User:Lériendil/Square and triangle superparticulars by prime subgroup: Difference between revisions
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== 2- and 3-prime subgroups (2.3 and 2.3.p) == | == 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. | 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) === | === 2-prime subgroup (2.3) === | ||
| Line 32: | Line 21: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #BFD7FF;" | ||
| S2 | | S2 | ||
| [[3-limit|2.3]] | | [[3-limit|2.3]] | ||
| [[4/3]] | | [[4/3]] | ||
| {{monzo| 2 -1 }} | | {{monzo| 2 -1 }} | ||
|- | |- style="background-color: #BFD7FF;" | ||
| S3 | | S3 | ||
| [[3-limit|2.3]] | | [[3-limit|2.3]] | ||
| Line 57: | Line 46: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #BFD7FF;" | ||
| T2 | | T2 | ||
| [[3-limit|2.3]] | | [[3-limit|2.3]] | ||
| Line 78: | Line 67: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #FFBFEF;" | ||
| S4 | | S4 | ||
| [[5-limit|L5]] | | [[5-limit|L5]] | ||
| [[16/15]] | | [[16/15]] | ||
| {{monzo| 4 -1 -1 }} | | {{monzo| 4 -1 -1 }} | ||
|- | |- style="background-color: #FFBFEF;" | ||
| S5 | | S5 | ||
| [[5-limit|L5]] | | [[5-limit|L5]] | ||
| [[25/24]] | | [[25/24]] | ||
| {{monzo| -3 -1 2 }} | | {{monzo| -3 -1 2 }} | ||
|- | |- style="background-color: #FFBFEF;" | ||
| S9 | | S9 | ||
| [[5-limit|L5]] | | [[5-limit|L5]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{monzo| -4 4 -1 }} | | {{monzo| -4 4 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S7 | | S7 | ||
| [[2.3.7 subgroup|2.3.7]] | | [[2.3.7 subgroup|2.3.7]] | ||
| [[49/48]] | | [[49/48]] | ||
| {{monzo| -4 -1 2 }} | | {{monzo| -4 -1 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S8 | | S8 | ||
| [[2.3.7 subgroup|2.3.7]] | | [[2.3.7 subgroup|2.3.7]] | ||
| [[64/63]] | | [[64/63]] | ||
| {{monzo| 6 -2 -1 }} | | {{monzo| 6 -2 -1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S17 | | S17 | ||
| 2.3.17 | | 2.3.17 | ||
| Line 123: | Line 112: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #FFBFEF;" | ||
| T3 | | T3 | ||
| [[5-limit|L5]] | | [[5-limit|L5]] | ||
| Line 132: | Line 121: | ||
| [[135/128]] | | [[135/128]] | ||
| {{monzo| -7 3 1 }} | | {{monzo| -7 3 1 }} | ||
|- | |- style="background-color: #FFBFEF;" | ||
| T4 | | T4 | ||
| [[5-limit|L5]] | | [[5-limit|L5]] | ||
| Line 141: | Line 130: | ||
| [[128/125]] | | [[128/125]] | ||
| {{monzo| 7 -3 }} | | {{monzo| 7 -3 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| T7 | | T7 | ||
| [[2.3.7 subgroup|2.3.7]] | | [[2.3.7 subgroup|2.3.7]] | ||
| Line 165: | Line 154: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S6 = T8 | | S6 = T8 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| [[36/35]] | | [[36/35]] | ||
| {{monzo| 2 2 -1 -1 }} | | {{monzo| 2 2 -1 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S15 | | S15 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| [[225/224]] | | [[225/224]] | ||
| {{monzo| -5 2 2 -1 }} | | {{monzo| -5 2 2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S49 | | S49 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| [[2401/2400]] | | [[2401/2400]] | ||
| {{monzo| -5 -1 -2 4 }} | | {{monzo| -5 -1 -2 4 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S10 | | S10 | ||
| L5.11 | | L5.11 | ||
| [[100/99]] | | [[100/99]] | ||
| {{monzo| 2 -2 2 -1 }} | | {{monzo| 2 -2 2 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S11 | | S11 | ||
| L5.11 | | L5.11 | ||
| [[121/120]] | | [[121/120]] | ||
| {{monzo| -3 -1 -1 2 }} | | {{monzo| -3 -1 -1 2 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S25 | | S25 | ||
| L5.13 | | L5.13 | ||
| [[625/624]] | | [[625/624]] | ||
| {{monzo| -4 -1 4 -1 }} | | {{monzo| -4 -1 4 -1 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S26 | | S26 | ||
| L5.13 | | L5.13 | ||
| [[676/675]] | | [[676/675]] | ||
| {{monzo| 2 -3 -2 2 }} | | {{monzo| 2 -3 -2 2 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S16 | | S16 | ||
| L5.17 | | L5.17 | ||
| [[256/255]] | | [[256/255]] | ||
| {{monzo| 8 -1 -1 -1 }} | | {{monzo| 8 -1 -1 -1 }} | ||
|- | |- style="background-color: #DFD7BF;" | ||
| S19 | | S19 | ||
| L5.19 | | L5.19 | ||
| [[361/360]] | | [[361/360]] | ||
| {{monzo| -3 -2 -1 2 }} | | {{monzo| -3 -2 -1 2 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S24 | | S24 | ||
| L5.23 | | L5.23 | ||
| [[576/575]] | | [[576/575]] | ||
| {{monzo| 6 2 -2 -1 }} | | {{monzo| 6 2 -2 -1 }} | ||
|- | |- style="background-color: #CFC7F7;" | ||
| S31 | | S31 | ||
| L5.31 | | L5.31 | ||
| [[961/960]] | | [[961/960]] | ||
| {{monzo| -6 -1 -1 2 }} | | {{monzo| -6 -1 -1 2 }} | ||
|- | |- style="background-color: #FFBFEF;" | ||
| S81 | | S81 | ||
| L5.41 | | L5.41 | ||
| [[6561/6560]] | | [[6561/6560]] | ||
| {{monzo| -5 8 -1 -1 }} | | {{monzo| -5 8 -1 -1 }} | ||
|- | |- style="background-color: #FFBFEF;" | ||
| S80 | | S80 | ||
| L5.79 | | L5.79 | ||
| Line 245: | Line 234: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #C7BFDF;" | ||
| T5 | | T5 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| Line 254: | Line 243: | ||
| [[875/864]] | | [[875/864]] | ||
| {{monzo| -5 -3 3 1 }} | | {{monzo| -5 -3 3 1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| T6 | | T6 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| Line 263: | Line 252: | ||
| [[1728/1715]] | | [[1728/1715]] | ||
| {{monzo| 6 3 -1 -3 }} | | {{monzo| 6 3 -1 -3 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| T8 = S6 | | T8 = S6 | ||
| [[7-limit|L7]] | | [[7-limit|L7]] | ||
| Line 272: | Line 261: | ||
| [[5120/5103]] | | [[5120/5103]] | ||
| {{monzo| 10 -6 1 -1 }} | | {{monzo| 10 -6 1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| T9 | | T9 | ||
| L5.11 | | L5.11 | ||
| Line 281: | Line 270: | ||
| [[8019/8000]] | | [[8019/8000]] | ||
| {{monzo| -6 6 -3 1 }} | | {{monzo| -6 6 -3 1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| T10 | | T10 | ||
| L5.11 | | L5.11 | ||
| Line 290: | Line 279: | ||
| [[4000/3993]] | | [[4000/3993]] | ||
| {{monzo| 5 -1 3 -3 }} | | {{monzo| 5 -1 3 -3 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T25 | | T25 | ||
| L5.13 | | L5.13 | ||
| Line 299: | Line 288: | ||
| [[140625/140608]] | | [[140625/140608]] | ||
| {{monzo| -6 2 6 -3 }} | | {{monzo| -6 2 6 -3 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| T16 | | T16 | ||
| L5.17 | | L5.17 | ||
| Line 320: | Line 309: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S13 | | S13 | ||
| 2.3.7.13 | | 2.3.7.13 | ||
| [[169/168]] | | [[169/168]] | ||
| {{monzo| -3 -1 -1 2 }} | | {{monzo| -3 -1 -1 2 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S27 | | S27 | ||
| 2.3.7.13 | | 2.3.7.13 | ||
| [[729/728]] | | [[729/728]] | ||
| {{monzo| -3 6 -1 -1 }} | | {{monzo| -3 6 -1 -1 }} | ||
|- | |- style="background-color: #D7DFCF;" | ||
| S28 | | S28 | ||
| 2.3.7.29 | | 2.3.7.29 | ||
| [[784/783]] | | [[784/783]] | ||
| {{monzo| 4 -3 2 -1 }} | | {{monzo| 4 -3 2 -1 }} | ||
|- | |- style="background-color: #CFC7F7;" | ||
| S63 | | S63 | ||
| 2.3.7.31 | | 2.3.7.31 | ||
| [[3969/3968]] | | [[3969/3968]] | ||
| {{monzo| -7 4 2 -1 }} | | {{monzo| -7 4 2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S48 | | S48 | ||
| 2.3.7.47 | | 2.3.7.47 | ||
| [[2304/2303]] | | [[2304/2303]] | ||
| {{monzo| 8 2 -2 -1 }} | | {{monzo| 8 2 -2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S97 | | S97 | ||
| 2.3.7.97 | | 2.3.7.97 | ||
| [[9409/9408]] | | [[9409/9408]] | ||
| {{monzo| -6 -1 -2 2 }} | | {{monzo| -6 -1 -2 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S127 | | S127 | ||
| 2.3.7.127 | | 2.3.7.127 | ||
| Line 367: | Line 356: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S12 | | S12 | ||
| 2.3.11.13 | | 2.3.11.13 | ||
| [[144/143]] | | [[144/143]] | ||
| {{monzo| 4 2 -1 -1 }} | | {{monzo| 4 2 -1 -1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S33 | | S33 | ||
| 2.3.11.17 | | 2.3.11.17 | ||
| [[1089/1088]] | | [[1089/1088]] | ||
| {{monzo| -6 2 -1 2 }} | | {{monzo| -6 2 -1 2 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S23 | | S23 | ||
| 2.3.11.23 | | 2.3.11.23 | ||
| [[529/528]] | | [[529/528]] | ||
| {{monzo| -4 -1 -1 2 }} | | {{monzo| -4 -1 -1 2 }} | ||
|- | |- style="background-color: #CFC7F7;" | ||
| S32 | | S32 | ||
| 2.3.11.31 | | 2.3.11.31 | ||
| [[1024/1023]] | | [[1024/1023]] | ||
| {{monzo| 10 -1 -1 -1 }} | | {{monzo| 10 -1 -1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S243 | | S243 | ||
| 2.3.11.61 | | 2.3.11.61 | ||
| [[59049/59048]] | | [[59049/59048]] | ||
| {{monzo| -3 10 -2 -1 }} | | {{monzo| -3 10 -2 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S242 | | S242 | ||
| 2.3.11.241 | | 2.3.11.241 | ||
| Line 409: | Line 398: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S53 | | S53 | ||
| 2.3.13.53 | | 2.3.13.53 | ||
| [[2809/2808]] | | [[2809/2808]] | ||
| {{monzo| -3 -3 -1 2 }} | | {{monzo| -3 -3 -1 2 }} | ||
|- | |- style="background-color: #DFD7BF;" | ||
| S18 | | S18 | ||
| 2.3.17.19 | | 2.3.17.19 | ||
| [[324/323]] | | [[324/323]] | ||
| {{monzo| 2 4 -1 -1 }} | | {{monzo| 2 4 -1 -1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S577 | | S577 | ||
| 2.3.17.577 | | 2.3.17.577 | ||
| [[332929/332928]] | | [[332929/332928]] | ||
| {{monzo| -7 -2 -2 2 }} | | {{monzo| -7 -2 -2 2 }} | ||
|- | |- style="background-color: #DFD7BF;" | ||
| S37 | | S37 | ||
| 2.3.19.37 | | 2.3.19.37 | ||
| [[1369/1368]] | | [[1369/1368]] | ||
| {{monzo| -3 -2 -1 2 }} | | {{monzo| -3 -2 -1 2 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S47 | | S47 | ||
| 2.3.23.47 | | 2.3.23.47 | ||
| Line 449: | Line 438: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #DFD7BF;" | ||
| T17 | | T17 | ||
| 2.3.17.19 | | 2.3.17.19 | ||
| Line 471: | Line 460: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S21 | | S21 | ||
| [[11-limit|L11]] | | [[11-limit|L11]] | ||
| [[441/440]] | | [[441/440]] | ||
| {{monzo| -3 2 -1 2 -1 }} | | {{monzo| -3 2 -1 2 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S55 | | S55 | ||
| [[11-limit|L11]] | | [[11-limit|L11]] | ||
| [[3025/3024]] | | [[3025/3024]] | ||
| {{monzo| -4 -3 2 -1 2 }} | | {{monzo| -4 -3 2 -1 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S99 | | S99 | ||
| [[11-limit|L11]] | | [[11-limit|L11]] | ||
| [[9801/9800]] | | [[9801/9800]] | ||
| {{monzo| -3 4 -2 -2 2 }} | | {{monzo| -3 4 -2 -2 2 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S14 | | S14 | ||
| L7.13 | | L7.13 | ||
| [[196/195]] | | [[196/195]] | ||
| {{monzo| 2 -1 -1 2 -1 }} | | {{monzo| 2 -1 -1 2 -1 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S64 | | S64 | ||
| L7.13 | | L7.13 | ||
| [[4096/4095]] | | [[4096/4095]] | ||
| {{monzo| 12 -2 -1 -1 -1 }} | | {{monzo| 12 -2 -1 -1 -1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S35 = T49 | | S35 = T49 | ||
| L7.17 | | L7.17 | ||
| [[1225/1224]] | | [[1225/1224]] | ||
| {{monzo| -3 -2 2 2 -1 }} | | {{monzo| -3 -2 2 2 -1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| S50 | | S50 | ||
| L7.17 | | L7.17 | ||
| [[2500/2499]] | | [[2500/2499]] | ||
| {{monzo| 2 -1 4 -2 -1 }} | | {{monzo| 2 -1 4 -2 -1 }} | ||
|- | |- style="background-color: #DFD7BF;" | ||
| S20 | | S20 | ||
| L7.19 | | L7.19 | ||
| [[400/399]] | | [[400/399]] | ||
| {{monzo| 4 -1 2 -1 -1 }} | | {{monzo| 4 -1 2 -1 -1 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S161 | | S161 | ||
| L7.23 | | L7.23 | ||
| [[25921/25920]] | | [[25921/25920]] | ||
| {{monzo| -6 -4 -1 2 2 }} | | {{monzo| -6 -4 -1 2 2 }} | ||
|- | |- style="background-color: #D7DFCF;" | ||
| S29 | | S29 | ||
| L7.29 | | L7.29 | ||
| [[841/840]] | | [[841/840]] | ||
| {{monzo| -3 -1 -1 -1 2 }} | | {{monzo| -3 -1 -1 -1 2 }} | ||
|- | |- style="background-color: #CFC7F7;" | ||
| S125 | | S125 | ||
| L7.31 | | L7.31 | ||
| [[15625/15624]] | | [[15625/15624]] | ||
| {{monzo| -3 -2 6 -1 -1 }} | | {{monzo| -3 -2 6 -1 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S36 | | S36 | ||
| L7.37 | | L7.37 | ||
| [[1296/1295]] | | [[1296/1295]] | ||
| {{monzo| 4 4 -1 -1 -1 }} | | {{monzo| 4 4 -1 -1 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S41 | | S41 | ||
| L7.41 | | L7.41 | ||
| [[1681/1680]] | | [[1681/1680]] | ||
| {{monzo| -4 -1 -1 -1 2 }} | | {{monzo| -4 -1 -1 -1 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S244 | | S244 | ||
| L7.61 | | L7.61 | ||
| [[59536/59535]] | | [[59536/59535]] | ||
| {{monzo| 4 -5 -1 -2 2 }} | | {{monzo| 4 -5 -1 -2 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S71 | | S71 | ||
| L7.71 | | L7.71 | ||
| [[5041/5040]] | | [[5041/5040]] | ||
| {{monzo| -4 -2 -1 -1 2 }} | | {{monzo| -4 -2 -1 -1 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S225 | | S225 | ||
| L7.113 | | L7.113 | ||
| [[50625/50624]] | | [[50625/50624]] | ||
| {{monzo| -6 4 -1 4 -1 }} | | {{monzo| -6 4 -1 4 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S126 | | S126 | ||
| L7.127 | | L7.127 | ||
| [[15876/15875]] | | [[15876/15875]] | ||
| {{monzo| 2 4 -3 2 -1 }} | | {{monzo| 2 4 -3 2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S224 | | S224 | ||
| L7.223 | | L7.223 | ||
| [[50176/50175]] | | [[50176/50175]] | ||
| {{monzo| 10 -2 -2 2 -1 }} | | {{monzo| 10 -2 -2 2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S251 | | S251 | ||
| L7.251 | | L7.251 | ||
| [[59536/59535]] | | [[59536/59535]] | ||
| {{monzo| -3 -2 -3 -1 2 }} | | {{monzo| -3 -2 -3 -1 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S449 | | S449 | ||
| L7.449 | | L7.449 | ||
| [[201601/201600]] | | [[201601/201600]] | ||
| {{monzo| -7 -2 -2 -1 2 }} | | {{monzo| -7 -2 -2 -1 2 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S4375 | | S4375 | ||
| L7.547 | | L7.547 | ||
| [[19140625/19140624]] | | [[19140625/19140624]] | ||
| {{monzo| -4 -7 8 2 -1 }} | | {{monzo| -4 -7 8 2 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S2401 | | S2401 | ||
| L7.1201 | | L7.1201 | ||
| [[5764801/5764800]] | | [[5764801/5764800]] | ||
| {{monzo| -5 -1 -2 8 -1 }} | | {{monzo| -5 -1 -2 8 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S2400 | | S2400 | ||
| L7.2399 | | L7.2399 | ||
| [[5760000/5759999]] | | [[5760000/5759999]] | ||
| {{monzo| 10 2 4 -4 -1 }} | | {{monzo| 10 2 4 -4 -1 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| S4801 | | S4801 | ||
| L7.4801 | | L7.4801 | ||
| Line 606: | Line 595: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T13 | | T13 | ||
| L7.13 | | L7.13 | ||
| Line 615: | Line 604: | ||
| [[10985/10976]] | | [[10985/10976]] | ||
| {{monzo| -5 1 -3 3 }} | | {{monzo| -5 1 -3 3 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T14 | | T14 | ||
| L7.13 | | L7.13 | ||
| Line 624: | Line 613: | ||
| [[43904/43875]] | | [[43904/43875]] | ||
| {{monzo| 7 -3 -3 3 -1 }} | | {{monzo| 7 -3 -3 3 -1 }} | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T26 | | T26 | ||
| L7.13 | | L7.13 | ||
| Line 633: | Line 622: | ||
| [[492128/492075]] | | [[492128/492075]] | ||
| {{monzo| 5 -9 -2 1 3 }} | | {{monzo| 5 -9 -2 1 3 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| T15 | | T15 | ||
| L7.17 | | L7.17 | ||
| Line 642: | Line 631: | ||
| [[57375/57344]] | | [[57375/57344]] | ||
| {{monzo| -13 3 3 -1 1 }} | | {{monzo| -13 3 3 -1 1 }} | ||
|- | |- style="background-color: #F7F7BF;" | ||
| T49 = S35 | | T49 = S35 | ||
| L7.17 | | L7.17 | ||
| Line 651: | Line 640: | ||
| [[2000033/2000000]] | | [[2000033/2000000]] | ||
| {{monzo| -7 -6 6 1 }} | | {{monzo| -7 -6 6 1 }} | ||
|- | |- style="background-color: #DFD7BF;" | ||
| T19 | | T19 | ||
| L7.19 | | L7.19 | ||
| Line 660: | Line 649: | ||
| [[48013/48000]] | | [[48013/48000]] | ||
| {{monzo| -7 -1 -3 1 3 }} | | {{monzo| -7 -1 -3 1 3 }} | ||
|- | |- style="background-color: #D7DFCF;" | ||
| T28 | | T28 | ||
| L7.29 | | L7.29 | ||
| Line 669: | Line 658: | ||
| [[219520/219501]] | | [[219520/219501]] | ||
| {{monzo| 7 -2 1 3 -3 }} | | {{monzo| 7 -2 1 3 -3 }} | ||
|- | |- style="background-color: #C7BFDF;" | ||
| T126 | | T126 | ||
| L7.127 | | L7.127 | ||
| Line 692: | Line 681: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| S65 | | S65 | ||
| L13(-7) | | L13(-7) | ||
| [[4225/4224]] | | [[4225/4224]] | ||
| {{monzo| -7 -1 2 -1 2 }} | | {{monzo| -7 -1 2 -1 2 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S45 | | S45 | ||
| L5.11.23 | | L5.11.23 | ||
| [[2025/2024]] | | [[2025/2024]] | ||
| {{monzo| -3 4 2 -1 -1 }} | | {{monzo| -3 4 2 -1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S44 | | S44 | ||
| L5.11.43 | | L5.11.43 | ||
| [[1936/1935]] | | [[1936/1935]] | ||
| {{monzo| 4 -2 -1 2 -1 }} | | {{monzo| 4 -2 -1 2 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S54 | | S54 | ||
| L5.11.53 | | L5.11.53 | ||
| [[2916/2915]] | | [[2916/2915]] | ||
| {{monzo| 2 6 -1 -1 -1 }} | | {{monzo| 2 6 -1 -1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S121 | | S121 | ||
| L5.11.61 | | L5.11.61 | ||
| [[14641/14640]] | | [[14641/14640]] | ||
| {{monzo| -4 -1 -1 4 -1 }} | | {{monzo| -4 -1 -1 4 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S89 | | S89 | ||
| L5.11.89 | | L5.11.89 | ||
| [[7921/7920]] | | [[7921/7920]] | ||
| {{monzo| -4 -2 -1 -1 2 }} | | {{monzo| -4 -2 -1 -1 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S485 | | S485 | ||
| L5.11.97 | | L5.11.97 | ||
| [[235225/235224]] | | [[235225/235224]] | ||
| {{monzo| -3 -5 2 -2 2 }} | | {{monzo| -3 -5 2 -2 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S100 | | S100 | ||
| L5.11.101 | | L5.11.101 | ||
| [[10000/9999]] | | [[10000/9999]] | ||
| {{monzo| 4 -2 4 -1 -1 }} | | {{monzo| 4 -2 4 -1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S109 | | S109 | ||
| L5.11.109 | | L5.11.109 | ||
| [[11881/11880]] | | [[11881/11880]] | ||
| {{monzo| -3 -3 -1 -1 2 }} | | {{monzo| -3 -3 -1 -1 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S199 | | S199 | ||
| L5.11.199 | | L5.11.199 | ||
| [[39601/39600]] | | [[39601/39600]] | ||
| {{monzo| -4 -2 -2 -1 2 }} | | {{monzo| -4 -2 -2 -1 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S241 | | S241 | ||
| L5.11.241 | | L5.11.241 | ||
| Line 762: | Line 751: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T11 | | T11 | ||
| L13(-7) | | L13(-7) | ||
| Line 771: | Line 760: | ||
| [[17303/17280]] | | [[17303/17280]] | ||
| {{monzo| 7 3 1 -3 -1 }} | | {{monzo| 7 3 1 -3 -1 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| T23 | | T23 | ||
| L5.11.23 | | L5.11.23 | ||
| Line 780: | Line 769: | ||
| [[304175/304128]] | | [[304175/304128]] | ||
| {{monzo| -10 -3 2 -1 3 }} | | {{monzo| -10 -3 2 -1 3 }} | ||
|- | |- style="background-color: #CFC7F7;" | ||
| T31 | | T31 | ||
| L5.11.31 | | L5.11.31 | ||
| Line 789: | Line 778: | ||
| [[327701/327680]] | | [[327701/327680]] | ||
| {{monzo| -16 -1 1 3 }} | | {{monzo| -16 -1 1 3 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| T241 | | T241 | ||
| L5.11.241 | | L5.11.241 | ||
| Line 810: | Line 799: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #FFDFBF;" | ||
| S22 | | S22 | ||
| L11(-5).23 | | L11(-5).23 | ||
| [[484/483]] | | [[484/483]] | ||
| {{monzo| 2 -1 -1 2 -1 }} | | {{monzo| 2 -1 -1 2 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S43 | | S43 | ||
| L11(-5).43 | | L11(-5).43 | ||
| [[1849/1848]] | | [[1849/1848]] | ||
| {{monzo| -3 -1 -1 -1 2 }} | | {{monzo| -3 -1 -1 -1 2 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S98 | | S98 | ||
| L11(-5).97 | | L11(-5).97 | ||
| [[9604/9603]] | | [[9604/9603]] | ||
| {{monzo| 2 -2 4 -1 -1 }} | | {{monzo| 2 -2 4 -1 -1 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| S197 | | S197 | ||
| L11(-5).197 | | L11(-5).197 | ||
| Line 845: | Line 834: | ||
! Ratio | ! Ratio | ||
! Smonzo | ! Smonzo | ||
|- | |- style="background-color: #D7BFEF;" | ||
| T12 | | T12 | ||
| L13(-5) | | L13(-5) | ||
| Line 854: | Line 843: | ||
| [[24192/24167]] | | [[24192/24167]] | ||
| {{monzo| 7 3 1 -1 -3 }} | | {{monzo| 7 3 1 -1 -3 }} | ||
|- | |- style="background-color: #FFDFBF;" | ||
| T22 | | T22 | ||
| L11(-5).23 | | L11(-5).23 | ||
| Line 863: | Line 852: | ||
| [[85184/85169]] | | [[85184/85169]] | ||
| {{monzo| 6 -1 3 -3 }} | | {{monzo| 6 -1 3 -3 }} | ||
|- | |- style="background-color: #F7C7BF;" | ||
| T97 | | T97 | ||
| L11(-5).97 | | L11(-5).97 | ||
Latest revision as of 18:12, 28 July 2024
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⟩ |