From: Dave Rusin Date: Thu, 2 Jul 1998 10:59:23 -0500 (CDT) To: kesslert@nevada.edu Subject: Re: Diophantine Equation Newsgroups: sci.math In article <6ndmes$3ct$1@news.nevada.edu> you write: >I saw a website that was pondering the equation > 3y^2=2x^3+386x^2+256x-58195 I'm curious about where this equation arises, since it evidently leads to some hard numerics; see below. >and they said it could be transformed to y^2=x^3-440067x+106074110. >Then they list many solutions to the last equation I ran this through Cremona's program mwrank. It implies this curve has rank 4. Here's an excerpt. I annotate it so you can get a little sense of what these mean, but the punchline is that there are infinitely many rational points on the curves, including many "small" ones which are given explicitly. ================================================================ Input curve [0,0,0,-440067,106074110] Working with minimal curve [1,-1,0,-27504,1664284] [That is, a change of variables to the curve [ 2 3 2 [ Y + XY = X - X - 27504 X + 1664284 [has been performed so as to make the coefficients smaller. [You can see the coordinate changes if you use the Maple package APECS. [In this case, each X,Y gives a point [4X-1, 8Y+4X] on the curve I put in. [ -- djr.] Height Constant = 12.635811952999 No points of order 2 [in fact there is no torsion at all -- djr] [A search through a set of equivalence classes of quartics yields many points. [All locally solvable quartics actually had points, so the rank analysis is [complete -- djr] Summary of results (all should be powers of 2): n0 = #E(Q)[2] = 1 n1 = #E(Q)/2E(Q) = 16 n2 = #S^(2)(E/Q) = 16 #III(E/Q)[2] = 1 rank = 4 [Now the found points are expressed as members of this free abelian group [The format is: [a:b:c] means X=a/c,Y=b/c --djr] P1 = [62 : 410 : 1] is generator number 1 P2 = [125 : 347 : 1] is generator number 2 P3 = [188 : 1670 : 1] is generator number 3 P4 = [-190 : 158 : 1] is generator number 4 P5 = [251 : 3119 : 1] = 1*P1 + -1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [68 : 290 : 1] = 0*P1 + -1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [-4 : 1334 : 1] = -1*P1 + 1*P2 + 0*P3 + 0*P4 (mod torsion) P5 = [23 : 1010 : 1] = 0*P1 + 0*P2 + -1*P3 + -1*P4 (mod torsion) P5 = [1322 : 47030 : 1] = -1*P1 + 0*P2 + -1*P3 + -1*P4 (mod torsion) P5 = [-890 : 15061 : 8] = 0*P1 + -1*P2 + 1*P3 + 1*P4 (mod torsion) P5 = [57 : 500 : 1] = -1*P1 + 0*P2 + 1*P3 + 0*P4 (mod torsion) P5 = [878 : 25130 : 1] = 0*P1 + 1*P2 + -1*P3 + 0*P4 (mod torsion) P5 = [-11404 : 65867 : 64] = 1*P1 + -1*P2 + -1*P3 + 0*P4 (mod torsion) P5 = [5882 : 447968 : 1] = -1*P1 + 1*P2 + 1*P3 + -1*P4 (mod torsion) P5 = [3726 : 73915 : 8] = -1*P1 + 2*P2 + 0*P3 + -1*P4 (mod torsion) Generator 1 is [62 : 410 : 1]; height 2.04031122815301 Generator 2 is [125 : 347 : 1]; height 2.44212115515056 Generator 3 is [188 : 1670 : 1]; height 2.56575825933078 Generator 4 is [-190 : 158 : 1]; height 2.95905178655165 [These correspond respectively to the (integer!) points [ [247, 3528], [499, 3276], [751, 14112], [-761, 504] [on the curve I entered as input, after using the transformations mentioned [earlier -- djr.] regulator is 23.619061641953 Max height = 7.56158713544573 Bound on naive height of extra generators = 20.1973990884448 [Now a brute-force search for other small points is initiated to see if [our generators merely generate a subgroup of finite index in Z^4. This [would take approximately forever, so I interrupted the job the next [morning. _Probably_ the generators don't miss anything. --djr] P5 = [-186 : 658 : 1] = -1*P1 + -1*P2 + 0*P3 + 0*P4 (mod torsion) P5 = [-139 : 1739 : 1] = 1*P1 + 0*P2 + 1*P3 + 0*P4 (mod torsion) P5 = [-4 : 1334 : 1] = -1*P1 + 1*P2 + 0*P3 + 0*P4 (mod torsion) P5 = [23 : 1010 : 1] = 0*P1 + 0*P2 + -1*P3 + -1*P4 (mod torsion) P5 = [57 : 500 : 1] = -1*P1 + 0*P2 + 1*P3 + 0*P4 (mod torsion) P5 = [62 : 410 : 1] = 1*P1 + 0*P2 + 0*P3 + 0*P4 (mod torsion) P5 = [68 : 290 : 1] = 0*P1 + -1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [114 : -32 : 1] = -1*P1 + 1*P2 + -1*P3 + -1*P4 (mod torsion) P5 = [125 : 347 : 1] = 0*P1 + 1*P2 + 0*P3 + 0*P4 (mod torsion) P5 = [131 : 479 : 1] = -1*P1 + 0*P2 + 0*P3 + -1*P4 (mod torsion) P5 = [150 : 872 : 1] = 0*P1 + -1*P2 + -1*P3 + 0*P4 (mod torsion) P5 = [188 : 1670 : 1] = 0*P1 + 0*P2 + 1*P3 + 0*P4 (mod torsion) P5 = [251 : 3119 : 1] = 1*P1 + -1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [878 : 25130 : 1] = 0*P1 + 1*P2 + -1*P3 + 0*P4 (mod torsion) P5 = [1322 : 47030 : 1] = -1*P1 + 0*P2 + -1*P3 + -1*P4 (mod torsion) P5 = [5882 : 447968 : 1] = -1*P1 + 1*P2 + 1*P3 + -1*P4 (mod torsion) P5 = [16127 : 2039783 : 1] = 1*P1 + 1*P2 + 0*P3 + -1*P4 (mod torsion) P5 = [414473 : 266628467 : 1] = 0*P1 + 0*P2 + 0*P3 + -2*P4 (mod torsion) P5 = [-890 : 15061 : 8] = 0*P1 + -1*P2 + 1*P3 + 1*P4 (mod torsion) P5 = [1062 : 4129 : 8] = -1*P1 + 0*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [3726 : 73915 : 8] = -1*P1 + 2*P2 + 0*P3 + -1*P4 (mod torsion) P5 = [200206 : 31570375 : 8] = 1*P1 + 1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [-2490 : 51104 : 27] = 2*P1 + -1*P2 + 0*P3 + 1*P4 (mod torsion) P5 = [705 : 26149 : 27] = 0*P1 + 0*P2 + -1*P3 + 1*P4 (mod torsion) P5 = [204654 : 17710130 : 27] = -1*P1 + 1*P2 + 1*P3 + 1*P4 (mod torsion) P5 = [-11404 : 65867 : 64] = 1*P1 + -1*P2 + -1*P3 + 0*P4 (mod torsion) P5 = [8370 : 39022 : 125] = 0*P1 + -1*P2 + 0*P3 + -1*P4 (mod torsion) P5 = [100470 : 2740168 : 125] = 0*P1 + 1*P2 + -1*P3 + -2*P4 (mod torsion) P5 = [4684870 : 904606258 : 125] = -2*P1 + 0*P2 + 1*P3 + 0*P4 (mod torsion) [and here I killed it -- djr] ================================================================ >I took the original multiplied by 108. Then i made the substitutions >a=18y c=386+6x and i got a^2=c^3-442380c+106961164. Does anyone know how >they got their answer???? You're right, not them, as I posted previously. I ran this curve too: Curve [0,0,0,-442380,106961164] : Height Constant = 13.5639398836324 No points of order 2 Looking for Type 1 quartics: Trying positive a from 1 up to 406 (39,0,1050,1374,480) --nontrivial--new #1 locally soluble... but no rational point found (limit 5) [Uh-oh: this means the analysis will be incomplete -- either there are no [points of a certain type, or they're really huge. -- djr Warning: Selmer rank = 1 and program finds lower bound for rank = 0 which differs by an odd integer from the Selmer rank. Hence the rank must be 1 more than reported here. Try rerunning with a higher bound for quartic point search. Summary of results (all should be powers of 2): n0 = #E(Q)[2] = 1 n1 = #E(Q)/2E(Q) >= 1 n2 = #S^(2)(E/Q) = 2 #III(E/Q)[2] <= 2 0 <= rank <= selmer-rank = 1 Rank = 0 Rank of points found is 0 [There must be a point -- in fact infinitely many of them forming a cyclic [group -- unless some very trusted conjectures fail, but evidently the [size of the coordinates is very large. These things happen. Sorry. -- djr] ============================================================== It's always of interest to me to find "small" equations with "large" solutions. Try y^2=x^3+877x; the solution with fewest total digits has simu x = (612776083187947368101 / 7884153586063900210) ^ 2 Perhaps your equation is another of this quality. I'll let my machine play with it a bit longer to see if anything results, and will let you know if it has any luck. dave