*k2^2 - 409/170*k2*k3 + 1121/170*k3^2 103/144*k1^2 - 25/288*k1*k2 + 307/192*k1*k3 + 11/32*k2^2 - 343/192*k2*k3 + 485/96*k3^2 [ [ 0, 1, 85/576, 425/576 ] ] t= -4 {@ (0 : 0 : 1 : -4 : 1 : 0), (-1/15 : 0 : -4/15 : 1 : 0 : 0) @} Q1A: 4*Ak2^2 - 22*Ak2*Ak3 - 10*Ak2 + 46*Ak3^2 - 4*Ak3 + 22 -63 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 49/18*k1^2*k2*k3 + 13/4*k1^2*k3^2 + 1/18*k1*k2^3 - 25/18*k1*k2^2*k3 + 49/18*k1*k2*k3^2 - 11/6*k1*k3^3 + 1/36*k2^4 - 2/9*k2^3*k3 + 1/2*k2^2*k3^2 - 2/9*k2*k3^3 + 1/36*k3^4 22*k1^2 - 10*k1*k2 - 4*k1*k3 + 4*k2^2 - 22*k2*k3 + 46*k3^2 18/25*k1^2 + 46/25*k1*k3 + 14/25*k2^2 - 42/25*k2*k3 + 92/25*k3^2 275/72*k1^2 - 125/72*k1*k2 - 25/36*k1*k3 + 25/36*k2^2 - 275/72*k2*k3 + 575/72*k3^2 k1^2 - 11/12*k1*k2 + 17/12*k1*k3 + 7/12*k2^2 - 5/2*k2*k3 + 65/12*k3^2 18/25*k1^2 + 46/25*k1*k3 + 14/25*k2^2 - 42/25*k2*k3 + 92/25*k3^2 [ [ 0, 1, 4/25, 16/25 ] ] t= -3 {@ (0 : 0 : 1 : -3 : 1 : 0), (-1/8 : 0 : -3/8 : 1 : 0 : 0) @} Q1A: 3*Ak2^2 - 7*Ak2*Ak3 - 11*Ak2 + 10*Ak3^2 + Ak3 + 16 -71/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 43/18*k1^2*k2*k3 + 35/12*k1^2*k3^2 + 1/18*k1*k2^3 - 4/3*k1*k2^2*k3 + 19/9*k1*k2*k3^2 - 17/18*k1*k3^3 + 1/36*k2^4 - 1/6*k2^3*k3 + 11/36*k2^2*k3^2 - 1/6*k2*k3^3 + 1/36*k3^4 16*k1^2 - 11*k1*k2 + k1*k3 + 3*k2^2 - 7*k2*k3 + 10*k3^2 5/8*k1^2 + 11/16*k1*k2 + 79/64*k1*k3 + 19/16*k2^2 - 95/64*k2*k3 + 57/32*k3^2 1024*k1^2 - 704*k1*k2 + 64*k1*k3 + 192*k2^2 - 448*k2*k3 + 640*k3^2 k1^2 + 51/2*k1*k2 - 45/2*k1*k3 - 29/2*k2^2 + 49/2*k2*k3 - 67/2*k3^2 5/8*k1^2 + 11/16*k1*k2 + 79/64*k1*k3 + 19/16*k2^2 - 95/64*k2*k3 + 57/32*k3^2 [ [ 0, 1, -1/64, -3/64 ] ] t= -1 {@ (0 : 0 : 1 : -1 : 1 : 0), (1 : 0 : 1 : 0 : 0 : 0) @} Q1A: Ak2^2 + 11*Ak2*Ak3 - 13*Ak2 - 8*Ak3^2 + 5*Ak3 + 4 153/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 31/18*k1^2*k2*k3 + 9/4*k1^2*k3^2 + 1/18*k1*k2^3 - 11/9*k1*k2^2*k3 + 8/9*k1*k2*k3^2 + 1/6*k1*k3^3 + 1/36*k2^4 - 1/18*k2^3*k3 + 1/12*k2^2*k3^2 - 1/18*k2*k3^3 + 1/36*k3^4 4*k1^2 - 13*k1*k2 + 5*k1*k3 + k2^2 + 11*k2*k3 - 8*k3^2 36*k1^2 + 90*k1*k2 - 99*k1*k3 + 18*k2^2 - 9*k2*k3 - 18*k3^2 4/1521*k1^2 - 1/117*k1*k2 + 5/1521*k1*k3 + 1/1521*k2^2 + 11/1521*k2*k3 - 8/1521*k3^2 k1^2 + 3/26*k1*k2 + 23/26*k1*k3 - 5/26*k2^2 - 5/26*k2*k3 + 9/26*k3^2 36*k1^2 + 90*k1*k2 - 99*k1*k3 + 18*k2^2 - 9*k2*k3 - 18*k3^2 [ [ 0, 1, -39, -39 ] ] t= 1 {@ (0 : 0 : 1 : 1 : 1 : 0), (-1 : 0 : 1 : 0 : 0 : 0) @} Q1A: -Ak2^2 + 13*Ak2*Ak3 - 15*Ak2 + 6*Ak3^2 + Ak3 - 8 193/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 19/18*k1^2*k2*k3 + 19/12*k1^2*k3^2 + 1/18*k1*k2^3 - 10/9*k1*k2^2*k3 - 1/3*k1*k2*k3^2 + 7/18*k1*k3^3 + 1/36*k2^4 + 1/18*k2^3*k3 + 1/12*k2^2*k3^2 + 1/18*k2*k3^3 + 1/36*k3^4 -8*k1^2 - 15*k1*k2 + k1*k3 - k2^2 + 13*k2*k3 + 6*k3^2 112*k1^2 + 40*k1*k2 + 99*k1*k3 - 24*k2^2 - 3*k2*k3 + 18*k3^2 -8/841*k1^2 - 15/841*k1*k2 + 1/841*k1*k3 - 1/841*k2^2 + 13/841*k2*k3 + 6/841*k3^2 k1^2 - 49/58*k1*k2 + 103/58*k1*k3 - 17/58*k2^2 + 25/58*k2*k3 + 25/58*k3^2 112*k1^2 + 40*k1*k2 + 99*k1*k3 - 24*k2^2 - 3*k2*k3 + 18*k3^2 [ [ 0, 1, -29, 29 ] ] t= 4 {@ (0 : 0 : 1 : 4 : 1 : 0), (-1/15 : 0 : 4/15 : 1 : 0 : 0) @} Q1A: -4*Ak2^2 - 14*Ak2*Ak3 - 18*Ak2 - 18*Ak3^2 - 20*Ak3 - 26 -23 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 - 1/18*k1^2*k2*k3 + 7/12*k1^2*k3^2 + 1/18*k1*k2^3 - 17/18*k1*k2^2*k3 - 13/6*k1*k2*k3^2 - 17/18*k1*k3^3 + 1/36*k2^4 + 2/9*k2^3*k3 + 1/2*k2^2*k3^2 + 2/9*k2*k3^3 + 1/36*k3^4 -26*k1^2 - 18*k1*k2 - 20*k1*k3 - 4*k2^2 - 14*k2*k3 - 18*k3^2 -254/225*k1^2 - 64/45*k1*k2 - 158/75*k1*k3 - 18/25*k2^2 - 118/75*k2*k3 - 116/75*k3^2 -2925/2888*k1^2 - 2025/2888*k1*k2 - 1125/1444*k1*k3 - 225/1444*k2^2 - 1575/2888*k2*k3 - 2025/2888*k3^2 k1^2 + 83/76*k1*k2 + 103/76*k1*k3 + 25/76*k2^2 + 35/38*k2*k3 + 79/76*k3^2 -254/225*k1^2 - 64/45*k1*k2 - 158/75*k1*k3 - 18/25*k2^2 - 118/75*k2*k3 - 116/75*k3^2 [ [ 0, 1, 76/225, -304/225 ] ] t= 5 {@ (0 : 0 : 1 : 5 : 1 : 0), (-1/24 : 0 : 5/24 : 1 : 0 : 0) @} Q1A: -5*Ak2^2 - 31*Ak2*Ak3 - 19*Ak2 - 62*Ak3^2 - 31*Ak3 - 32 -279/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 5/18*k1^2*k2*k3 + 1/4*k1^2*k3^2 + 1/18*k1*k2^3 - 8/9*k1*k2^2*k3 - 25/9*k1*k2*k3^2 - 11/6*k1*k3^3 + 1/36*k2^4 + 5/18*k2^3*k3 + 3/4*k2^2*k3^2 + 5/18*k2*k3^3 + 1/36*k3^4 -32*k1^2 - 19*k1*k2 - 31*k1*k3 - 5*k2^2 - 31*k2*k3 - 62*k3^2 -9/8*k1^2 - 15/16*k1*k2 - 161/64*k1*k3 - 7/16*k2^2 - 111/64*k2*k3 - 95/32*k3^2 -2048/2025*k1^2 - 1216/2025*k1*k2 - 1984/2025*k1*k3 - 64/405*k2^2 - 1984/2025*k2*k3 - 3968/2025*k3^2 k1^2 + 71/90*k1*k2 + 151/90*k1*k3 + 23/90*k2^2 + 13/10*k2*k3 + 217/90*k3^2 -9/8*k1^2 - 15/16*k1*k2 - 161/64*k1*k3 - 7/16*k2^2 - 111/64*k2*k3 - 95/32*k3^2 [ [ 0, 1, 15/64, -75/64 ] ] t= 6 {@ (0 : 0 : 1 : 6 : 1 : 0), (-1/35 : 0 : 6/35 : 1 : 0 : 0) @} Q1A: -6*Ak2^2 - 52*Ak2*Ak3 - 20*Ak2 - 134*Ak3^2 - 44*Ak3 - 38 -128 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 11/18*k1^2*k2*k3 - 1/12*k1^2*k3^2 + 1/18*k1*k2^3 - 5/6*k1*k2^2*k3 - 61/18*k1*k2*k3^2 - 53/18*k1*k3^3 + 1/36*k2^4 + 1/3*k2^3*k3 + 19/18*k2^2*k3^2 + 1/3*k2*k3^3 + 1/36*k3^4 -38*k1^2 - 20*k1*k2 - 44*k1*k3 - 6*k2^2 - 52*k2*k3 - 134*k3^2 -1238/1225*k1^2 - 164/245*k1*k2 - 3116/1225*k1*k3 - 374/1225*k2^2 - 2228/1225*k2*k3 - 5142/1225*k3^2 -23275/21218*k1^2 - 6125/10609*k1*k2 - 13475/10609*k1*k3 - 3675/21218*k2^2 - 15925/10609*k2*k3 - 82075/21218*k3^2 k1^2 + 66/103*k1*k2 + 198/103*k1*k3 + 23/103*k2^2 + 170/103*k2*k3 + 415/103*k3^2 -1238/1225*k1^2 - 164/245*k1*k2 - 3116/1225*k1*k3 - 374/1225*k2^2 - 2228/1225*k2*k3 - 5142/1225*k3^2 [ [ 0, 1, 206/1225, -1236/1225 ] ] t= 7 {@ (0 : 0 : 1 : 7 : 1 : 0), (-1/48 : 0 : 7/48 : 1 : 0 : 0) @} Q1A: -7*Ak2^2 - 77*Ak2*Ak3 - 21*Ak2 - 240*Ak3^2 - 59*Ak3 - 44 -791/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 17/18*k1^2*k2*k3 - 5/12*k1^2*k3^2 + 1/18*k1*k2^3 - 7/9*k1*k2^2*k3 - 4*k1*k2*k3^2 - 77/18*k1*k3^3 + 1/36*k2^4 + 7/18*k2^3*k3 + 17/12*k2^2*k3^2 + 7/18*k2*k3^3 + 1/36*k3^4 -44*k1^2 - 21*k1*k2 - 59*k1*k3 - 7*k2^2 - 77*k2*k3 - 240*k3^2 -515/576*k1^2 - 583/1152*k1*k2 - 1897/768*k1*k3 - 89/384*k2^2 - 1435/768*k2*k3 - 2051/384*k3^2 -101376/83521*k1^2 - 48384/83521*k1*k2 - 135936/83521*k1*k3 - 16128/83521*k2^2 - 177408/83521*k2*k3 - 552960/83521*k3^2 k1^2 + 319/578*k1*k2 + 1235/578*k1*k3 + 7/34*k2^2 + 1151/578*k2*k3 + 3437/578*k3^2 -515/576*k1^2 - 583/1152*k1*k2 - 1897/768*k1*k3 - 89/384*k2^2 - 1435/768*k2*k3 - 2051/384*k3^2 [ [ 0, 1, 289/2304, -2023/2304 ] ] t= 8 {@ (0 : 0 : 1 : 8 : 1 : 0), (-1/63 : 0 : 8/63 : 1 : 0 : 0) @} Q1A: -8*Ak2^2 - 106*Ak2*Ak3 - 22*Ak2 - 386*Ak3^2 - 76*Ak3 - 50 -279 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 23/18*k1^2*k2*k3 - 3/4*k1^2*k3^2 + 1/18*k1*k2^3 - 13/18*k1*k2^2*k3 - 83/18*k1*k2*k3^2 - 35/6*k1*k3^3 + 1/36*k2^4 + 4/9*k2^3*k3 + 11/6*k2^2*k3^2 + 4/9*k2*k3^3 + 1/36*k3^4 -50*k1^2 - 22*k1*k2 - 76*k1*k3 - 8*k2^2 - 106*k2*k3 - 386*k3^2 -50/63*k1^2 - 176/441*k1*k2 - 1046/441*k1*k3 - 82/441*k2^2 - 838/441*k2*k3 - 2840/441*k3^2 -11025/8192*k1^2 - 4851/8192*k1*k2 - 8379/4096*k1*k3 - 441/2048*k2^2 - 23373/8192*k2*k3 - 85113/8192*k3^2 k1^2 + 63/128*k1*k2 + 299/128*k1*k3 + 25/128*k2^2 + 149/64*k2*k3 + 1047/128*k3^2 -50/63*k1^2 - 176/441*k1*k2 - 1046/441*k1*k3 - 82/441*k2^2 - 838/441*k2*k3 - 2840/441*k3^2 [ [ 0, 1, 128/1323, -1024/1323 ] ] t= 9 {@ (0 : 0 : 1 : 9 : 1 : 0), (-1/80 : 0 : 9/80 : 1 : 0 : 0) @} Q1A: -9*Ak2^2 - 139*Ak2*Ak3 - 23*Ak2 - 578*Ak3^2 - 95*Ak3 - 56 -1487/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 29/18*k1^2*k2*k3 - 13/12*k1^2*k3^2 + 1/18*k1*k2^3 - 2/3*k1*k2^2*k3 - 47/9*k1*k2*k3^2 - 137/18*k1*k3^3 + 1/36*k2^4 + 1/2*k2^3*k3 + 83/36*k2^2*k3^2 + 1/2*k2*k3^3 + 1/36*k3^4 -56*k1^2 - 23*k1*k2 - 95*k1*k3 - 9*k2^2 - 139*k2*k3 - 578*k3^2 -71/100*k1^2 - 13/40*k1*k2 - 14549/6400*k1*k3 - 31/200*k2^2 - 12299/6400*k2*k3 - 24039/3200*k3^2 -358400/241081*k1^2 - 147200/241081*k1*k2 - 608000/241081*k1*k3 - 57600/241081*k2^2 - 889600/241081*k2*k3 - 3699200/241081*k3^2 k1^2 + 441/982*k1*k2 + 2481/982*k1*k3 + 185/982*k2^2 + 2615/982*k2*k3 + 10543/982*k3^2 -71/100*k1^2 - 13/40*k1*k2 - 14549/6400*k1*k3 - 31/200*k2^2 - 12299/6400*k2*k3 - 24039/3200*k3^2 [ [ 0, 1, 491/6400, -4419/6400 ] ] t= 10 {@ (0 : 0 : 1 : 10 : 1 : 0), (-1/99 : 0 : 10/99 : 1 : 0 : 0) @} Q1A: -10*Ak2^2 - 176*Ak2*Ak3 - 24*Ak2 - 822*Ak3^2 - 116*Ak3 - 62 -476 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 35/18*k1^2*k2*k3 - 17/12*k1^2*k3^2 + 1/18*k1*k2^3 - 11/18*k1*k2^2*k3 - 35/6*k1*k2*k3^2 - 173/18*k1*k3^3 + 1/36*k2^4 + 5/9*k2^3*k3 + 17/6*k2^2*k3^2 + 5/9*k2*k3^3 + 1/36*k3^4 -62*k1^2 - 24*k1*k2 - 116*k1*k3 - 10*k2^2 - 176*k2*k3 - 822*k3^2 -6278/9801*k1^2 - 2660/9801*k1*k2 - 24/11*k1*k3 - 434/3267*k2^2 - 6328/3267*k2*k3 - 3110/363*k3^2 -303831/186050*k1^2 - 58806/93025*k1*k2 - 284229/93025*k1*k3 - 9801/37210*k2^2 - 431244/93025*k2*k3 - 4028211/186050*k3^2 k1^2 + 127/305*k1*k2 + 827/305*k1*k3 + 56/305*k2^2 + 914/305*k2*k3 + 4154/305*k3^2 -6278/9801*k1^2 - 2660/9801*k1*k2 - 24/11*k1*k3 - 434/3267*k2^2 - 6328/3267*k2*k3 - 3110/363*k3^2 [ [ 0, 1, 610/9801, -6100/9801 ] ] t= 11 {@ (0 : 0 : 1 : 11 : 1 : 0), (-1/120 : 0 : 11/120 : 1 : 0 : 0) @} Q1A: -11*Ak2^2 - 217*Ak2*Ak3 - 25*Ak2 - 1124*Ak3^2 - 139*Ak3 - 68 -2367/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 41/18*k1^2*k2*k3 - 7/4*k1^2*k3^2 + 1/18*k1*k2^3 - 5/9*k1*k2^2*k3 - 58/9*k1*k2*k3^2 - 71/6*k1*k3^3 + 1/36*k2^4 + 11/18*k2^3*k3 + 41/12*k2^2*k3^2 + 11/18*k2*k3^3 + 1/36*k3^4 -68*k1^2 - 25*k1*k2 - 139*k1*k3 - 11*k2^2 - 217*k2*k3 - 1124*k3^2 -233/400*k1^2 - 37/160*k1*k2 - 3359/1600*k1*k3 - 93/800*k2^2 - 3117/1600*k2*k3 - 7689/800*k3^2 -108800/61009*k1^2 - 40000/61009*k1*k2 - 222400/61009*k1*k3 - 17600/61009*k2^2 - 347200/61009*k2*k3 - 1798400/61009*k3^2 k1^2 + 193/494*k1*k2 + 1429/494*k1*k3 + 89/494*k2^2 + 1645/494*k2*k3 + 8315/494*k3^2 -233/400*k1^2 - 37/160*k1*k2 - 3359/1600*k1*k3 - 93/800*k2^2 - 3117/1600*k2*k3 - 7689/800*k3^2 [ [ 0, 1, 247/4800, -2717/4800 ] ] t= 12 {@ (0 : 0 : 1 : 12 : 1 : 0), (-1/143 : 0 : 12/143 : 1 : 0 : 0) @} Q1A: -12*Ak2^2 - 262*Ak2*Ak3 - 26*Ak2 - 1490*Ak3^2 - 164*Ak3 - 74 -719 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 47/18*k1^2*k2*k3 - 25/12*k1^2*k3^2 + 1/18*k1*k2^3 - 1/2*k1*k2^2*k3 - 127/18*k1*k2*k3^2 - 257/18*k1*k3^3 + 1/36*k2^4 + 2/3*k2^3*k3 + 73/18*k2^2*k3^2 + 2/3*k2*k3^3 + 1/36*k3^4 -74*k1^2 - 26*k1*k2 - 164*k1*k3 - 12*k2^2 - 262*k2*k3 - 1490*k3^2 -10910/20449*k1^2 - 4096/20449*k1*k2 - 3766/1859*k1*k3 - 2114/20449*k2^2 - 40010/20449*k2*k3 - 217716/20449*k3^2 -4477/2312*k1^2 - 1573/2312*k1*k2 - 4961/1156*k1*k3 - 363/1156*k2^2 - 15851/2312*k2*k3 - 90145/2312*k3^2 k1^2 + 327/884*k1*k2 + 2715/884*k1*k3 + 157/884*k2^2 + 1619/442*k2*k3 + 18011/884*k3^2 -10910/20449*k1^2 - 4096/20449*k1*k2 - 3766/1859*k1*k3 - 2114/20449*k2^2 - 40010/20449*k2*k3 - 217716/20449*k3^2 [ [ 0, 1, 68/1573, -816/1573 ] ] t= 13 {@ (0 : 0 : 1 : 13 : 1 : 0), (-1/168 : 0 : 13/168 : 1 : 0 : 0) @} Q1A: -13*Ak2^2 - 311*Ak2*Ak3 - 27*Ak2 - 1926*Ak3^2 - 191*Ak3 - 80 -3431/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 53/18*k1^2*k2*k3 - 29/12*k1^2*k3^2 + 1/18*k1*k2^3 - 4/9*k1*k2^2*k3 - 23/3*k1*k2*k3^2 - 305/18*k1*k3^3 + 1/36*k2^4 + 13/18*k2^3*k3 + 19/4*k2^2*k3^2 + 13/18*k2*k3^3 + 1/36*k3^4 -80*k1^2 - 27*k1*k2 - 191*k1*k3 - 13*k2^2 - 311*k2*k3 - 1926*k3^2 -1735/3528*k1^2 - 1241/7056*k1*k2 - 18443/9408*k1*k3 - 73/784*k2^2 - 18469/9408*k2*k3 - 54925/4704*k3^2 -2257920/1079521*k1^2 - 762048/1079521*k1*k2 - 5390784/1079521*k1*k3 - 366912/1079521*k2^2 - 8777664/1079521*k2*k3 - 54359424/1079521*k3^2 k1^2 + 733/2078*k1*k2 + 6749/2078*k1*k3 + 365/2078*k2^2 + 8303/2078*k2*k3 + 50387/2078*k3^2 -1735/3528*k1^2 - 1241/7056*k1*k2 - 18443/9408*k1*k3 - 73/784*k2^2 - 18469/9408*k2*k3 - 54925/4704*k3^2 [ [ 0, 1, 1039/28224, -13507/28224 ] ] t= 14 {@ (0 : 0 : 1 : 14 : 1 : 0), (-1/195 : 0 : 14/195 : 1 : 0 : 0) @} Q1A: -14*Ak2^2 - 364*Ak2*Ak3 - 28*Ak2 - 2438*Ak3^2 - 220*Ak3 - 86 -1008 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 59/18*k1^2*k2*k3 - 11/4*k1^2*k3^2 + 1/18*k1*k2^3 - 7/18*k1*k2^2*k3 - 149/18*k1*k2*k3^2 - 119/6*k1*k3^3 + 1/36*k2^4 + 7/9*k2^3*k3 + 11/2*k2^2*k3^2 + 7/9*k2*k3^3 + 1/36*k3^4 -86*k1^2 - 28*k1*k2 - 220*k1*k3 - 14*k2^2 - 364*k2*k3 - 2438*k3^2 -1926/4225*k1^2 - 132/845*k1*k2 - 8036/4225*k1*k3 - 358/4225*k2^2 - 8316/4225*k2*k3 - 53662/4225*k3^2 -181675/80802*k1^2 - 29575/40401*k1*k2 - 232375/40401*k1*k3 - 29575/80802*k2^2 - 384475/40401*k2*k3 - 5150275/80802*k3^2 k1^2 + 68/201*k1*k2 + 688/201*k1*k3 + 35/201*k2^2 + 290/67*k2*k3 + 5719/201*k3^2 -1926/4225*k1^2 - 132/845*k1*k2 - 8036/4225*k1*k3 - 358/4225*k2^2 - 8316/4225*k2*k3 - 53662/4225*k3^2 [ [ 0, 1, 134/4225, -1876/4225 ] ] t= 15 {@ (0 : 0 : 1 : 15 : 1 : 0), (-1/224 : 0 : 15/224 : 1 : 0 : 0) @} Q1A: -15*Ak2^2 - 421*Ak2*Ak3 - 29*Ak2 - 3032*Ak3^2 - 251*Ak3 - 92 -4679/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 65/18*k1^2*k2*k3 - 37/12*k1^2*k3^2 + 1/18*k1*k2^3 - 1/3*k1*k2^2*k3 - 80/9*k1*k2*k3^2 - 413/18*k1*k3^3 + 1/36*k2^4 + 5/6*k2^3*k3 + 227/36*k2^2*k3^2 + 5/6*k2*k3^3 + 1/36*k3^4 -92*k1^2 - 29*k1*k2 - 251*k1*k3 - 15*k2^2 - 421*k2*k3 - 3032*k3^2 -761/1792*k1^2 - 3515/25088*k1*k2 - 92819/50176*k1*k3 - 1951/25088*k2^2 - 98969/50176*k2*k3 - 344265/25088*k3^2 -4616192/1918225*k1^2 - 1455104/1918225*k1*k2 - 12594176/1918225*k1*k3 - 150528/383645*k2^2 - 21124096/1918225*k2*k3 - 152133632/1918225*k3^2 k1^2 + 903/2770*k1*k2 + 9963/2770*k1*k3 + 479/2770*k2^2 + 12911/2770*k2*k3 + 91381/2770*k3^2 -761/1792*k1^2 - 3515/25088*k1*k2 - 92819/50176*k1*k3 - 1951/25088*k2^2 - 98969/50176*k2*k3 - 344265/25088*k3^2 [ [ 0, 1, 1385/50176, -20775/50176 ] ] t= 16 {@ (0 : 0 : 1 : 16 : 1 : 0), (-1/255 : 0 : 16/255 : 1 : 0 : 0) @} Q1A: -16*Ak2^2 - 482*Ak2*Ak3 - 30*Ak2 - 3714*Ak3^2 - 284*Ak3 - 98 -1343 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 71/18*k1^2*k2*k3 - 41/12*k1^2*k3^2 + 1/18*k1*k2^3 - 5/18*k1*k2^2*k3 - 19/2*k1*k2*k3^2 - 473/18*k1*k3^3 + 1/36*k2^4 + 8/9*k2^3*k3 + 43/6*k2^2*k3^2 + 8/9*k2*k3^3 + 1/36*k3^4 -98*k1^2 - 30*k1*k2 - 284*k1*k3 - 16*k2^2 - 482*k2*k3 - 3714*k3^2 -25838/65025*k1^2 - 1648/13005*k1*k2 - 39082/21675*k1*k3 - 1558/21675*k2^2 - 42826/21675*k2*k3 - 319504/21675*k3^2 -3186225/1241888*k1^2 - 975375/1241888*k1*k2 - 4616775/620944*k1*k3 - 65025/155236*k2^2 - 15671025/1241888*k2*k3 - 120751425/1241888*k3^2 k1^2 + 497/1576*k1*k2 + 5941/1576*k1*k3 + 271/1576*k2^2 + 3935/788*k2*k3 + 59665/1576*k3^2 -25838/65025*k1^2 - 1648/13005*k1*k2 - 39082/21675*k1*k3 - 1558/21675*k2^2 - 42826/21675*k2*k3 - 319504/21675*k3^2 [ [ 0, 1, 1576/65025, -25216/65025 ] ] t= 17 {@ (0 : 0 : 1 : 17 : 1 : 0), (-1/288 : 0 : 17/288 : 1 : 0 : 0) @} Q1A: -17*Ak2^2 - 547*Ak2*Ak3 - 31*Ak2 - 4490*Ak3^2 - 319*Ak3 - 104 -6111/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 77/18*k1^2*k2*k3 - 15/4*k1^2*k3^2 + 1/18*k1*k2^3 - 2/9*k1*k2^2*k3 - 91/9*k1*k2*k3^2 - 179/6*k1*k3^3 + 1/36*k2^4 + 17/18*k2^3*k3 + 97/12*k2^2*k3^2 + 17/18*k2*k3^3 + 1/36*k3^4 -104*k1^2 - 31*k1*k2 - 319*k1*k3 - 17*k2^2 - 547*k2*k3 - 4490*k3^2 -215/576*k1^2 - 133/1152*k1*k2 - 16229/9216*k1*k3 - 77/1152*k2^2 - 18235/9216*k2*k3 - 72607/4608*k3^2 -958464/351649*k1^2 - 285696/351649*k1*k2 - 2939904/351649*k1*k3 - 156672/351649*k2^2 - 5041152/351649*k2*k3 - 41379840/351649*k3^2 k1^2 + 363/1186*k1*k2 + 4675/1186*k1*k3 + 203/1186*k2^2 + 6317/1186*k2*k3 + 51069/1186*k3^2 -215/576*k1^2 - 133/1152*k1*k2 - 16229/9216*k1*k3 - 77/1152*k2^2 - 18235/9216*k2*k3 - 72607/4608*k3^2 [ [ 0, 1, 593/27648, -10081/27648 ] ] t= 18 {@ (0 : 0 : 1 : 18 : 1 : 0), (-1/323 : 0 : 18/323 : 1 : 0 : 0) @} Q1A: -18*Ak2^2 - 616*Ak2*Ak3 - 32*Ak2 - 5366*Ak3^2 - 356*Ak3 - 110 -1724 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 83/18*k1^2*k2*k3 - 49/12*k1^2*k3^2 + 1/18*k1*k2^3 - 1/6*k1*k2^2*k3 - 193/18*k1*k2*k3^2 - 605/18*k1*k3^3 + 1/36*k2^4 + k2^3*k3 + 163/18*k2^2*k3^2 + k2*k3^3 + 1/36*k3^4 -110*k1^2 - 32*k1*k2 - 356*k1*k3 - 18*k2^2 - 616*k2*k3 - 5366*k3^2 -36710/104329*k1^2 - 11044/104329*k1*k2 - 179744/104329*k1*k3 - 6518/104329*k2^2 - 206672/104329*k2*k3 - 1749690/104329*k3^2 -5738095/1988018*k1^2 - 834632/994009*k1*k2 - 9285281/994009*k1*k3 - 938961/1988018*k2^2 - 16066666/994009*k2*k3 - 279914707/1988018*k3^2 k1^2 + 297/997*k1*k2 + 4101/997*k1*k3 + 170/997*k2^2 + 5642/997*k2*k3 + 48448/997*k3^2 -36710/104329*k1^2 - 11044/104329*k1*k2 - 179744/104329*k1*k3 - 6518/104329*k2^2 - 206672/104329*k2*k3 - 1749690/104329*k3^2 [ [ 0, 1, 1994/104329, -35892/104329 ] ] t= 19 {@ (0 : 0 : 1 : 19 : 1 : 0), (-1/360 : 0 : 19/360 : 1 : 0 : 0) @} Q1A: -19*Ak2^2 - 689*Ak2*Ak3 - 33*Ak2 - 6348*Ak3^2 - 395*Ak3 - 116 -7727/4 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 89/18*k1^2*k2*k3 - 53/12*k1^2*k3^2 + 1/18*k1*k2^3 - 1/9*k1*k2^2*k3 - 34/3*k1*k2*k3^2 - 677/18*k1*k3^3 + 1/36*k2^4 + 19/18*k2^3*k3 + 121/12*k2^2*k3^2 + 19/18*k2*k3^3 + 1/36*k3^4 -116*k1^2 - 33*k1*k2 - 395*k1*k3 - 19*k2^2 - 689*k2*k3 - 6348*k3^2 -10781/32400*k1^2 - 253/2592*k1*k2 - 24311/14400*k1*k3 - 1267/21600*k2^2 - 85663/43200*k2*k3 - 128041/7200*k3^2 -15033600/4932841*k1^2 - 4276800/4932841*k1*k2 - 51192000/4932841*k1*k3 - 2462400/4932841*k2^2 - 89294400/4932841*k2*k3 - 822700800/4932841*k3^2 k1^2 + 1291/4442*k1*k2 + 19031/4442*k1*k3 + 755/4442*k2^2 + 26615/4442*k2*k3 + 241913/4442*k3^2 -10781/32400*k1^2 - 253/2592*k1*k2 - 24311/14400*k1*k3 - 1267/21600*k2^2 - 85663/43200*k2*k3 - 128041/7200*k3^2 [ [ 0, 1, 2221/129600, -42199/129600 ] ] t= 20 {@ (0 : 0 : 1 : 20 : 1 : 0), (-1/399 : 0 : 20/399 : 1 : 0 : 0) @} Q1A: -20*Ak2^2 - 766*Ak2*Ak3 - 34*Ak2 - 7442*Ak3^2 - 436*Ak3 - 122 -2151 Curve over Rational Field defined by k1^4 + 1/3*k1^3*k2 + 19/9*k1^3*k3 + 13/36*k1^2*k2^2 + 95/18*k1^2*k2*k3 - 19/4*k1^2*k3^2 + 1/18*k1*k2^3 - 1/18*k1*k2^2*k3 - 215/18*k1*k2*k3^2 - 251/6*k1*k3^3 + 1/36*k2^4 + 10/9*k2^3*k3 + 67/6*k2^2*k3^2 + 10/9*k2*k3^3 + 1/36*k3^4 -122*k1^2 - 34*k1*k2 - 436*k1*k3 - 20*k2^2 - 766*k2*k3 - 7442*k3^2 -5582/17689*k1^2 - 1600/17689*k1*k2 - 29306/17689*k1*k3 - 978/17689*k2^2 - 35106/17689*k2*k3 - 332460/17689*k3^2 -1079029/336200*k1^2 - 300713/336200*k1*k2 - 1928101/168100*k1*k3 - 17689/33620*k2^2 - 6774887/336200*k2*k3 - 65820769/336200*k3^2 k1^2 + 233/820*k1*k2 + 3653/820*k1*k3 + 139/820*k2^2 + 2593/410*k2*k3 + 49741/820*k3^2 -5582/17689*k1^2 - 1600/17689*k1*k2 - 29306/17689*k1*k3 - 978/17689*k2^2 - 35106/17689*k2*k3 - 332460/17689*k3^2 [ [ 0, 1, 820/53067, -16400/53067 ] ] > > //Now we pass to the function field. > > Qt:=RationalFunctionField(Rationals()); > P:=PolynomialRing(Qt); > f:=X*(X-1)*(X-2)*(X+1)*(X+3); > F:=HyperellipticCurve(f); > J:=Jacobian(F); > Km:=KummerSurface(J); > KJ:=Scheme(Km); > R:=Parent(DefiningPolynomial(Km)); > V:=k1+k2+t*k3+k4; > P2t:=ProjectiveSpace(Qt,2); > mp:=homCoordinateRing(P2t)|[U,V,W]>; > > // A routine that interpolates a rational function with numerator and > // denominator or degree at most d through the points described by L > // (your should really apply this to an overdetermined system) > > function ratint(L,d) function> M:=[]; function> for c in L do function|for> t0:=c[1]; function|for> C:=c[2]; function|for> M cat:=[[t0^i:i in [0..d]] cat [-C*t0^i:i in [0..d]]]; function|for> end for; function> V:=Kernel(Transpose(Matrix(M))); function> if Dimension(V) gt 0 then function|if> print Basis(V); function|if> v:=Eltseq(Basis(V)[1]); function|if> return (Parent(x)!v[1..d+1])/(Parent(x)!v[d+2..2*d+2]); function|if> else function|if> error "no solution"; function|if> end if; function> end function; > > //We now solve Q1,Q2,Q3 over Qt via interpolation: > mons:=MonomialsOfDegree(Parent(u),2); > cf:=[:l in L]; > Q1t:=&+[ratint([:c in cf],5)*mp(mons[i]):i in [1..#mons]]; [ (1 0 0 0 0 0 1 0 0 0 0 0), (0 1 0 0 0 0 0 1 0 0 0 0), (0 0 1 0 0 0 0 0 1 0 0 0), (0 0 0 1 0 0 0 0 0 1 0 0), (0 0 0 0 1 0 0 0 0 0 1 0), (0 0 0 0 0 1 0 0 0 0 0 1) ] [ (1 0 0 0 0 -243/102400000 41/40 -597/1600 -1791/64000 -5373/2560000 -16119/102400000 -729/51200000), (0 1 0 0 0 -81/2560000 0 41/40 -597/1600 -1791/64000 -5373/2560000 -243/1280000), (0 0 1 0 0 -27/64000 0 0 41/40 -597/1600 -1791/64000 -81/32000), (0 0 0 1 0 -9/1600 0 0 0 41/40 -597/1600 -27/800), ( 0 0 0 0 1 -3/40 0 0 0 0 41/40 -9/20) ] [ (1 0 0 0 0 -161051/537824 -41/7 -199/98 -2189/1372 -24079/19208 -264869/268912 131769/134456), (0 1 0 0 0 -14641/38416 0 -41/7 -199/98 -2189/1372 -24079/19208 11979/9604), (0 0 1 0 0 -1331/2744 0 0 -41/7 -199/98 -2189/1372 1089/686), (0 0 0 1 0 -121/196 0 0 0 -41/7 -199/98 99/49), ( 0 0 0 0 1 -11/14 0 0 0 0 -41/7 18/7) ] [ ( 1 0 0 0 0 243 -41 141 -423 1269 -3807 1458), ( 0 1 0 0 0 -81 0 -41 141 -423 1269 -486), ( 0 0 1 0 0 27 0 0 -41 141 -423 162), ( 0 0 0 1 0 -9 0 0 0 -41 141 -54), ( 0 0 0 0 1 3 0 0 0 0 -41 18) ] [ (1 0 0 0 0 1/537824 -41/7 293/98 -293/1372 293/19208 -293/268912 9/134456), (0 1 0 0 0 -1/38416 0 -41/7 293/98 -293/1372 293/19208 -9/9604), (0 0 1 0 0 1/2744 0 0 -41/7 293/98 -293/1372 9/686), ( 0 0 0 1 0 -1/196 0 0 0 -41/7 293/98 -9/49), ( 0 0 0 0 1 1/14 0 0 0 0 -41/7 18/7) ] [ ( 0 1 0 0 0 0 -82 36 0 0 0 0), ( 0 0 1 0 0 0 0 -82 36 0 0 0), ( 0 0 0 1 0 0 0 0 -82 36 0 0), ( 0 0 0 0 1 0 0 0 0 -82 36 0), ( 0 0 0 0 0 1 0 0 0 0 -82 36) ] > > cf:=[:l in L]; > Q2t:=&+[ratint([:c in cf],3)*mp(mons[i]):i in [1..#mons]]; [ ( 1 0 0 -27/64000 -41/20 597/800 1791/32000 81/16000), ( 0 1 0 -9/1600 0 -41/20 597/800 27/400), ( 0 0 1 -3/40 0 0 -41/20 9/10) ] [ ( 1 0 0 1/5832 82/9 -365/81 365/1458 -1/81), ( 0 1 0 -1/324 0 82/9 -365/81 2/9), ( 0 0 1 1/18 0 0 82/9 -4) ] [ ( 1 0 -1345/10404 11/867 -82/51 2287/2601 -22/289 0), ( 0 1 11/102 -2/17 0 -82/51 12/17 0) ] [ ( 1 0 0 1/2744 82/7 -293/49 293/686 -9/343), ( 0 1 0 -1/196 0 82/7 -293/49 18/49), ( 0 0 1 1/14 0 0 82/7 -36/7) ] [ ( 1 0 -5/4 -1/4 -41 -23 18 0), ( 0 1 -1 -1/4 0 -41 18 0) ] [ ( 1 0 -29/196 1/98 -82/7 293/49 -18/49 0), ( 0 1 1/14 -1/7 0 -82/7 36/7 0) ] > > cf:=[:l in L]; > Q3t:=&+[ratint([:c in cf],3)*mp(mons[i]):i in [1..#mons]]; [ ( 1 0 0 27 -41 141 -423 162), ( 0 1 0 -9 0 -41 141 -54), ( 0 0 1 3 0 0 -41 18) ] [ ( 1 0 0 1/2744 -41/7 293/98 -293/1372 9/686), ( 0 1 0 -1/196 0 -41/7 293/98 -9/49), ( 0 0 1 1/14 0 0 -41/7 18/7) ] [ ( 1 0 -1/2 -1/8 41/2 5/4 -9/2 0), ( 0 1 -1/2 -1/4 0 41/2 -9 0) ] [ ( 0 1 0 0 -82 36 0 0), ( 0 0 1 0 0 -82 36 0), ( 0 0 0 1 0 0 -82 36) ] [ ( 1 0 -29/196 1/98 41/7 -293/98 9/49 0), ( 0 1 1/14 -1/7 0 41/7 -18/7 0) ] [ ( 1 -4 -1/2 1/2 -41 18 0 0) ] > > //a help function that computes the splitting discriminant of a singular > //plane conic. > > function dsc(M) function> B:=Basis(Kernel(M)); function> B:=ExtendBasis(B,Generic(Universe(B))); function> B:=Matrix([B[2],B[3]]); function> return -Determinant(B*M*Transpose(B)); function> end function; > > //make the answers a bit nicer: > > Q1:=36*(t-41/18)*Q1t;Q1; (36*t - 82)*U^2 + (6*t - 80)*U*V + (-11*t + 14)*U*W + (6*t + 2)*V^2 + (t + 14)*V*W + t*W^2 > Q2:=36*(t-41/18)*Q2t;Q2; (-3*t + 40)*U^2 + (-1/2*t - 9)*U*V + (-6*t^2 + 11/2*t + 51)*U*W + (-1/2*t - 7)*V^2 + (-1/2*t^2 - 2*t + 2)*V*W + (-t^2 + 1/2*t + 7)*W^2 > Q3:=36*(t-41/18)*Q3t;Q3; (6*t + 2)*U^2 + (t + 14)*U*V + (t^2 + 2*t - 4)*U*W + t*V^2 + (2*t^2 - t - 14)*V*W + (t^3 - t^2 - 8*t + 2)*W^2 > > //The curve is: > Ct:=Curve(P2t,Q1*Q3-Q2^2);Ct; Curve over Univariate rational function field over Rational Field defined by (207*t^2 - 180*t - 1764)*U^4 + (69*t^2 - 60*t - 588)*U^3*V + (437*t^2 - 380*t - 3724)*U^3*W + (299/4*t^2 - 65*t - 637)*U^2*V^2 + (69*t^3 - 695/2*t^2 - 338*t + 2450)*U^2*V*W + (-69*t^3 + 1827/4*t^2 + 243*t - 3381)*U^2*W^2 + (23/2*t^2 - 10*t - 98)*U*V^3 + (23/2*t^3 - 503/2*t^2 + 112*t + 2058)*U*V^2*W + (-253/2*t^3 + 335/2*t^2 + 1028*t - 490)*U*V*W^2 + (-23*t^4 + 43*t^3 + 513/2*t^2 - 266*t - 686)*U*W^3 + (23/4*t^2 - 5*t - 49)*V^4 + (23/2*t^3 - 10*t^2 - 98*t)*V^3*W + (23/4*t^4 - 5*t^3 - 75/2*t^2 - 10*t - 98)*V^2*W^2 + (23/2*t^3 - 10*t^2 - 98*t)*V*W^3 + (23/4*t^2 - 5*t - 49)*W^4 > > //compute the nodes of the Kummer surface: > nodes:=RationalPoints(SingularSubscheme(KJ)); > > //Here, we're lazy: We compute the tropes by taking all possible planes through > //3 nodes and only takiong those that have 6 nodes on them. > > triples:=Subsets({a:a in nodes},3); > function plane(c) function> P:=Ambient(Scheme(Rep(c))); function> cf:=Basis(Kernel(Transpose(Matrix([Eltseq(p): p in c])))); function> return Scheme(P,[&+[v[i]*P.i:i in [1..Dimension(P)+1]]: v in cf]); function> end function; > tropes:={v: c in triples | #{a: a in nodes | a in v} eq 6 where v:=plane(c)}; > > //Check that we have them all. > assert #tropes eq 16; > > //Now compute the bitangents that these lines induce on Ct: > tng1t:=[Curve(P2t,Evaluate(DefiningEquation(trope),[U,V,W,-U-V-t*W])):trope in tropes]; > [DefiningPolynomial(t):t in tng1t]; [ U - 1/2*V + 1/4*W, 5/4*U + 3/4*V + (1/4*t + 1/4)*W, 4/3*U + 1/3*V + (1/3*t + 2/3)*W, 1/6*U + 7/6*V + (1/6*t - 1/6)*W, U - V + W, 7/8*U + 1/8*V + (-1/8*t - 3/8)*W, U + 1/3*V + 1/9*W, 7/6*U + 1/6*V + (1/6*t + 1/6)*W, W, 5/6*U - 1/6*V + (-1/6*t + 1/6)*W, 1/2*U - 1/2*V + (-1/2*t + 3/2)*W, 2/3*U - 4/3*V + (-1/3*t + 2/3)*W, 10/9*U - 2/9*V + (1/9*t - 4/9)*W, U + V + W, 2*U + 2*V + t*W, U ] > > //These are the singular quadrics > sngquads:=[Q1-2*x*Q2+x^2*Q3: x in [0,-3,-1,1,2]] cat [Q3]; > [Determinant(SymmetricMatrix(Q)):Q in sngquads]; [ 0, 0, 0, 0, 0, 0 ] > > //get the splitting discriminants: > dscs:=[dsc(SymmetricMatrix(Q)):Q in sngquads];dscs; [ -23/4*t^2 + 5*t + 49, -207/4*t^4 + 390*t^3 - 434*t^2 - 2440*t + 4900, -23/4*t^4 + 28*t^3 + 6*t^2 - 176*t + 196, -23/4*t^4 - 18*t^3 + 46*t^2 + 216*t + 196, -23*t^4 - 95*t^3 + 609/4*t^2 + 1105*t + 1225, -23/4*t^2 + 5*t + 49 ] > > //Check that all split when one splits; > [IsSquare(l*dscs[1]):l in dscs]; [ true, true, true, true, true, true ] > > //Therefore, it is sufficient to get the following to be a square: > dscs[1]; -23/4*t^2 + 5*t + 49 > > //parametrize the corresponding conic: > L:=DefiningEquations(Parametrization(Conic(HyperellipticCurve( > Numerator(dscs[1]))))); > > //and extract the desired substitution from that > sval:=Evaluate(L[1],[t,1])/Evaluate(L[3],[t,1]);sval; (2*t^2 - 5*t - 3)/(t^2 + 1/2*t + 3/2) > > //Create the induced maps: > scov:=homQt| sval>; > lft:=homParent(U)| scov*Bang(Qt,Parent(U)), [U,V,W]>; > > //Make sure that things are really printed with s now: > //(this ruins printing with t) > _:=Codomain(scov); > > tng1s:=[Scheme(P2t,lft(DefiningEquation(l))):l in tng1t]; > > //Take the primary components of the singular conics. These are bitangents as > //well. > tng2s:=&cat[PrimaryComponents(Scheme(P2t,lft(Q))):Q in sngquads]; > Cs:= Curve(P2t,lft(DefiningEquation(Ct))); > tngs:=tng1s cat tng2s; > > //we now really have 28 bitangents > #tngs eq 28; true > > //Make sure that things are really printed with s now: > _:=Codomain(scov); > > //Check that all these really are bitangents > [SingularSubscheme(I) eq ReducedSubscheme(I) where I:= Cs meet T: T in tngs]; [ true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true ] > > //The defining equations of all 28 bitangents: > [DefiningEquation(T):T in tngs]; [ U - 1/2*V + 1/4*W, 5/4*U + 3/4*V + (3/4*s^2 - 9/8*s - 3/8)/(s^2 + 1/2*s + 3/2)*W, 4/3*U + 1/3*V + (4/3*s^2 - 4/3*s)/(s^2 + 1/2*s + 3/2)*W, 1/6*U + 7/6*V + (1/6*s^2 - 11/12*s - 3/4)/(s^2 + 1/2*s + 3/2)*W, U - V + W, 7/8*U + 1/8*V + (-5/8*s^2 + 7/16*s - 3/16)/(s^2 + 1/2*s + 3/2)*W, U + 1/3*V + 1/9*W, 7/6*U + 1/6*V + (1/2*s^2 - 3/4*s - 1/4)/(s^2 + 1/2*s + 3/2)*W, W, 5/6*U - 1/6*V + (-1/6*s^2 + 11/12*s + 3/4)/(s^2 + 1/2*s + 3/2)*W, 1/2*U - 1/2*V + (1/2*s^2 + 13/4*s + 15/4)/(s^2 + 1/2*s + 3/2)*W, 2/3*U - 4/3*V + (2*s + 2)/(s^2 + 1/2*s + 3/2)*W, 10/9*U - 2/9*V + (-2/9*s^2 - 7/9*s - 1)/(s^2 + 1/2*s + 3/2)*W, U + V + W, 2*U + 2*V + (2*s^2 - 5*s - 3)/(s^2 + 1/2*s + 3/2)*W, U, U + (7*s + 3)/(s + 21)*V + (s - 3)/(s + 21)*W, U + (-1/5*s + 1/2)/(s + 11/10)*V + (-1/5*s - 1/10)/(s + 11/10)*W, U + (1/4*s + 1/2)/(s - 1/4)*V + (s^3 + 17/8*s^2 - 1/8*s - 3/4)/(s^3 + 1/4*s^2 + 11/8*s - 3/8)*W, U + (-1/5*s - 3/5)/(s - 3/5)*V + (-1/5*s^3 + 1/2*s^2 + 3*s - 9/10)/(s^3 - 1/10*s^2 + 6/5*s - 9/10)*W, U + 1/2/(s + 1/2)*V + (1/2*s^2 + 3/4*s - 3/4)/(s^3 + s^2 + 7/4*s + 3/4)*W, U + (-s - 1)/(s - 3)*V + (s^3 + 7/2*s^2 + 2*s - 1/2)/(s^3 - 5/2*s^2 - 9/2)*W, U + s/(s + 3)*V + (s^3 + 1/2*s^2 - 3/2*s)/(s^3 + 7/2*s^2 + 3*s + 9/2)*W, U + (-1/2*s + 1/2)/(s + 2)*V + (1/4*s^3 + s^2 - 5/4*s)/(s^3 + 5/2*s^2 + 5/2*s + 3)*W, U + (-s + 1/2)/(s + 7/2)*V + (s^3 + 4*s^2 - 3/4*s - 3/4)/(s^3 + 4*s^2 + 13/4*s + 21/4)*W, U + (3/5*s - 1/5)/(s + 9/5)*V + (3/5*s^3 - 1/2*s^2 - 2*s + 7/10)/(s^3 + 23/10*s^2 + 12/5*s + 27/10)*W, U + (1/7*s - 3/7)/(s + 3/7)*V + (-5/7*s^3 - 5/2*s^2 + 9/14)/(s^3 + 13/14*s^2 + 12/7*s + 9/14)*W, U + (s + 1/2)/(s - 5/2)*V + (s^3 - 2*s^2 - 23/4*s + 9/4)/(s^3 - 2*s^2 + 1/4*s - 15/4)*W ] Total time: 16.369 seconds, Total memory usage: 8.28MB