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Rational Points of Projective Curves: X^3+Y^3+Z^3 = nXYZ (n \in [0..100])

[2004.12.04]X^3+Y^3+Z^3=nXYZ(n \in [0..100])の有理点


■整数nを固定したとき、射影曲線Cn: X3+Y3+Z3=nXYZの有理点[X:Y:Z]を求める。
Cnの自明な有理点、つまり、XYZ=0である有理点[X:Y:Z]を求める。
Z=0とすると、X3+Y3=0より、X+Y=0となる。よって、[X:Y:Z]=[X:-X:0]=[-1:1:0]である。
同様にして、Y=0とすると、[X:Y:Z]=[-1:0:1]である。X=0とすると、[X:Y:Z]=[0:-1:1] である。
Cnは、３個の自明な有理点[-1:1:0],[-1:0:1],[0:-1:1]を持つ。

■n=0の場合
X,Y,Zを整数とするとき、
     C0: X3+Y3+Z3 = 0
ならば、Fermatの最終定理(cubic case)より、XYZ=0であるので、曲線C0の有理点は、自明なものに限る。

■n=3の場合
X,Y,Zを整数とするとき、
     C3: X3+Y3+Z3 = 3XYZ
より、
     (X+Y+Z)(X2+Y2+Z2-XY-YZ-ZX) = 0
     (X+Y+Z)((X-Y)2+(Y-Z)2+(Z-X)2)/2 = 0
となるので、
     X+Y+Z=0 または X=Y=Z
を得る。
C3の自明でない有理点は、[1:1:1],[u:v:-(u+v)](u,vは互いに素な整数,uv!=0)に限る。

■n > 0, n!=3の場合
射影曲線Cnの自明でない有理点[X:Y:Z]を求める。
そのためには、x=X/Z, y=Y/Zとして、曲線
     C'n: x3+y3+1 = nxy
の有理点を求めれば良い。

曲線C'nは、双有理変換
     S = x+y,
     T = (3(x+y)+n)(x-y)
[逆変換は、
     x = (S+T/{3S+n})/2,
     y = (S-T/{3S+n})/2
]
によって、曲線
     An: -T2 = (3S+n)(S3-nS2+4)
に写される。

曲線Anは、双有理変換
     U = 3S+n,
     V = T
[逆変換は、
     S = (U-n)/3,
     T = V
]
によって、曲線
     A'n: -81V2 = 3U4-18nU3+27n2U2-12(n3-27)U
に写される。

曲線A'nは、双有理変換
     u = 12(n3-27)/U,
     v = 108(n3-27)V/U2
[逆変換は、
     U = 12(n3-27)/u,
     V = 4(n3-27)v/{3u2}
]
によって、曲線
     Bn: v2 = u3-27n2u2+216n(n3-27)u-432(n3-27)2
に写される。

曲線Bnは、双有理変換
     X = u+9n2
     Y = v
[逆変換は、
     u = X-9n2,
     v = Y
]
によって、曲線
     En: Y2 = X3-27(n4+216n)X+54(n6-540n3-5832)
に写される。

これらの各変換を合成すると、曲線C'nから楕円曲線Enへの間の双有理変換は、
     X = {3(-9n2(x+y)+n3-108)}/{3(x+y)+n},
     Y = {108(n3-27)(x-y)}/{3(x+y)+n},
[逆変換は、
     x = {9n3-3nX+Y-972}/{18(X+9n2)},
     y = {9n3-3nX-Y-972}/{18(X+9n2)},
]
となる。

■n!=3のとき、楕円曲線Enの判別式Δ(En)は、
     Δ(En) = 212・312(n-3)3(n2+3n+9)3 > 0
となるので、Enは非特異楕円曲線である。

[pari/gpによる計算]
gp>  en=ec(n)
time = 4 ms.
%96 = [0, 0, 0, -27*n^4 - 5832*n, 54*n^6 - 29160*n^3 - 314928, 0, -54*n^4 - 11664*n, 216*n^6 - 116640*n^3 - 1259712, -729*n^8 - 314928*n^5 - 34012224*n^2, 1296*n^4 + 279936*n, -46656*n^6 + 25194240*n^3 + 272097792, 2176782336*n^9 - 176319369216*n^6 + 4760622968832*n^3 - 42845606719488, (n^12 + 648*n^9 + 139968*n^6 + 10077696*n^3)/(n^9 - 81*n^6 + 2187*n^3 - 19683), 0, 0, 0, 0, 0, 0]
gp>  en.disc
time = 0 ms.
%97 = 2176782336*n^9 - 176319369216*n^6 + 4760622968832*n^3 - 42845606719488
gp>  factor(en.disc)
time = 28 ms.
%98 = 
[n - 3 3]

[n^2 + 3*n + 9 3]

gp>  factor(2176782336)
time = 0 ms.
%99 = 
[2 12]

[3 12]

■楕円曲線Enは、位数3のねじれ点Tn(3(n+6)2, 108(n2 + 3n + 9)を持つ。

[pari/gpによる計算]
gp>  T(n)
time = 0 ms.
%100 = [3*n^2 + 36*n + 108, 108*n^2 + 324*n + 972]
gp>  ellpow(en,T(n),2)
time = 0 ms.
%101 = [3*n^2 + 36*n + 108, -108*n^2 - 324*n - 972]
gp>  ellpow(en,T(n),3)
time = 0 ms.
%102 = [0]

■n=1,2,4,5,...,100に対して、楕円曲線Enのねじれ点群En(Q)torsを計算すると、
     E5(Q)tors =Z/6Z,
     En(Q)tors =Z/3Z if n=1,2,6,...,100
となる。

[pari/gpによる計算]
gp>  for(n=1,100,if(n!=3,print("E_",n,"_{tors}=",elltors(ec(n),1))))
E_1_{tors}=[3, [3], [[147, 1404]]]
E_2_{tors}=[3, [3], [[192, 2052]]]
E_4_{tors}=[3, [3], [[300, 3996]]]
E_5_{tors}=[6, [6], [[-141, 756]]]
E_6_{tors}=[3, [3], [[432, 6804]]]
E_7_{tors}=[3, [3], [[507, 8532]]]
E_8_{tors}=[3, [3], [[588, 10476]]]
E_9_{tors}=[3, [3], [[675, 12636]]]
E_10_{tors}=[3, [3], [[768, 15012]]]
E_11_{tors}=[3, [3], [[867, 17604]]]
E_12_{tors}=[3, [3], [[972, 20412]]]
E_13_{tors}=[3, [3], [[1083, 23436]]]
E_14_{tors}=[3, [3], [[1200, 26676]]]
E_15_{tors}=[3, [3], [[1323, 30132]]]
E_16_{tors}=[3, [3], [[1452, 33804]]]
E_17_{tors}=[3, [3], [[1587, 37692]]]
E_18_{tors}=[3, [3], [[1728, 41796]]]
E_19_{tors}=[3, [3], [[1875, 46116]]]
E_20_{tors}=[3, [3], [[2028, 50652]]]
E_21_{tors}=[3, [3], [[2187, 55404]]]
E_22_{tors}=[3, [3], [[2352, 60372]]]
E_23_{tors}=[3, [3], [[2523, 65556]]]
E_24_{tors}=[3, [3], [[2700, 70956]]]
E_25_{tors}=[3, [3], [[2883, 76572]]]
E_26_{tors}=[3, [3], [[3072, 82404]]]
E_27_{tors}=[3, [3], [[3267, 88452]]]
E_28_{tors}=[3, [3], [[3468, 94716]]]
E_29_{tors}=[3, [3], [[3675, 101196]]]
E_30_{tors}=[3, [3], [[3888, 107892]]]
E_31_{tors}=[3, [3], [[4107, 114804]]]
E_32_{tors}=[3, [3], [[4332, 121932]]]
E_33_{tors}=[3, [3], [[4563, 129276]]]
E_34_{tors}=[3, [3], [[4800, 136836]]]
E_35_{tors}=[3, [3], [[5043, 144612]]]
E_36_{tors}=[3, [3], [[5292, 152604]]]
E_37_{tors}=[3, [3], [[5547, 160812]]]
E_38_{tors}=[3, [3], [[5808, 169236]]]
E_39_{tors}=[3, [3], [[6075, 177876]]]
E_40_{tors}=[3, [3], [[6348, 186732]]]
E_41_{tors}=[3, [3], [[6627, 195804]]]
E_42_{tors}=[3, [3], [[6912, 205092]]]
E_43_{tors}=[3, [3], [[7203, 214596]]]
E_44_{tors}=[3, [3], [[7500, 224316]]]
E_45_{tors}=[3, [3], [[7803, 234252]]]
E_46_{tors}=[3, [3], [[8112, 244404]]]
E_47_{tors}=[3, [3], [[8427, 254772]]]
E_48_{tors}=[3, [3], [[8748, 265356]]]
E_49_{tors}=[3, [3], [[9075, 276156]]]
E_50_{tors}=[3, [3], [[9408, 287172]]]
E_51_{tors}=[3, [3], [[9747, 298404]]]
E_52_{tors}=[3, [3], [[10092, 309852]]]
E_53_{tors}=[3, [3], [[10443, 321516]]]
E_54_{tors}=[3, [3], [[10800, 333396]]]
E_55_{tors}=[3, [3], [[11163, 345492]]]
E_56_{tors}=[3, [3], [[11532, 357804]]]
E_57_{tors}=[3, [3], [[11907, 370332]]]
E_58_{tors}=[3, [3], [[12288, 383076]]]
E_59_{tors}=[3, [3], [[12675, 396036]]]
E_60_{tors}=[3, [3], [[13068, 409212]]]
E_61_{tors}=[3, [3], [[13467, 422604]]]
E_62_{tors}=[3, [3], [[13872, 436212]]]
E_63_{tors}=[3, [3], [[14283, 450036]]]
E_64_{tors}=[3, [3], [[14700, 464076]]]
E_65_{tors}=[3, [3], [[15123, 478332]]]
E_66_{tors}=[3, [3], [[15552, 492804]]]
E_67_{tors}=[3, [3], [[15987, 507492]]]
E_68_{tors}=[3, [3], [[16428, 522396]]]
E_69_{tors}=[3, [3], [[16875, 537516]]]
E_70_{tors}=[3, [3], [[17328, 552852]]]
E_71_{tors}=[3, [3], [[17787, 568404]]]
E_72_{tors}=[3, [3], [[18252, 584172]]]
E_73_{tors}=[3, [3], [[18723, 600156]]]
E_74_{tors}=[3, [3], [[19200, 616356]]]
E_75_{tors}=[3, [3], [[19683, 632772]]]
E_76_{tors}=[3, [3], [[20172, 649404]]]
E_77_{tors}=[3, [3], [[20667, 666252]]]
E_78_{tors}=[3, [3], [[21168, 683316]]]
E_79_{tors}=[3, [3], [[21675, 700596]]]
E_80_{tors}=[3, [3], [[22188, 718092]]]
E_81_{tors}=[3, [3], [[22707, 735804]]]
E_82_{tors}=[3, [3], [[23232, 753732]]]
E_83_{tors}=[3, [3], [[23763, 771876]]]
E_84_{tors}=[3, [3], [[24300, 790236]]]
E_85_{tors}=[3, [3], [[24843, 808812]]]
E_86_{tors}=[3, [3], [[25392, 827604]]]
E_87_{tors}=[3, [3], [[25947, 846612]]]
E_88_{tors}=[3, [3], [[26508, 865836]]]
E_89_{tors}=[3, [3], [[27075, 885276]]]
E_90_{tors}=[3, [3], [[27648, 904932]]]
E_91_{tors}=[3, [3], [[28227, 924804]]]
E_92_{tors}=[3, [3], [[28812, 944892]]]
E_93_{tors}=[3, [3], [[29403, 965196]]]
E_94_{tors}=[3, [3], [[30000, 985716]]]
E_95_{tors}=[3, [3], [[30603, 1006452]]]
E_96_{tors}=[3, [3], [[31212, 1027404]]]
E_97_{tors}=[3, [3], [[31827, 1048572]]]
E_98_{tors}=[3, [3], [[32448, 1069956]]]
E_99_{tors}=[3, [3], [[33075, 1091556]]]
E_100_{tors}=[3, [3], [[33708, 1113372]]]
time = 2mn, 54,832 ms.

■楕円曲線E6のMordell-Weil群E6(Q)のrankは1であり、その生成元は(-72, 756)である。
     E6(Q) = Z+Z/3Z
[mwrank3による計算]
bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0, 0, 0, -69984, -4094064]

Curve [0,0,0,-69984,-4094064] : Working with minimal curve [0,0,1,-54,-88]
        [u,r,s,t] = [6,0,0,108]
No points of order 2
Basic pair: I=2592, J=151632
disc=46664771328
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = 2592, J = 151632
Looking for Type 2 quartics:
Trying positive a from 1 up to 6 (square a first...)
(1,0,-288,2628,-6696)   --trivial
(1,0,-240,1988,-4584)   --trivial
(1,0,-72,252,-216)      --trivial
(1,-1,-48,32,16)        --trivial
Trying positive a from 1 up to 6 (...then non-square a)
Trying negative a from -1 down to -13
(-3,0,54,84,9)  --trivial
(-6,-6,36,24,-12)       --trivial
(-11,-12,36,36,0)       --trivial
Finished looking for Type 2 quartics.
Looking for Type 1 quartics:
Trying positive a from 1 up to 19 (square a first...)
(1,0,12,28,204) --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [-2 : 3 : 1]
        height = 0.621081184070788
Rank of B=im(eps) increases to 1 (The previous point is on the egg)
Exiting search for Type 1 quartics after finding one which is globally soluble.
Mordell rank contribution from B=im(eps) = 1
Selmer  rank contribution from B=im(eps) = 1
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 1
Points generating E(Q)/2E(Q):
Point [-72 : 756 : 1], height = 0.621081184070788
After descent, rank of points found is 1
Transferring points back to original curve [0,0,0,-69984,-4094064]

Generator 1 is [-72 : 756 : 1]; height 0.621081184070788

The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
 (modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 0.621081184070788

 (2.8 seconds)
Enter curve: 
[0,0,0,0,0]

bash-2.05a$

よって、E6(Q)の有理点をいくつか求めると、以下のようになる。
[pari/gpにおる計算]
gp>  rpE([-72,756],6,10)
[-72, 756]
[1440, 53676]
[-143, 1729]
[2030544/5041, 2058290892/357911]
[-551647224/2505889, 3182984139636/3966822287]
[3494518273/11957764, -26542978919423/41349947912]
[-1555752240978840/7570240479649, -26419142596839574241268/20828812647389616143]
[18960879673407511152/37366860526179409, -68922457551136410882341289588/7223204598598581696147673]
[-3238593975006752614031/27086596717186175625, -225743134292184826056518737255297/140971610929930949467190953125]
[117619779875881832108980192512/44420883071343155073512929, -40132271189535597136143092415057942654320148/296060713794572486917377420761020609967]
time = 5 ms.

■曲線C'6の有理点をいくつか計算すると、以下のようになる。
[pari/gpによる計算]
gp> rpC([-72,756],6,10)
[2/3, 1/3]
[19/21, -52/21]
[5275/3258, 1817/3258]
[-124904/3096807, -2847511/3096807]
[15051171563/4904676969, 10840875082/4904676969]
[-203863624933571/458665691607396, -150777667094725/458665691607396]
[46875396961726681714/29381282043563909553, 81821352777652044467/29381282043563909553]
[-84573893519721693268169777/71494678266896520634735569, 6593378373315887264027696/71494678266896520634735569]
[214674741825814609600406026499153/518752964649250744517842481308050, 666161010410184261713443501009747/518752964649250744517842481308050]
[-35684611132352443884094017956885819475716/10474376943941514032799536181621296708451, 17400403668620568200751342380492411548517/10474376943941514032799536181621296708451]
time = 7 ms.

■射影曲線C6の有理点をいくつか計算すると、以下のようになる。
[pari/gpによる計算]
gp> rpCC([-72,756],6,10)
[2, 1, 3]
[19, -52, 21]
[5275, 1817, 3258]
[-124904, -2847511, 3096807]
[15051171563, 10840875082, 4904676969]
[-203863624933571, -150777667094725, 458665691607396]
[46875396961726681714, 81821352777652044467, 29381282043563909553]
[-84573893519721693268169777, 6593378373315887264027696, 71494678266896520634735569]
[214674741825814609600406026499153, 666161010410184261713443501009747, 518752964649250744517842481308050]
[-35684611132352443884094017956885819475716, 17400403668620568200751342380492411548517, 10474376943941514032799536181621296708451]
time = 15 ms.

- E~n: Y2Z=X3-27(n4+216n)XZ2+54(n6-540n3-5832)Z3
En: y2=x3-27(n4+216n)x+54(n6-540n3-5832)
 C~n: X3+Y3+Z3=nXYZ
Cn: x3+y3+1=nxy
 -
n [a1,a2,a3,a4,a6]
j(En) Complex Multiplication.
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		 Cn(Q)/Cn(Q)torsの生成元
[x,y] 		C~nの自明でない有理点[X:Y:Z]
 n
0 [0, 0, 0, 0, -314928]
0 CM
27
 Z/3Z
[108, 972]
0
 -
 -
 -
 -
 0
1 [0, 0, 0, -5859, -344034]
-10218313/17576
26
 Z/3Z
[147, 1404]
0
 -
 -
 -
 -
 1
2 [0, 0, 0, -12096, -544752]
-89915392/6859
19
 Z/3Z
[192, 2052]
0
 -
 -
 -
 -
 2
4 [0, 0, 0, -30240, -1959984]
1404928000/50653
37
 Z/3Z
[300, 3996]
0
 -
 -
 -
 -
 4
5 [0, 0, 0, -46035, -3116178]
4956477625/941192
14
 Z/6Z
[-141, 756]
0
 -
 -
 -
 [2 : 1 : 1]
 5
6 [0, 0, 0, -69984, -4094064]
884736/343
189
 Z/3Z
[432, 6804]
1
 [-72 : 756 : 1]
 0.621081184070788
 [2/3, 1/3]
 [2 : 1 : 3],
[19 : -52 : 21],
...
 6
7 [0, 0, 0, -105651, -3963762]
59914169497/31554496
158
 Z/3Z
[507, 8532]
0
 -
 -
 -
 -
 7
8 [0, 0, 0, -157248, -1089072]
197544116224/114084125
485
 Z/3Z
[588, 10476]
0
 -
 -
 -
 -
 8
9 [0, 0, 0, -229635, 7125246]
31255875/17576
702
 Z/3Z
[675, 12636]
1
 [-378 : 6318 : 1]
 1.12173272561309
 [7/2, 3/2]
 [7 : 3 : 2],
[-1005 : 133 : 632],
...
 9
10 [0, 0, 0, -328320, 24525072]
1798045696000/921167317
973
 Z/3Z
[768, 15012]
1
 [-312 : 9828 : 1]
 1.73976255464624
 [18/7, 5/7]
 [18 : 5 : 7],
[-27445 : -3924 : 39949],
...
 10
11 [0, 0, 0, -459459, 56537406]
4927753743913/2217342464
326
 Z/3Z
[867, 17604]
0
 -
 -
 -
 -
 11
12 [0, 0, 0, -629856, 110539728]
884736/343
189
 Z/3Z
[972, 20412]
0
 -
 -
 -
 -
 12
13 [0, 0, 0, -846963, 196268238]
30867540216409/10218313000
2170
 Z/3Z
[1083, 23436]
1
 [246 : 1674 : 1]
 2.21622549111872
 [13/38, 9/38]
 [13 : 9 : 38],
[474075 : -703859 : 55784],
...
 13
14 [0, 0, 0, -1118880, 326264976]
71163817984000/20057135813
2717
 Z/3Z
[1200, 26676]
1
 [264 : 7020 : 1]
 1.55136626557583
 [7/13, 2/13]
 [7 : 2 : 13],
[3708 : -15323 : 4355],
...
 14
15 [0, 0, 0, -1454355, 516363822]
7940149875/1906624
1674
 Z/3Z
[1323, 30132]
1
 [11367 : 1205280 : 1]
 0.988016162029668
 [3, -7]
 [3 : -7 : 1],
[91 : -516 : 185],
...
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 -
 -
 89
90 [0, 0, 0, -1771994880, 28676556045072]
14361665363509248000/19680813080999
104139
 Z/3Z
[27648, 904932]
0
 -
 -
 -
 -
 90
91 [0, 0, 0, -1852054659, 30642965164926]
322752865415206614683113/427883804368413184
188386
 Z/3Z
[28227, 924804]
0
 -
 -
 -
 -
 91
92 [0, 0, 0, -1934797536, 32720463215568]
367972433096436139589632/472112250234418781
778661
 Z/3Z
[28812, 944892]
1
 [76139824482838284 : -1446282127025483796 : 2781210948047]
 20.6407890205949
 [-20446843218005/35661385544981, -548624531286/35661385544981]
 [-20446843218005 : -548624531286 : 35661385544981],
[29570913061065806338271638543465089110883785414630076 : -927304754271808106563036706213981027673433469999994985 : 304837663357987553057411465416672141266939247168964089],
...
 92
93 [0, 0, 0, -2020282803, 34914014541198]
29196203893939467/36264691000
89370
 Z/3Z
[29403, 965196]
0
 -
 -
 -
 -
 93
94 [0, 0, 0, -2108570400, 37228748032656]
476291712014161408000000/572938924592818693
830557
 Z/3Z
[30000, 985716]
1
 [22305 : 1137591 : 1],
[136272 : 47754252 : 1]
 4.49273733329212,
6.90026024772712
 [945/746, 19/746],
[11111/2569, -52056/2569]
 [945 : 19 : 746],
[11111 : -52056 : 2569],
[-8146206091 : -392320602765 : 629550717436],
[70522666866122832 : -1567536726449591375 : 365914133782922543],
...
 94
95 [0, 0, 0, -2199720915, 39669960723822]
540768592178120621373625/630189869156528192
428674
 Z/3Z
[30603, 1006452]
0
 -
 -
 -
 -
 95
96 [0, 0, 0, -2293795584, 42243121426896]
31151685664229031936/35181150961663
884709
 Z/3Z
[31212, 1027404]
1
 [85572 : 21738564 : 1]
 2.04942566352733
 [5/3, -38/3]
 [5 : -38 : 3],
[3724 : -274495 : 164991],
...
 96
97 [0, 0, 0, -2390856291, 44953874406558]
694335107320069385499337/760163589918162136
912646
 Z/3Z
[31827, 1048572]
0
 -
 -
 -
 -
 97
98 [0, 0, 0, -2490965568, 47808043093008]
785257116283731782139904/833676010955667125
941165
 Z/3Z
[32448, 1069956]
1
 [3505966464 : 23792987196 : 117649]
 11.6099349488318
 [-2559169/59978401, -14154192/59978401]
 [-2559169 : -14154192 : 59978401],
[-3054242160141791718350449429920 : 559440694955858975865297275041 : 169073488605624583423710715679],
...
 98
99 [0, 0, 0, -2594186595, 50811633833886]
45063098241089803875/46407610105856
60642
 Z/3Z
[33075, 1091556]
1
 [11533302848898 : -2130798476829492 : 184220009]
 15.2176919537895
 [-14466072543/1832602198, 1150522313/1832602198]
 [-14466072543 : 1150522313 : 1832602198],
[-3490028850205447028882735266461990051887 : -67002789957082521176183340772951815585 : 5550582306991336944516169719922718858192],
...
 99
100 [0, 0, 0, -2700583200, 53970839685072]
1000648139978077696000000/999919002186980317
76921
 Z/3Z
[33708, 1113372]
0
 -
 -
 -
 -
 100

[参考文献]

Last Update: 2020.06.21
H.Nakao

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