群構造が　はいつたら　どんな　美味しい　事　が　ありますか?

   　中学生知悉の　易し過ぎの　直線　L; y=2*(x - 3)- 67/4
           上には　無論 ●有理点が　無限に在り●
                      例えば
{{-1, -(99/4)}, {-(68/69), -(6823/276)}, {-(67/69), -(6815/276)}, {-(
   22/23), -(2269/92)}, {-(65/69), -(6799/276)}, {-(64/69), -(6791/
   276)}, {-(21/23), -(2261/92)}, {-(62/69), -(6775/276)}, {-(61/
   69), -(6767/276)}, {-(20/23), -(2253/92)}, {-(59/69), -(6751/
   276)}, {-(58/69), -(6743/276)}, {-(19/23), -(2245/92)}, {-(56/
   69), -(6727/276)}, {-(55/69), -(6719/276)}, {-(18/23), -(2237/
   92)}, {-(53/69), -(6703/276)}, {-(52/69), -(6695/276)}, {-(17/
   23), -(2229/92)}, {-(50/69), -(6679/276)}, {-(49/69), -(6671/
   276)}, {-(16/23), -(2221/92)}, {-(47/69), -(6655/276)}, {-(2/3), -(
   289/12)}, {-(15/23), -(2213/92)}, {-(44/69), -(6631/276)}, {-(43/
   69), -(6623/276)}, {-(14/23), -(2205/92)}, {-(41/69), -(6607/
   276)}, {-(40/69), -(6599/276)}, {-(13/23), -(2197/92)}, {-(38/
   69), -(6583/276)}, {-(37/69), -(6575/276)}, {-(12/23), -(2189/
   92)}, {-(35/69), -(6559/276)}, {-(34/69), -(6551/276)}, {-(11/
   23), -(2181/92)}, {-(32/69), -(6535/276)}, {-(31/69), -(6527/
   276)}, {-(10/23), -(2173/92)}, {-(29/69), -(6511/276)}, {-(28/
   69), -(6503/276)}, {-(9/23), -(2165/92)}, {-(26/69), -(6487/
   276)}, {-(25/69), -(6479/276)}, {-(8/23), -(2157/92)}, {-(1/3), -(
   281/12)}, {-(22/69), -(6455/276)}, {-(7/23), -(2149/92)}, {-(20/
   69), -(6439/276)}, {-(19/69), -(6431/276)}, {-(6/23), -(2141/
   92)}, {-(17/69), -(6415/276)}, {-(16/69), -(6407/276)}, {-(5/
   23), -(2133/92)}, {-(14/69), -(6391/276)}, {-(13/69), -(6383/
   276)}, {-(4/23), -(2125/92)}, {-(11/69), -(6367/276)}, {-(10/
   69), -(6359/276)}, {-(3/23), -(2117/92)}, {-(8/69), -(6343/
   276)}, {-(7/69), -(6335/276)}, {-(2/23), -(2109/92)}, {-(5/69), -(
   6319/276)}, {-(4/69), -(6311/276)}, {-(1/23), -(2101/92)}, {-(2/
   69), -(6295/276)}, {-(1/69), -(6287/276)}, {0, -(91/4)}, {1/
69, -(6271/276)}, {2/69, -(6263/276)}, {1/23, -(2085/92)}, {4/
69, -(6247/276)}, {5/69, -(6239/276)}, {2/23, -(2077/92)}, {7/
69, -(6223/276)}, {8/69, -(6215/276)}, {3/23, -(2069/92)}, {10/
69, -(6199/276)}, {11/69, -(6191/276)}, {4/23, -(2061/92)}, {13/
69, -(6175/276)}, {14/69, -(6167/276)}, {5/23, -(2053/92)}, {16/
69, -(6151/276)}, {17/69, -(6143/276)}, {6/23, -(2045/92)}, {19/
69, -(6127/276)}, {20/69, -(6119/276)}, {7/23, -(2037/92)}, {22/
69, -(6103/276)}, {1/3, -(265/12)}, {8/23, -(2029/92)}, {25/
69, -(6079/276)}, {26/69, -(6071/276)}, {9/23, -(2021/92)}, {28/
69, -(6055/276)}, {29/69, -(6047/276)}, {10/23, -(2013/92)}, {31/
69, -(6031/276)}, {32/69, -(6023/276)}, {11/23, -(2005/92)}, {34/
69, -(6007/276)}, {35/69, -(5999/276)}, {12/23, -(1997/92)}, {37/
69, -(5983/276)}, {38/69, -(5975/276)}, {13/23, -(1989/92)}, {40/
69, -(5959/276)}, {41/69, -(5951/276)}, {14/23, -(1981/92)}, {43/
69, -(5935/276)}, {44/69, -(5927/276)}, {15/23, -(1973/92)}, {2/
3, -(257/12)}, {47/69, -(5903/276)}, {16/23, -(1965/92)}, {49/
69, -(5887/276)}, {50/69, -(5879/276)}, {17/23, -(1957/92)}, {52/
69, -(5863/276)}, {53/69, -(5855/276)}, {18/23, -(1949/92)}, {55/
69, -(5839/276)}, {56/69, -(5831/276)}, {19/23, -(1941/92)}, {58/
69, -(5815/276)}, {59/69, -(5807/276)}, {20/23, -(1933/92)}, {61/
69, -(5791/276)}, {62/69, -(5783/276)}, {21/23, -(1925/92)}, {64/
69, -(5767/276)}, {65/69, -(5759/276)}, {22/23, -(1917/92)}, {67/
69, -(5743/276)}, {68/69, -(5735/276)}, {1, -(83/4)}}

https://userdisk.webry.biglobe.ne.jp/020/691/47/N000/000/007/154115859347244523177.gif

　この有理点全体の集合 L∩Q^2 に　群構造が　入る　ように
　　　　和を　定義して　単位元も　求めて下さい;
　　　　　「あなたならどう　定義　する」
　https://www.youtube.com/watch?v=2-C9PAFmL5c&start_radio=1&list=RD2-C9PAFmL5c#t=36
　
　　群構造が　はいつたら　どんな　美味しい　事　が　ありますか?
　
　
　https://artofproblemsolving.com/community/c7t177f7h1674375_the_rational_points_on_curvex4y4x2y25x25y2130
The rational points on curve: c; x^4+y^4-x^2y^2-5x^2-5y^2+13=0
c∩Q^2 が　空でなければ　群構造を　ゐれて下さい;

c∩Q^2 が　空ならば　其の証明を願います;

cの双対曲線 c^★を　多様な発想で是非　求めて下さい；
斎次化( Homogenization　; 同次化 )　は　しておきます;
X^4 - X^2 Y^2 + Y^4 - 5 X^2 Z^2 - 5 Y^2 Z^2 + 13 Z^4=0

c^★　の　特異点達を　求めて下さい;

c には　無論　高校生がよく指導され　邂逅する　二重接線　達が　在る。
　　それ等を　全てモトメテ下さい;

c^★の　有理点全体の集合 c^★∩Q^2 に　群構造が　入る　ように
　　和を　定義して　単位元も　求めて下さい;
　　
　　