User:Lériendil/Square and triangle superparticulars by prime subgroup

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Some shorthand notation used here:

  • Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
  • Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
  • Uk = Sk/S(k+1) stands for the kth ultraparticular, which has the same subgroup as Tk except in the case where k is congruent to 4 (mod 9), in which case the subgroup of Uk lacks prime 3 from that of Tk.
  • Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
  • Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.

This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.

2- and 3-prime subgroups (2.3 and 2.3.p)

Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.

2-prime subgroup (2.3)

Square-particular Subgroup Comma
Ratio Smonzo
S2 2.3 4/3 [2 -1⟩
S3 2.3 9/8 [-3 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T2 2.3 3/2 [-1 1⟩ U2 2.3 32/27 [5 -3⟩

3-prime subgroups (2.3.p)

Square-particular Subgroup Comma
Ratio Smonzo
S4 L5 16/15 [4 -1 -1⟩
S5 L5 25/24 [-3 -1 2⟩
S9 L5 81/80 [-4 4 -1⟩
S7 2.3.7 49/48 [-4 -1 2⟩
S8 2.3.7 64/63 [6 -2 -1⟩
S17 2.3.17 289/288 [-5 -2 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T3 L5 6/5 [1 1 -1⟩ U3 L5 135/128 [-7 3 1⟩
T4 L5 10/9 [1 -2 1⟩ U4 2.5 128/125 [7 -3⟩
T7 2.3.7 28/27 [2 -3 1⟩ U7 2.3.7 1029/1024 [-10 1 3⟩

4-prime subgroups

Note that the lists of triangle-particulars are complete and the insertion of higher primes will add no new inclusions to them. The lists of square particulars other than the "Higher primes" table are likewise complete.

5-add-one-limit (L5.p)

Square-particular Subgroup Comma
Ratio Smonzo
S6 = T8 L7 36/35 [2 2 -1 -1⟩
S15 L7 225/224 [-5 2 2 -1⟩
S49 L7 2401/2400 [-5 -1 -2 4⟩
S10 L5.11 100/99 [2 -2 2 -1⟩
S11 L5.11 121/120 [-3 -1 -1 2⟩
S25 L5.13 625/624 [-4 -1 4 -1⟩
S26 L5.13 676/675 [2 -3 -2 2⟩
S16 L5.17 256/255 [8 -1 -1 -1⟩
S19 L5.19 361/360 [-3 -2 -1 2⟩
S24 L5.23 576/575 [6 2 -2 -1⟩
S31 L5.31 961/960 [-6 -1 -1 2⟩
S81 L5.41 6561/6560 [-5 8 -1 -1⟩
S80 L5.79 6400/6399 [8 -4 2 -1⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T5 L7 15/14 [-1 1 1 -1⟩ U5 L7 875/864 [-5 -3 3 1⟩
T6 L7 21/20 [-2 1 -1 1⟩ U6 L7 1728/1715 [6 3 -1 -3⟩
T8 = S6 L7 36/35 [2 2 -1 -1⟩ U8 L7 5120/5103 [10 -6 1 -1⟩
T9 L5.11 45/44 [-2 2 1 -1⟩ U9 L5.11 8019/8000 [-6 6 -3 1⟩
T10 L5.11 55/54 [-1 -3 1 1⟩ U10 L5.11 4000/3993 [5 -1 3 -3⟩
T25 L5.13 325/324 [-2 -4 2 1⟩ U25 L5.13 140625/140608 [-6 2 6 -3⟩
T16 L5.17 136/135 [3 -3 -1 1⟩ U16 L5.17 24576/24565 [13 1 -1 -3⟩

2.3.7.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S13 2.3.7.13 169/168 [-3 -1 -1 2⟩
S27 2.3.7.13 729/728 [-3 6 -1 -1⟩
S28 2.3.7.29 784/783 [4 -3 2 -1⟩
S63 2.3.7.31 3969/3968 [-7 4 2 -1⟩
S48 2.3.7.47 2304/2303 [8 2 -2 -1⟩
S97 2.3.7.97 9409/9408 [-6 -1 -2 2⟩
S127 2.3.7.127 16129/16128 [-8 -2 -1 2⟩

2.3.11.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S12 2.3.11.13 144/143 [4 2 -1 -1⟩
S33 2.3.11.17 1089/1088 [-6 2 -1 2⟩
S23 2.3.11.23 529/528 [-4 -1 -1 2⟩
S32 2.3.11.31 1024/1023 [10 -1 -1 -1⟩
S243 2.3.11.61 59049/59048 [-3 10 -2 -1⟩
S242 2.3.11.241 58564/58563 [2 -5 4 -1⟩

Higher primes

Square-particular Subgroup Comma
Ratio Smonzo
S53 2.3.13.53 2809/2808 [-3 -3 -1 2⟩
S18 2.3.17.19 324/323 [2 4 -1 -1⟩
S577 2.3.17.577 332929/332928 [-7 -2 -2 2⟩
S37 2.3.19.37 1369/1368 [-3 -2 -1 2⟩
S47 2.3.23.47 2209/2208 [-5 -1 -1 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T17 2.3.17.19 153/152 [-3 2 1 -1⟩ U17 2.3.17.19 93347/93312 [-7 -6 3 1⟩

5-prime subgroups

7-add-one-limit (L7.p)

Square-particular Subgroup Comma
Ratio Smonzo
S21 L11 441/440 [-3 2 -1 2 -1⟩
S55 L11 3025/3024 [-4 -3 2 -1 2⟩
S99 L11 9801/9800 [-3 4 -2 -2 2⟩
S14 L7.13 196/195 [2 -1 -1 2 -1⟩
S64 L7.13 4096/4095 [12 -2 -1 -1 -1⟩
S35 = T49 L7.17 1225/1224 [-3 -2 2 2 -1⟩
S50 L7.17 2500/2499 [2 -1 4 -2 -1⟩
S20 L7.19 400/399 [4 -1 2 -1 -1⟩
S161 L7.23 25921/25920 [-6 -4 -1 2 2⟩
S29 L7.29 841/840 [-3 -1 -1 -1 2⟩
S125 L7.31 15625/15624 [-3 -2 6 -1 -1⟩
S36 L7.37 1296/1295 [4 4 -1 -1 -1⟩
S41 L7.41 1681/1680 [-4 -1 -1 -1 2⟩
S244 L7.61 59536/59535 [4 -5 -1 -2 2⟩
S71 L7.71 5041/5040 [-4 -2 -1 -1 2⟩
S225 L7.113 50625/50624 [-6 4 -1 4 -1⟩
S126 L7.127 15876/15875 [2 4 -3 2 -1⟩
S224 L7.223 50176/50175 [10 -2 -2 2 -1⟩
S251 L7.251 59536/59535 [-3 -2 -3 -1 2⟩
S449 L7.449 201601/201600 [-7 -2 -2 -1 2⟩
S4375 L7.547 19140625/19140624 [-4 -7 8 2 -1⟩
S2401 L7.1201 5764801/5764800 [-5 -1 -2 8 -1⟩
S2400 L7.2399 5760000/5759999 [10 2 4 -4 -1⟩
S4801 L7.4801 23049601/23049600 [-7 -1 -2 -4 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T13 L7.13 91/90 [-1 -2 -1 1 1⟩ U13 2.5.7.13 10985/10976 [-5 1 -3 3⟩
T14 L7.13 105/104 [-3 1 1 1 -1⟩ U14 L7.13 43904/43875 [7 -3 -3 3 -1⟩
T26 L7.13 351/350 [-1 3 -2 -1 1⟩ U26 L7.13 492128/492075 [5 -9 -2 1 3⟩
T15 L7.17 120/119 [3 1 1 -1 -1⟩ U15 L7.17 57375/57344 [-13 3 3 -1 1⟩
T49 = S35 L7.17 1225/1224 [-3 -2 2 2 -1⟩ U49 2.5.7.17 2000033/2000000 [-7 -6 6 1⟩
T19 L7.19 190/189 [1 -3 1 -1 1⟩ U19 L7.19 48013/48000 [-7 -1 -3 1 3⟩
T28 L7.29 406/405 [1 -4 -1 1 1⟩ U28 L7.29 219520/219501 [7 -2 1 3 -3⟩
T126 L7.127 8001/8000 [-6 2 -3 1 1⟩ U126 L7.127 256048128/256047875 [10 6 -3 3 -3⟩

11-add-one-limit

L5.11.p subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S65 L13(-7) 4225/4224 [-7 -1 2 -1 2⟩
S45 L5.11.23 2025/2024 [-3 4 2 -1 -1⟩
S44 L5.11.43 1936/1935 [4 -2 -1 2 -1⟩
S54 L5.11.53 2916/2915 [2 6 -1 -1 -1⟩
S121 L5.11.61 14641/14640 [-4 -1 -1 4 -1⟩
S89 L5.11.89 7921/7920 [-4 -2 -1 -1 2⟩
S485 L5.11.97 235225/235224 [-3 -5 2 -2 2⟩
S100 L5.11.101 10000/9999 [4 -2 4 -1 -1⟩
S109 L5.11.109 11881/11880 [-3 -3 -1 -1 2⟩
S199 L5.11.199 39601/39600 [-4 -2 -2 -1 2⟩
S241 L5.11.241 58081/58080 [-5 -1 -1 -2 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T11 L13(-7) 66/65 [1 1 -1 1 -1⟩ U11 L13(-7) 17303/17280 [7 3 1 -3 -1⟩
T23 L5.11.23 276/275 [2 1 -2 -1 1⟩ U23 L5.11.23 304175/304128 [-10 -3 2 -1 3⟩
T31 L5.11.31 496/495 [4 -2 -1 -1 1⟩ U31 2.5.11.31 327701/327680 [-16 -1 1 3⟩
T241 L5.11.241 29161/29160 [-4 -6 -1 2 1⟩ U241 L5.11.241 1133799201/1133799040 [-7 4 -1 -6 3⟩

No-fives (L11(-5).p) subgroups

Square-particular Subgroup Comma
Ratio Smonzo
S22 L11(-5).23 484/483 [2 -1 -1 2 -1⟩
S43 L11(-5).43 1849/1848 [-3 -1 -1 -1 2⟩
S98 L11(-5).97 9604/9603 [2 -2 4 -1 -1⟩
S197 L11(-5).197 38809/38808 [-3 -2 -2 -1 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T12 L13(-5) 78/77 [1 1 -1 -1 1⟩ U12 L13(-5) 24192/24167 [7 3 1 -1 -3⟩
T22 L11(-5).23 253/252 [-2 -2 -1 1 1⟩ U22 2.7.11.23 85184/85169 [6 -1 3 -3⟩
T97 L11(-5).97 4753/4752 [-4 -3 2 -1 1⟩ U97 L11(-5).97 30118209/30118144 [-8 1 -6 1 3⟩

See also

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