一个有趣的高秩曲线

wayne · 发表于 2025-7-22 20:11:32

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在mathoverflow上看到了一个椭圆曲线,$y^2 = x^3 + x^2 -15662264585 x + 746984342506759.$, 一开始还以为是没法计算的.没想到 PARI/Gp 瞬间计算出了全部的8个生成元.
然后又简单跑了一下,好家伙, 有60个整数点, 就算是有理数解,前2000个解的分母都超级的小,才15位数.常规情况下椭圆曲线的解都是指数级别增长的.

https://mathoverflow.net/questio ... curve/477130#477130

wayne · 发表于 2025-7-22 20:12:33

前1004组解,舍弃了y为负值的解

(16131, 22327876)
(17023, 22029500)
(31635, 16827628)
(52797, 8199964)
(65823, 1113820)
(66259, 341548)
(66305, 49728)
(66306, 1)
(78077, 294276)
(78271, 777724)
(91186, 8775799)
(100099, 13497788)
(115587, 21930044)
(116479, 22427332)
(-135565, 19464372)
(-139583, 14616304)
(-140353, 13432836)
(-143366, 6761923)
(-143759, 5251624)
(-144321, 1183868)
(177315, 59537828)
(249345, 111104672)
(5695267, 13588363996)
(6053715, 14891584172)
(11010161, 36531086424)
(15055773, 58417053100)
(52676723, 382319900340)
(72219471, 613734879724)
(2472641917, 122953767408196)
(2472840829, 122968604254268)
(88021/4, 162576929/8)
(781333/4, 573584767/8)
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(25204438717/4, 4001432965117675/8)
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(396283/9, 322557812/27)
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(707431/9, 33441596/27)
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复制代码

wayne · 发表于 2025-7-22 22:19:37

出于好奇, 我生成了所有的 $\sum_{i=1}^8a_i G_i$ 的解. 其中$-3<= a_i <=3$,抛弃对称的y为负的解, 得到了$(7^8-1)/2=2882400$组. 得到的文件有2.03GB, 其中第2882400组解如下, 分母也才1094位数,这种增速慢如蜗牛
(120632051280384509433005487465469482849524776118198284658765697931722968386929463949238011281408039024105936888562818866286038770251148187516280048455307327436549387373616809608723910238302580347590051224231599783983943553206646546592301498787696960827076138040798239033223551631931820016452724425617975168732874781855294655666693725973045589584007654053777496110850377698167488420314711301343491601652868181422510298671538287948430496547523658502180615193730803642339608390954765985584000725534733127949216660038898691514659972236696339062428666169317347683301154280944343062201895269604242525683380178595524818618356873798662122431629851000830388701174485795847747458697180834204656919371871296361419817981315573744695004970795438850/1843593458133305832397525270465511068013702835665186944036521134556209518335937292205886725295165125014166622990942575665253594356901075651867751671133807887754765311039963590713081636921382942970001211502282114526275556713059498374412985311745771631739988730410733719065343283622569327426061035454053291807596040475885038892783521235791662865154299517289273490731798371105133252661560849001747785740620965092619959599628246760464490435060774015360093648065300331161493369454518699708782905064054488869409872561569098456470094559151250722888040542594360923572873497521830115854022207991721296888761754360221032719273394950314040425637596432508659845496806037196331920186597744997588467903768552912990330053529742724427748346920849, 120291684392049288310918634797853922290122524621846025454870699126557092749108881859415263834296125087742430807336685769064836312804226100503985043026602740419314474513971076830693682145528319767494252204192634952202601755693993773876634846866617189557754274493479043200467789782338587627751126496824901361118542111635478025370496047490228821314675965645311459597481357529584404222128518363809468135581721266311914422447011150805226786407135160908078696456222417999289089698216499990992240509553645037408136673606570324809295912020576497417671394972930101290068879356714954499143100602037074307964287216054941766829423195200274226084041402084668537736024554478552448959318048293240410593575714439546449785378123137709550388739934775578629233019542219973549722521963669700080616338058091834281232591180055389797027942960129233128847828359616815809794389338779812224507957925347934121794116413697736759080748626880544916319179894311910168271344703688258972648856163742283138071939424648262560642328668770704149375864367291062905283717189994671189706603667871109038828832815352493060839339118876924120521/79158533092817977983512853079443942429132431943937959949220329133757653634223140201189974000549423250603801323756282351843008716316107956349711699634538796603480084824494508434351884476458318086404133192013701530015380769846299614894946960306861500441665576311775870952476999794082941860917901810050366019083365175496689823222231081833220475683411675391126712685558864977233878368701816227500619147795913317104322543288995246051352240377998937028192979441037671176543540998298497756230708640934385969424340026613863994306524491525591254659006433877464749255812776159868276793814316977690986946696991424803067008655543899959951025829995074696484781151363040101069079281890510347970586625187593252083626120698561551055756203910482178207128202775651509601515236691536142999409132310710155629094074034556478318991225131114531632749688059624964183147456289019289781293656725034649040268049865105789331564619439874317994435189527682332318636610319497187870391417603681143926253860083414598165918606435843696848545221673254098760078092161019189015100579986499363275814608198322945992363856192561480007)

nyy · 发表于 2025-7-22 23:52:45

如何计算出有理点了？

nyy · 发表于 2025-7-23 00:01:35

https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html

Rank ≥ 29

Elkies - Klagsbrun (2024)

y2 + xy = x3 - 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497 Independent points of infinite order: P1 = [2891195474228537189458255536634, 1159930748096124706459835910727318679593425283] P2 = [3402542165322127811451484642234, 1661508223164691055862657623730465560755290883] P3 = [4298760026558467240422107564794, 4313142249890236204790986787384907722927474563] P4 = [3728756667770947009884455714554, 2530180219584734091116528693531660545660397443] P5 = [5991744132052078230511185130234, 10418901628842034362301273055728300669218858883] P6 = [3236493534632768520540227223034, 1324626796262167243658687198416201825373745283] P7 = [78226686134991174232380689386234, 690394210062759896503429654125516779999512554883] P8 = [11492605643548859374635605140234, 35536316911450952155461624238308456029618940883] P9 = [-5143303362384229804906088118566, 7622356511107986864120352355674305680222368483] P10 = [443985655575065435281568435002, 6584468124388858623214803939643557365635620355] P11 = [-979565018904269680752629749766, 8987348422104537684966706438714038633832170883] P12 = [5184894285212178249566461261834, 7390536788003150201273204464695859875505480483] P13 = [-4469171023687146502067179612166, 9310658892841458934133221137392081403414455683] P14 = [3606405835110925482450522970234, 2183644666981703632482662193390480040127898883] P15 = [16151744576785317732688993162234, 61908882092472338946519909276455831463747210883] P16 = [3573684355943766387962362869754, 2094467155115749424853047283659077805560259203] P17 = [-759376049938858166436491644166, 8679171135458197195914024161800061810952119683] P18 = [-5328058719935886182106003119366, 6920588147379497633202935557367499676224350083] P19 = [5380268474895377355583039694554, 8105660240030025092450118297303424395856037443] P20 = [17069233487425098088940203248484, 67583677272795299213867443505411893525786510633] P21 = [5215432542403430758248050783794, 7501515746204716855921710958364078294243814643] P22 = [2838942178046024039763692432122, 1212346280964590308944175800544505700108208003] P23 = [243146882395382015946366404808154/81, 811625272160726332199288136187427505366582108107/729] P24 = [2558229016839511149831260080762, 1706598395830079994387505244133382709649637123] P25 = [2361253942905600810977556672634, 2157503396243552448798851089310708763298766083] P26 = [2678312077644931683114439906234, 1462722361020796436741527433473386115047618883] P27 = [3379397084927230910084852603902, 1608494167359575995485655188349208450365853755] P28 = [3632407730870998917912491355514, 2255654937037700801978158381185619053396712963] P29 = [2428778263277521959543043930234, 1998325023610603606161737305486867803334410883] Rank records historyAndrej Dujella home page

nyy · 发表于 2025-7-23 00:03:35

那这条椭圆曲线的纯文本完整方程就是：

y² + xy = x³ - 27006183241630922218434652145297453784768054621836357954737385x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497

这就是去掉格式符号后，仅用文字和数字表示的完整方程啦。

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