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Rational Points on Ellptic Curves: y^2=x^3+n (n \in [2001..3000])


[2021.08.22]y^2=x^3+n (n \in [2001..3000])の有理点



■n=2001,2002,...,3000に対して、
     En: y2 = x3+n
の有理点を求める。

■n=2001,2002,...,3000に対して、楕円曲線Enのねじれ点群En(Q)torsを計算すると、
     En(Q)tors=Z/2Z if n=2197,2744
                 =Z/3Z if n=2025,2116,2209,2304,2401,2500,2601,2704,2809,2916
                 ={O} if nが平方数,立方数でない
となる。

[pari/gpによる計算]
gp> ec(n)=ellinit([0,0,0,0,n]);
gp> tg(n1,n2)=local(e,g,o);for(i=n1,n2,e=ec(i);g=elltors(e);o=g[1];if(o>1,print("E_",i,"(Q)_{tors}=",g)));
gp> tg(2001,3000)
E_2025(Q)_{tors}=[3, [3], [[0, 45]]]
E_2116(Q)_{tors}=[3, [3], [[0, 46]]]
E_2197(Q)_{tors}=[2, [2], [[-13, 0]]]
E_2209(Q)_{tors}=[3, [3], [[0, 47]]]
E_2304(Q)_{tors}=[3, [3], [[0, 48]]]
E_2401(Q)_{tors}=[3, [3], [[0, 49]]]
E_2500(Q)_{tors}=[3, [3], [[0, 50]]]
E_2601(Q)_{tors}=[3, [3], [[0, 51]]]
E_2704(Q)_{tors}=[3, [3], [[0, 52]]]
E_2744(Q)_{tors}=[2, [2], [[-14, 0]]]
E_2809(Q)_{tors}=[3, [3], [[0, 53]]]
E_2916(Q)_{tors}=[3, [3], [[0, 54]]]
time = 7 ms.

■楕円曲線EnのMordell-Weil群のrankと生成元を求めると以下のようになる。

- E~n: Y2Z=X3+nZ3
En: y2=x3+n
-
n [a1,a2,a3,a4,a6]
j(En)
Conductor of En
En(Q)tors
En(Q)torsの生成元
rank(En(Q)) En(Q)/En(Q)tors
の生成元 [X:Y:Z]
En(Q)/En(Q)tors
の生成元の高さ
En(Z)
Enの整点[x,y]
(where y >= 0)
n
2001 [0, 0, 0, 0, 2001]
0
432432108
{O}
-
2 [100 : 1001 : 1],
[-240 : 973 : 27]
4.22081709728170,
4.78200147016608
[100, 1001] 2001
2002 [0, 0, 0, 0, 2002]
0
1731457728
{O}
-
1 [463631703 : 2740813837 : 13997521] 14.7092295498433
- 2002
2003 [0, 0, 0, 0, 2003]
0
1733187888
{O}
-
0 - - - 2003
2004 [0, 0, 0, 0, 2004]
0
27108108
{O}
-
1 [7245 : 39833 : 729] 7.51210218232332
- 2004
2005 [0, 0, 0, 0, 2005]
0
434162700
{O}
-
1 [-25233 : 48632 : 2197] 8.20234061142673
- 2005
2006 [0, 0, 0, 0, 2006]
0
579461184
{O}
-
1 [-6379275 : 21808673 : 658503] 11.9869057709548
- 2006
2007 [0, 0, 0, 0, 2007]
0
193346352
{O}
-
2 [-11 : 26 : 1],
[-46 : 341 : 8]
3.06792985631483,
4.40776533995089
[-11, 26] 2007
2008 [0, 0, 0, 0, 2008]
0
36288576
{O}
-
1 [174 : 1288 : 27] 4.55062773646770
- 2008
2009 [0, 0, 0, 0, 2009]
0
8895852
{O}
-
0 - - - 2009
2010 [0, 0, 0, 0, 2010]
0
1745323200
{O}
-
1 [52551 : -447373 : 729] 8.78061810635509
- 2010
2011 [0, 0, 0, 0, 2011]
0
1747060272
{O}
-
1 [264 : 22961 : 512] 7.15670592301725
- 2011
2012 [0, 0, 0, 0, 2012]
0
109299888
{O}
-
0 - - - 2012
2013 [0, 0, 0, 0, 2013]
0
437634252
{O}
-
1 [115563 : 582490 : 9261] 9.27398397536137
- 2013
2014 [0, 0, 0, 0, 2014]
0
1752276672
{O}
-
0 - - - 2014
2015 [0, 0, 0, 0, 2015]
0
584672400
{O}
-
1 [-1849937 : -12697398 : 300763] 10.8858424073370
- 2015
2016 [0, 0, 0, 0, 2016]
0
762048
{O}
-
1 [-32914 : 46745 : 2744] 8.35726645864951
- 2016
2017 [0, 0, 0, 0, 2017]
0
146458404
{O}
-
2 [2 : 45 : 1],
[-12 : 17 : 1]
1.99132820356079,
2.61725031350952
[-12, 17], [2, 45], [14, 69] 2017
2018 [0, 0, 0, 0, 2018]
0
1759243968
{O}
-
1 [1378 : 18089 : 8] 6.58041149262411
- 2018
2019 [0, 0, 0, 0, 2019]
0
1760987952
{O}
-
1 [615 : 3176 : 27] 5.67799718109606
- 2019
2020 [0, 0, 0, 0, 2020]
0
27542700
{O}
-
0 - - - 2020
2021 [0, 0, 0, 0, 2021]
0
441119628
{O}
-
1 [26447597175 : 157839641146 : 781229961] 17.4078046234275
- 2021
2022 [0, 0, 0, 0, 2022]
0
1766225088
{O}
-
0 - - - 2022
2023 [0, 0, 0, 0, 2023]
0
6117552
{O}
-
1 [20535 : 160028 : 3375] 8.45922725743587
- 2023
2024 [0, 0, 0, 0, 2024]
0
36869184
{O}
-
3 [-10 : 32 : 1],
[1 : 45 : 1],
[-7 : 41 : 1]
2.36761089380362,
2.45140428982595,
3.03392796994677
[-10, 32], [-7, 41], [1, 45], [26, 140], [58, 444] 2024
2025 [0, 0, 0, 0, 2025]
0
24300
Z/3Z
[0, 45]
1 [-9 : 36 : 1] 1.58672692908361
[-9, 36], [0, 45], [10, 55], [90, 855] 2025
2026 [0, 0, 0, 0, 2026]
0
591073344
{O}
-
1 [-1 : 45 : 1] 2.45165080811151
[-1, 45] 2026
2027 [0, 0, 0, 0, 2027]
0
1774970928
{O}
-
1 [17989157611001430 : -113443078629363317 : 468554286663000] 26.3747468611418
- 2027
2028 [0, 0, 0, 0, 2028]
0
657072
{O}
-
1 [13 : 65 : 1] 1.49104373429137
[13, 65] 2028
2029 [0, 0, 0, 0, 2029]
0
444618828
{O}
-
0 - - - 2029
2030 [0, 0, 0, 0, 2030]
0
1780228800
{O}
-
2 [-796 : 659 : 64],
[1497 : 11213 : 27]
5.85822178398087,
6.35440216830304
- 2030
2031 [0, 0, 0, 0, 2031]
0
1781983152
{O}
-
0 - - - 2031
2032 [0, 0, 0, 0, 2032]
0
6967728
{O}
-
0 - - - 2032
2033 [0, 0, 0, 0, 2033]
0
148791204
{O}
-
2 [-2 : 45 : 1],
[-8 : 39 : 1]
1.99329087081334,
2.03238940003828
[-8, 39], [-2, 45], [11, 58], [206, 2957] 2033
2034 [0, 0, 0, 0, 2034]
0
198583488
{O}
-
0 - - - 2034
2035 [0, 0, 0, 0, 2035]
0
596336400
{O}
-
1 [62096 : 304299 : 4096] 8.25922111223058
- 2035
2036 [0, 0, 0, 0, 2036]
0
27980748
{O}
-
0 - - - 2036
2037 [0, 0, 0, 0, 2037]
0
448131852
{O}
-
0 - - - 2037
2038 [0, 0, 0, 0, 2038]
0
1794287808
{O}
-
1 [51 : 367 : 1] 4.08642224156813
[51, 367] 2038
2039 [0, 0, 0, 0, 2039]
0
1796049072
{O}
-
0 - - - 2039
2040 [0, 0, 0, 0, 2040]
0
112363200
{O}
-
0 - - - 2040
2041 [0, 0, 0, 0, 2041]
0
449893548
{O}
-
1 [714 : 3869 : 27] 5.31490092460849
- 2041
2042 [0, 0, 0, 0, 2042]
0
600446016
{O}
-
1 [2795 : 14373 : 125] 6.14023544980056
- 2042
2043 [0, 0, 0, 0, 2043]
0
200344752
{O}
-
1 [-1600944046 : -40182823205 : 890277128] 16.7437047147533
- 2043
2044 [0, 0, 0, 0, 2044]
0
37601424
{O}
-
1 [-114 : 1198 : 27] 4.52113257930870
- 2044
2045 [0, 0, 0, 0, 2045]
0
451658700
{O}
-
0 - - - 2045
2046 [0, 0, 0, 0, 2046]
0
1808402112
{O}
-
1 [25485 : 167957 : 3375] 8.48758522187042
- 2046
2047 [0, 0, 0, 0, 2047]
0
1810170288
{O}
-
1 [93 : 898 : 1] 4.61687568359885
[93, 898] 2047
2048 [0, 0, 0, 0, 2048]
0
1728
{O}
-
0 - - - 2048
2049 [0, 0, 0, 0, 2049]
0
453427308
{O}
-
1 [40 : 257 : 1] 3.42540880882959
[40, 257] 2049
2050 [0, 0, 0, 0, 2050]
0
72619200
{O}
-
1 [-26013 : 43393 : 2197] 8.21241497022669
- 2050
2051 [0, 0, 0, 0, 2051]
0
605750544
{O}
-
1 [65 : 526 : 1] 4.29546909047769
[65, 526] 2051
2052 [0, 0, 0, 0, 2052]
0
38988
{O}
-
2 [-12 : 18 : 1],
[-3 : 45 : 1]
0.69538152334806,
1.54696805914760
[-12, 18], [-3, 45], [4, 46], [24, 126], [64, 514], [168, 2178] 2052
2053 [0, 0, 0, 0, 2053]
0
151733124
{O}
-
0 - - - 2053
2054 [0, 0, 0, 0, 2054]
0
1822571712
{O}
-
1 [20980 : 364157 : 8000] 9.00333692437987
- 2054
2055 [0, 0, 0, 0, 2055]
0
1824346800
{O}
-
1 [-18512283236759970 : -75846530161127557 : 2071209156723000] 26.5667487903427
- 2055
2056 [0, 0, 0, 0, 2056]
0
114132672
{O}
-
1 [1530 : 7796 : 125] 5.70882212287809
- 2056
2057 [0, 0, 0, 0, 2057]
0
3776652
{O}
-
0 - - - 2057
2058 [0, 0, 0, 0, 2058]
0
762048
{O}
-
2 [7 : 49 : 1],
[1778 : 26509 : 8]
1.11726518626281,
4.87941004354801
[7, 49] 2058
2059 [0, 0, 0, 0, 2059]
0
1831455792
{O}
-
2 [114 : 563 : 8],
[-8721 : 311080 : 6859]
4.62572073041143,
8.89575428164279
- 2059
2060 [0, 0, 0, 0, 2060]
0
38192400
{O}
-
1 [-11 : 27 : 1] 2.52637599273676
[-11, 27] 2060
2061 [0, 0, 0, 0, 2061]
0
50972652
{O}
-
1 [-5 : 44 : 1] 3.02468756390715
[-5, 44] 2061
2062 [0, 0, 0, 0, 2062]
0
612265536
{O}
-
1 [14553 : 77257 : 1331] 7.94522939622964
- 2062
2063 [0, 0, 0, 0, 2063]
0
1838578608
{O}
-
1 [2219 : 16570 : 343] 6.94684222419643
- 2063
2064 [0, 0, 0, 0, 2064]
0
449307
{O}
-
1 [25 : 133 : 1] 3.54332909075392
[25, 133] 2064
2065 [0, 0, 0, 0, 2065]
0
460536300
{O}
-
2 [-6 : 43 : 1],
[-1932 : 14897 : 343]
2.57031541243955,
6.45943666240103
[-6, 43], [55371, 13029374] 2065
2066 [0, 0, 0, 0, 2066]
0
1843929792
{O}
-
0 - - - 2066
2067 [0, 0, 0, 0, 2067]
0
1845715248
{O}
-
2 [49 : 346 : 1],
[-13481034 : -4366835 : 1061208]
4.05363438280788,
12.3435183398261
[49, 346] 2067
2068 [0, 0, 0, 0, 2068]
0
28867212
{O}
-
1 [-348 : 2795 : 64] 5.80137622319028
- 2068
2069 [0, 0, 0, 0, 2069]
0
154107396
{O}
-
2 [35 : 212 : 1],
[-3227 : 12060 : 343]
3.78407108147520,
6.40437584869771
[35, 212] 2069
2070 [0, 0, 0, 0, 2070]
0
205675200
{O}
-
1 [437047 : -6165017 : 2197] 10.4622818982176
- 2070
2071 [0, 0, 0, 0, 2071]
0
617621904
{O}
-
1 [-2790 : 45269 : 1000] 7.61903922495286
- 2071
2072 [0, 0, 0, 0, 2072]
0
115915968
{O}
-
1 [138 : 1268 : 27] 4.54117909986172
- 2072
2073 [0, 0, 0, 0, 2073]
0
464111532
{O}
-
2 [28 : 155 : 1],
[-294 : 755 : 27]
3.15872508549212,
4.81160344176695
[28, 155] 2073
2074 [0, 0, 0, 0, 2074]
0
1858237632
{O}
-
1 [-157450313655 : 905151903193 : 21928488399] 18.9171038222359
- 2074
2075 [0, 0, 0, 0, 2075]
0
74401200
{O}
-
1 [29647 : 556162 : 12167] 9.28517338181250
- 2075
2076 [0, 0, 0, 0, 2076]
0
116363952
{O}
-
0 - - - 2076
2077 [0, 0, 0, 0, 2077]
0
465904332
{O}
-
1 [-2404740 : -5703031 : 216000] 11.2681113489774
- 2077
2078 [0, 0, 0, 0, 2078]
0
621804096
{O}
-
1 [-505 : 5607 : 125] 5.69091573893883
- 2078
2079 [0, 0, 0, 0, 2079]
0
2561328
{O}
-
0 - - - 2079
2080 [0, 0, 0, 0, 2080]
0
2433600
{O}
-
1 [9 : 53 : 1] 3.10567188124081
[9, 53] 2080
2081 [0, 0, 0, 0, 2081]
0
467700588
{O}
-
0 - - - 2081
2082 [0, 0, 0, 0, 2082]
0
1872600768
{O}
-
1 [-764 : 1249 : 64] 5.86062416643334
- 2082
2083 [0, 0, 0, 0, 2083]
0
1874400048
{O}
-
1 [9341089296 : 74715061535 : 147197952] 16.8133045997222
- 2083
2084 [0, 0, 0, 0, 2084]
0
29315628
{O}
-
1 [5 : 47 : 1] 3.03755950728192
[5, 47] 2084
2085 [0, 0, 0, 0, 2085]
0
469500300
{O}
-
1 [2692264867183047 : 13231891554750946 : 170926154248473] 25.1330811240775
- 2085
2086 [0, 0, 0, 0, 2086]
0
1879803072
{O}
-
0 - - - 2086
2087 [0, 0, 0, 0, 2087]
0
627201936
{O}
-
1 [-14 : 365 : 8] 4.39889108263055
- 2087
2088 [0, 0, 0, 0, 2088]
0
13079232
{O}
-
2 [6 : 48 : 1],
[33 : 195 : 1]
1.62590330374827,
3.00796492065949
[6, 48], [33, 195], [42, 276] 2088
2089 [0, 0, 0, 0, 2089]
0
157101156
{O}
-
4 [-4 : 45 : 1],
[-10 : 33 : 1],
[-12 : 19 : 1],
[3 : 46 : 1]
2.01143112262341,
2.05889278037726,
2.62754246798052,
3.01975805440168
[-12, 19], [-10, 33], [-4, 45], [3, 46], [8, 51], [18, 89], [60, 467], [71, 600], [80, 717], [170, 2217], [183, 2476], [698, 18441], [9278, 893679], [129968, 46854861] 2089
2090 [0, 0, 0, 0, 2090]
0
1887019200
{O}
-
1 [1487892546595970232039749 : 10582385037046255021677047 : 29913091361805999448171] 38.5684980629649
- 2090
2091 [0, 0, 0, 0, 2091]
0
1888825392
{O}
-
1 [2546 : 45421 : 8] 7.17385187741377
- 2091
2092 [0, 0, 0, 0, 2092]
0
118164528
{O}
-
1 [35046 : 491155 : 10648] 9.20435982447848
- 2092
2093 [0, 0, 0, 0, 2093]
0
473110092
{O}
-
1 [243781658889 : -2437292362240 : 2444008923] 19.0930312760924
- 2093
2094 [0, 0, 0, 0, 2094]
0
1894249152
{O}
-
0 - - - 2094
2095 [0, 0, 0, 0, 2095]
0
1896058800
{O}
-
0 - - - 2095
2096 [0, 0, 0, 0, 2096]
0
2471184
{O}
-
1 [50 : 387 : 8] 3.89334308744205
- 2096
2097 [0, 0, 0, 0, 2097]
0
52768908
{O}
-
0 - - - 2097
2098 [0, 0, 0, 0, 2098]
0
633830976
{O}
-
1 [-9 : 37 : 1] 3.06238064297399
[-9, 37] 2098
2099 [0, 0, 0, 0, 2099]
0
1903306032
{O}
-
1 [6916785853836 : -114510871541773 : 2486712422592] 22.0479143721873
- 2099
2100 [0, 0, 0, 0, 2100]
0
1190700
{O}
-
1 [85 : 785 : 1] 3.46275948290544
[85, 785] 2100
2101 [0, 0, 0, 0, 2101]
0
476733708
{O}
-
2 [15 : 74 : 1],
[270480 : 2205979 : 4096]
3.26476170738230,
9.85549698987342
[15, 74] 2101
2102 [0, 0, 0, 0, 2102]
0
1908750528
{O}
-
0 - - - 2102
2103 [0, 0, 0, 0, 2103]
0
1910567088
{O}
-
0 - - - 2103
2104 [0, 0, 0, 0, 2104]
0
119524032
{O}
-
0 - - - 2104
2105 [0, 0, 0, 0, 2105]
0
159516900
{O}
-
0 - - - 2105
2106 [0, 0, 0, 0, 2106]
0
2628288
{O}
-
0 - - - 2106
2107 [0, 0, 0, 0, 2107]
0
13046544
{O}
-
1 [-7 : 42 : 1] 1.19998235158524
[-7, 42] 2107
2108 [0, 0, 0, 0, 2108]
0
119978928
{O}
-
1 [2 : 46 : 1] 2.32416909743345
[2, 46] 2108
2109 [0, 0, 0, 0, 2109]
0
480371148
{O}
-
0 - - - 2109
2110 [0, 0, 0, 0, 2110]
0
1923307200
{O}
-
1 [57 : 1243 : 27] 5.21533561657980
- 2110
2111 [0, 0, 0, 0, 2111]
0
1925130672
{O}
-
0 - - - 2111
2112 [0, 0, 0, 0, 2112]
0
117612
{O}
-
1 [-8 : 40 : 1] 1.20722936899394
[-8, 40] 2112
2113 [0, 0, 0, 0, 2113]
0
482195052
{O}
-
1 [40236 : -436073 : 343] 8.26194826315689
- 2113
2114 [0, 0, 0, 0, 2114]
0
643535424
{O}
-
1 [295 : 5067 : 1] 5.16443675546738
[295, 5067] 2114
2115 [0, 0, 0, 0, 2115]
0
214714800
{O}
-
2 [1 : 46 : 1],
[-11 : 28 : 1]
3.01534059401142,
3.08352937928456
[-11, 28], [1, 46] 2115
2116 [0, 0, 0, 0, 2116]
0
19044
Z/3Z
[0, 46]
1 [12 : 62 : 1] 2.25896026976769
[0, 46], [12, 62], [69, 575] 2116
2117 [0, 0, 0, 0, 2117]
0
484022412
{O}
-
2 [-1 : 46 : 1],
[1714 : 25091 : 8]
3.01557652153693,
6.79032434702357
[-1, 46] 2117
2118 [0, 0, 0, 0, 2118]
0
1937919168
{O}
-
1 [-2651767905966312 : -6065948886252515 : 234382961788416] 25.1456359460677
- 2118
2119 [0, 0, 0, 0, 2119]
0
1939749552
{O}
-
1 [898848 : 4943405 : 32768] 10.5402746581643
- 2119
2120 [0, 0, 0, 0, 2120]
0
121348800
{O}
-
0 - - - 2120
2121 [0, 0, 0, 0, 2121]
0
485853228
{O}
-
1 [22 : 113 : 1] 3.00352832595072
[22, 113] 2121
2122 [0, 0, 0, 0, 2122]
0
1945245888
{O}
-
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Last Update: 2021.08.31
H.Nakao

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