Nontorsion points of low height on elliptic curves over number fields

The first table lists representatives of elliptic curves in form y2+a1xy+a3y=x^3+a2x2+a4x+a6 over some number field K of degree d with a nontorsion point P of canonical height h(P) satisfying dh(P)<0.01. These are comparable with the points on curves over Q given by Elkies in the sense that they use the same scaling for h(P) (for the elliptic Lehmer problem, dh(P) values should be compared).

dh(P) K,d h(P) w P [a1,a2,a3,a4,a6] Method
0.0032192… Q(√6),2 0.0016096… √6 (0,0) [-12443w+30479, 230496005w+564597600,-7958566915120w+19494428025840,0,0] Elkies (A,u,c)=(w-3,w-3,1)
0.0039804… Q(√2),2 0.0019902… √2 (0,0) [-13w-23,49w+70,-1820w-2576,0,0] Elkies (A,u,c)=(w+1,w-1,1)
0.0050312… Q(√3),2 0.0025156… √3 (0,0) [1086w+1881,716035w+1240209,1277410855w+2212540503,0,0] Elkies (A,u,c)=(-2w-4,w+1,1)
0.0056482… Q(√3),2 0.0028241… √3 (0,0) [2856w-4944,42937344w-74369664,-746077879296w+1292244793344,0,0] Elkies (A,u,c)=(1,2w-4,1)
0.0056603… Q(√5),2 0.0028301… 1/2+(√5)/2 (0,0) [4-w,6w-18,60w-90,0,0] Elkies (A,u,c)=(w,w-2,1)
0.0061000… Q(√7),2 0.0030500… √7 (0,0) [-5922w+15669,-35749431w+94584105,-543103643331w+1436917176387,0,0] Elkies (A,u,c)=(2w-6,1-w,1)
0.0062374… Q(√6),2 0.0031187… √6 (0,0) [276w+678,48060w+117720,40956840w+100323360,0,0] Elkies (A,u,c)=(1,w+2,1)
0.0063291… Q(√3),2 0.0031645… √3 (0,0) [17298w+29961,332452269w+575824221,9670381784073w+16749592578603,0,0] Elkies (A,u,c)=(-2w-4,-w-3,1)
0.0066567… Q(√2),2 0.0033283… √2 (0,0) [4w-5,-6w,24w-48,0,0] Elkies (A,u,c)=(w-2,-w-1,1)
0.0068524… Q(√5),2 0.0034262… 1/2+(√5)/2 (0,0) [120w-205,2200w-3550,1067000-659500w,0,0] Elkies (A,u,c)=(w-1,2-w,1)
0.0072293… Q(√2),2 0.0036146… √2 (0,0) [21w-29,2715w-3840,-368940w+521760,0,0] Elkies (A,u,c)=(w-1,1-w,1)
0.0073153… Q(√5),2 0.0036576… 1/2+(√5)/2 (0,0) [-4w-1,4w-2,-12w-4,0,0] Elkies (A,u,c)=(w-2,-w-1,1)
0.0077126… Q(√3),2 0.0038563… √3 (497w+860,7523w+13029) [w,-w,1,-(1544879w+2675806),1407381720w+2437656645] EDS
(-(288w+498),8097w+14024) [1,w-1,0,-(365487w+633040),140434476w+243239647]
0.0077221… Q(√7),2 0.0038610… √7 (0,0) [-2134w-5647,-3313113w-8765673,42656703903w+112859030277,0,0] Elkies (A,u,c)=(-2,w+3,1)
0.0077642… Q(√2),2 0.0038821… √2 (0,0) [-1084w+1533,-1217471w+1721764,-1745861055w+2469020382,0,0] Elkies (A,u,c)=(w-2,w-2,1)
0.0077962… Q(√6),2 0.0038981… √6 (0,0) [46575w+114075,2013399585w+4931801640,298790194490820w+731883516649380,0,0] Elkies (A,u,c)=(2w+5,w+3,1)
0.0079803… Q(√2),2 0.0039901… √2 (0,0) [-14w-17,38w+54,-1256w-1776,0,0] Elkies (A,u,c)=(w-1,1,1)
0.0082965… Q(√-5),2 0.0041482… √-5 (0,0) [210w-255,-32625w-14625,2559375w + 20660625, 0, 0] Elkies (A,u,c)=(2,w-1,1)
0.0083201… Q(√-1),2 0.0041600… √-1 (0,0) [22w+21,161w+73,2589w-573,0,0] Elkies (A,u,c)=(2w-2,1-w,1)
0.0086982… Q(√5),2 0.0043491… 1/2+(√5)/2 (0,0) [-10w-3,14w+8,-232w-144,0,0] Elkies (A,u,c)=(w-2,1,1)
0.0087607… Q(√-3),2 0.0043803… 1/2+(√-3)/2 (0,0) [6w+21,18w+36,648w+540,0,0] Elkies (A,u,c)=(w-1,w,1)
0.0088450… Q(√7),2 0.0044225… √7 (0,0) [-1520w+4036,-2178528w+5763936,-6182979456w+16358626176,0,0] Elkies (A,u,c)=(w-4,1-w,1)
0.0090467… Q(√7),2 0.0045233… √7 (0,0) [-1449w+3762,-3260817w+8627715,-15940631844w+42174921564,0,0] Elkies (A,u,c)=(w,w-2,1)
0.0090698… Q(√3),2 0.0045349… √3 (0,0) [144w+249,6516w+11286,1487700w+2576772,0,0] Elkies (A,u,c)=(-2w-4,w,1)
0.0092420… Q(√-2),2 0.0046210… √-2 (0,0) [12w-3,3w-66,-465w+24,0,0] Elkies (A,u,c)=(w,-w,1)
0.0092730… Q(√6),2 0.0046365… √6 (0,0) [-690w-1689,-386235w-946080,1527073155w+3740550030,0,0] Elkies (A,u,c)=(-2,w+2,1)
0.0092754… Q(√3),2 0.0046377… √3 (0,0) [93w+150,-5103w-8829,-479196w-830088,0,0] Elkies (A,u,c)=(1,w+2,1)
0.0094336… Q(√-1),2 0.0047168… √-1 (0,0) [5w,w-3,-12w-4,0,0] Elkies (A,u,c)=(w,-w,1)
0.0094446… Q(√-7),2 0.0047223… 1/2+(√-7)/2 (2w-4,3w+6) [1,-2w,0,-(w+2),5w+2] EDS
(-(w+3),9-3w) [1,1-w,1-w,2w-2,9-3w]
0.0096287… Q(√3),2 0.0048143… √3 (0,0) [-54w+60,2484w-4212,-74520w+126360,0,0] Elkies (A,u,c)=(1,w-1,1)
0.0097879… Q(√-2),2 0.0048558… √-2 (0,0) [304w-488,-18816w+26880,15955968w-22579200,0,0] Elkies (A,u,c)=(w+1,2-w,1)
0.0099060… Q(√5),2 0.0049530… 1/2+(√5)/2 (0,0) [-35w-16,188w+106,-11910w-7360,0,0] Elkies (A,u,c)=(w-2,w,1)

Methods: the Elkies (A,u,c) parametrisation is as described in the page linked above; for the EDS approach, see the preprint below.

Further contributions (over any number field) would be much appreciated!

The second table lists representatives of elliptic curves in form y2+a1xy+a3y=x^3+a2x2+a4x+a6 over some quadratic ring Z[w] with a nontorsion point P of canonical height h(P) at most 0.01, as a supplement to the data given in this preprint. These examples were obtained via an approach using elliptic divisibility sequences suggested by Everest and Ward.

h(P) w P [a1,a2,a3,a4,a6]
0.0038563… √3 (497w+860,7523w+13029) [w,-w,1,-(1544879w+2675806),1407381720w+2437656645]
(-(288w+498),8097w+14024) [1,w-1,0,-(365487w+633040),140434476w+243239647]
0.0047223… 1/2+(√-7)/2 (2w-4,3w+6) [1,-2w,0,-(w+2),5w+2]
(-(w+3),9-3w) [1,1-w,1-w,2w-2,9-3w]
0.0053416… √3 (1,-w) [1,w,w,-(w+5),4-w]
(1,0) [1,-w,w,-5,w+4]
0.0054424… 1/2+(√-7)/2 (w-2,5-4w) [1,3w-2,-w,7w-5,-(12w+29)]
(-(2w+1),4w+1) [1,1,2w-1,8-8w,10w-25]
0.0058010… 1/2+(√-7)/2 (2,-(3w+1)) [w,0,2w,-2,3-w]
(2-w,2-2w) [w-1,3w,0,w-8,2-5w]
0.0060112… 1/2+(√-7)/2 (21,-2) [w,1,1,-w,-w]
(2-w,1) [w-1,3w-2,-w,-(w+5),1-w]
0.0061272… √2 (1,0) [1,w,w+1,-1,-w]
(1,-w) [1,-w,w+1,-(w+1),0]
0.0064724… √-2 (2w,2w+1) [0,-w,w,2w+6,w-3]
(-2w,1-3w) [0,w,w,6-2w,-(w+3)]
0.0069470… √2 (w,-1) [w+1,-w,w,-(w+3),w+1]
(-w,w+1) [w+1,0,w,-3,1-w]
0.0072803… 1/2+(√-7)/2 (0,2-w) [1,-2w,0,7w-6,2-3w]
(w-1,-2w) [1,1-w,w-1,3-7w,w-11]
0.0073349… 1/2+(√-7)/2 (-(w+3),4w+2) [1,w+1,w-1,7w+1,w-11]
(-3,5-4w) [1,2w-1,1,8w-12,-(10w+17)]
0.0073479… 1/2+(√-7)/2 (-w,w-1) [w-1,3w,-w,w-5,1-w]
(0,1-w) [w,0,w-1,w+1,0]
0.0074870… 1/2+(√-3)/2 (1,0) [1-w,-2w,1-w,2w-1,0]
(1,0) [w,2w-2,w,1-2w,0]
0.0074943… √-1 (w,1) [1,1,,w,2,w+2]
(w,w+1) [w,-1,1,2-w,w-2]
0.0076951… 1/2+(√-7)/2 (3-3w,2-2w) [1,3w-1,1-w,3w-1,3w-9]
(3-3w,2-2w) [1,3w-1,1-w,3w-1,3w-9]
0.0080799… 1/2+(√5)/2 (1,2w-1) [1-w,w,1-w,w-2,0]
(1,1-2w) [w,1-w,w,-(w+1),0]
0.0087764… √3 (0,0) [0,w+3,w+1,2w+2,0]
(0,0) [0,3-w,1-w,2-2w,0]
0.0087786… √-1 (1,-w) [w,-(w+1),1,0,0]
(-1,1-w) [1,1-w,w,-w,0]
0.0088447… √2 (1-w,1) [1,w-1,w+1,-(2w+2),1]
(w+1,1-w) [1,-(w+1),w+1,w-2,1-w]
0.0089008… 1/2+(√-7)/2 (2,-3) [w,-2w,0,w,1]
(2,3-w) [w-1,2w-2,2w,3-w,3-w]
0.0089933… 1/2+(√5)/2 (w+3,4w) [0,w,0,-(21w+16),61w+41]
(1-w,-w) [1,2w,2w,2w-1,2-2w]
(2w+3,3w+4) [0,-w,2w,-(6w+11),22w+21]
(4-w,8-5w) [0,2-2w,2w,5w-15,21-7w]
0.0090543… 1/2+(√-3)/2 (2w-4,4w+4) [0,1-2w,0,-12,36w-12]
(-(w+3),8-4w) [0,2-w,0,-(w+10),13-25w]
0.0091282… 1/2+(√5)/2 (5,2-2w) [1-w,-1,w-1,-(34w+45),158w+149]
(5,2w) [w,-1,-w,34w-79,307-158w]
0.0091781… √3 (0,0) [0,w,w,0,0]
(0,0) [0,-w,w,0,0]
0.0093444… √2 (1,-1) [w+1,-1,1,-(3w+4),2w+3]
(1,-1) [w+1,-(w+1),1,2w-4,3-2w]
0.0097150… √2 (w,w) [1,-w,w,w-2,2w+4]
(-w,2w) [1,w,w,-(2w+2),4-2w]
0.0097217… 1/2+(√5)/2 (w-1,4w) [0,1,0,-(15w+12),27w+20]
(-w,4-5w) [0,1,2w,15w-27,46-28w]
0.0097259… 1/2+(√5)/2 (2w,4w) [0,1-w,0,-(20w+16),76w+48]
(2-2w,4-4w) [0,w,0,20w-36,124-76w]
(-w,4w+4) [0,1-w,0,-(11w+13),27w+22]
(w-1,2-2w) [1,2-2w,2w-1,2-2w,-1]

One Comment

  1. hi Graeme,

    I added a new answer to your mathoverflow question about non-torsion points on elliptic curves with small heights:
    http://mathoverflow.net/questions/77365/records-for-low-height-points-on-elliptic-curves-over-number-fields/155767#155767

    I have found lots more quadratic examples than you have above (214 with dh(P)<0.01), but I could not find an email address for you. Please email me, if you are still interested and I will send you the file containing all such examples that I found.

    PV

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