MAGMA 2012



Pentadic Quantum Class Groups


5.1.1. Complex Base Objects

According to the following table, computed at the University of Manitoba, Winnipeg,
the existence of 357 pentadic quantum class groups G52(K)
with abelianizations of diamond type (5,5)
over complex base objects K = Q(D1/2) with 5-irregular discriminants D
has been discovered down to the lower bound -106 < D < 0.
(Output format is Counter.UnsignedDiscriminant:ClassNumber comma separated.)


1. 11199: 100, 2. 12451: 25, 3. 17944: 50, 4. 30263: 150, 5. 33531: 50,
6. 37363: 25, 7. 38047: 75, 8. 39947: 50, 9. 42871: 150, 10. 53079: 200,
11. 54211: 50, 12. 58424: 100, 13. 61556: 200, 14. 62632: 50, 15. 63411: 100,
16. 64103: 200, 17. 65784: 100, 18. 66328: 50, 19. 67031: 400, 20. 67063: 150,
21. 67128: 100, 22. 69811: 50, 23. 72084: 100, 24. 74051: 175, 25. 75688: 50,
26. 83767: 150, 27. 84271: 200, 28. 85099: 50, 29. 85279: 200, 30. 87971: 100,
31. 89751: 300, 32. 90795: 100, 33. 90868: 50, 34. 92263: 150, 35. 98591: 450,
36. 99031: 150, 37. 99743: 300, 38. 104503: 150, 39. 105151: 150, 40. 112643: 75,
41. 113140: 100, 42. 114395: 100, 43. 115912: 100, 44. 116187: 50, 45. 119191: 225,
46. 119915: 100, 47. 120463: 150, 48. 127103: 225, 49. 128680: 100, 50. 132520: 100,
51. 134312: 200, 52. 135176: 300, 53. 140696: 300, 54. 143508: 100, 55. 146184: 200,
56. 146776: 100, 57. 146863: 200, 58. 148507: 50, 59. 150347: 100, 60. 150647: 350,
61. 151627: 50, 62. 151879: 300, 63. 154408: 50, 64. 154824: 200, 65. 155867: 150,
66. 162452: 200, 67. 163659: 100, 68. 164011: 75, 69. 165391: 275, 70. 165711: 400,
71. 167172: 100, 72. 169220: 200, 73. 172968: 100, 74. 173203: 50, 75. 173243: 100,
76. 176183: 300, 77. 181752: 100, 78. 186904: 100, 79. 191807: 400, 80. 196339: 100,
81. 203567: 400, 82. 203656: 200, 83. 208307: 100, 84. 216452: 200, 85. 231032: 200,
86. 234276: 200, 87. 235796: 200, 88. 239524: 200, 89. 244103: 400, 90. 246511: 225,
91. 255624: 200, 92. 257428: 100, 93. 259679: 700, 94. 260879: 525, 95. 264995: 200,
96. 267719: 475, 97. 277135: 300, 98. 280847: 400, 99. 281396: 300, 100. 283956: 200,
101. 286187: 150, 102. 290068: 100, 103. 292387: 100, 104. 293380: 200, 105. 293848: 100,
106. 294379: 100, 107. 296851: 100, 108. 299748: 100, 109. 300388: 100, 110. 301807: 200,
111. 306799: 350, 112. 309399: 600, 113. 309659: 200, 114. 311699: 200, 115. 313176: 300,
116. 315231: 400, 117. 318479: 700, 118. 322251: 200, 119. 326468: 200, 120. 327235: 150,
121. 327503: 400, 122. 329039: 700, 123. 332404: 150, 124. 333083: 150, 125. 336135: 300,
126. 339171: 200, 127. 339627: 100, 128. 341407: 300, 129. 344499: 150, 130. 346683: 100,
131. 353767: 175, 132. 360115: 100, 133. 361887: 300, 134. 365672: 300, 135. 366803: 200,
136. 370932: 200, 137. 379735: 300, 138. 383243: 200, 139. 384324: 300, 140. 385183: 200,
141. 388583: 700, 142. 397191: 400, 143. 400127: 600, 144. 402468: 200, 145. 402756: 300,
146. 404916: 200, 147. 411835: 100, 148. 412323: 100, 149. 414851: 300, 150. 420376: 200,
151. 421579: 100, 152. 422120: 400, 153. 422279: 700, 154. 424723: 100, 155. 425480: 400,
156. 428187: 100, 157. 429835: 100, 158. 430996: 200, 159. 441067: 100, 160. 442835: 200,
161. 445499: 275, 162. 446084: 400, 163. 447432: 200, 164. 449003: 175, 165. 454872: 200,
166. 456424: 200, 167. 458695: 400, 168. 463439: 700, 169. 464579: 300, 170. 465976: 200,
171. 467572: 200, 172. 467620: 200, 173. 469871: 1050, 174. 472196: 400, 175. 473167: 225,
176. 473243: 150, 177. 480355: 100, 178. 480356: 400, 179. 482899: 175, 180. 485587: 75,
181. 485635: 150, 182. 486595: 100, 183. 487511: 950, 184. 488327: 650, 185. 490020: 200,
186. 494168: 300, 187. 495636: 200, 188. 499076: 400, 189. 509556: 300, 190. 511735: 300,
191. 513287: 600, 192. 515503: 300, 193. 516755: 200, 194. 521003: 200, 195. 524312: 350,
196. 528164: 600, 197. 529960: 200, 198. 531508: 200, 199. 531919: 525, 200. 537140: 400,
201. 552395: 200, 202. 557007: 600, 203. 566779: 150, 204. 569044: 200, 205. 569588: 300,
206. 569963: 150, 207. 571019: 225, 208. 571247: 950, 209. 572147: 200, 210. 577443: 200,
211. 577640: 400, 212. 579943: 300, 213. 584803: 100, 214. 587323: 100, 215. 592580: 400,
216. 593971: 200, 217. 594019: 200, 218. 596811: 300, 219. 599379: 200, 220. 599691: 200,
221. 600779: 300, 222. 601379: 275, 223. 601572: 300, 224. 603451: 150, 225. 604703: 550,
226. 606868: 150, 227. 610955: 200, 228. 613451: 275, 229. 626071: 700, 230. 626879: 800,
231. 627508: 200, 232. 628311: 800, 233. 629032: 300, 234. 631720: 200, 235. 637439: 1200,
236. 638772: 200, 237. 639163: 100, 238. 640491: 200, 239. 642356: 400, 240. 652168: 200,
241. 652335: 600, 242. 654232: 200, 243. 657143: 450, 244. 662523: 150, 245. 673687: 300,
246. 674983: 300, 247. 676123: 150, 248. 676884: 400, 249. 678279: 800, 250. 680791: 550,
251. 681544: 300, 252. 682824: 400, 253. 688091: 450, 254. 688539: 200, 255. 694011: 200,
256. 695071: 700, 257. 702067: 75, 258. 703747: 100, 259. 704423: 550, 260. 706523: 225,
261. 707071: 525, 262. 707811: 200, 263. 709103: 850, 264. 709208: 350, 265. 709796: 600,
266. 710479: 600, 267. 714019: 200, 268. 717991: 600, 269. 718504: 400, 270. 721499: 325,
271. 728164: 300, 272. 730319: 900, 273. 737079: 700, 274. 739387: 150, 275. 758159: 1425,
276. 758487: 300, 277. 759268: 200, 278. 761092: 200, 279. 761167: 450, 280. 762715: 100,
281. 762916: 200, 282. 766771: 150, 283. 769732: 200, 284. 772516: 400, 285. 775367: 925,
286. 778243: 100, 287. 778591: 400, 288. 780331: 200, 289. 780583: 325, 290. 780935: 700,
291. 788839: 550, 292. 799460: 400, 293. 802036: 200, 294. 805668: 300, 295. 809263: 300,
296. 811128: 300, 297. 811131: 300, 298. 812423: 1050, 299. 815016: 400, 300. 815767: 350,
301. 818728: 200, 302. 822392: 400, 303. 825519: 700, 304. 827236: 200, 305. 831819: 300,
306. 835319: 975, 307. 838451: 400, 308. 847923: 200, 309. 851899: 225, 310. 864552: 400,
311. 866292: 200, 312. 877783: 275, 313. 878331: 300, 314. 883643: 300, 315. 890808: 300,
316. 891731: 300, 317. 895147: 100, 318. 898395: 200, 319. 899567: 600, 320. 901223: 650,
321. 901811: 325, 322. 908003: 325, 323. 913636: 300, 324. 916360: 200, 325. 918420: 400,
326. 925048: 200, 327. 929560: 200, 328. 933476: 600, 329. 935063: 1075, 330. 935419: 150,
331. 936520: 200, 332. 938047: 400, 333. 939540: 400, 334. 946563: 150, 335. 947512: 200,
336. 950648: 600, 337. 954571: 175, 338. 958856: 700, 339. 961543: 300, 340. 961608: 200,
341. 961763: 300, 342. 968052: 300, 343. 969315: 200, 344. 970408: 300, 345. 970679: 1600,
346. 971143: 675, 347. 978168: 200, 348. 978407: 950, 349. 980607: 700, 350. 985115: 450,
351. 986623: 400, 352. 987031: 450, 353. 988132: 200, 354. 990923: 275, 355. 992203: 100,
356. 994183: 325, 357. 998359: 700.

5.1.2. Extended Complex Base Objects

Hoping to find a first example of coclass 4,
we extended table 5.1.1 to a total of
813 pentadic quantum class groups G52(K)
with abelianizations of diamond type (5,5)
over complex base objects K = Q(D1/2) with 5-irregular discriminants D
down to the lower bound -2*106 < D < 0.
(Output format is Counter:UnsignedDiscriminant separated by semicolons.)


358:1000795; 359:1002507; 360:1002616;
361:1005384; 362:1006931; 363:1007219; 364:1007327; 365:1008143; 366:1012755; 367:1015656; 368:1016195; 369:1016292; 370:1017115;
371:1018456; 372:1024616; 373:1024680; 374:1028443; 375:1030195; 376:1031295; 377:1031343; 378:1036580; 379:1038523; 380:1039751;
381:1044372; 382:1050891; 383:1054835; 384:1058763; 385:1059299; 386:1060984; 387:1061227; 388:1063144; 389:1064055; 390:1067243;
391:1067271; 392:1067363; 393:1069384; 394:1074327; 395:1083496; 396:1086148; 397:1086243; 398:1087768; 399:1089199; 400:1092324;
401:1093955; 402:1094035; 403:1096564; 404:1096763; 405:1097823; 406:1098827; 407:1100243; 408:1101179; 409:1102808; 410:1103447;
411:1104483; 412:1108191; 413:1110243; 414:1113171; 415:1115403; 416:1119076; 417:1119467; 418:1121155; 419:1121203; 420:1124740;
421:1126180; 422:1126932; 423:1129108; 424:1131895; 425:1134696; 426:1143188; 427:1147011; 428:1147688; 429:1150939; 430:1151428;
431:1151731; 432:1153240; 433:1154632; 434:1156919; 435:1157995; 436:1160547; 437:1165636; 438:1165908; 439:1166051; 440:1172008;
441:1177135; 442:1178264; 443:1179599; 444:1180955; 445:1182399; 446:1182695; 447:1182904; 448:1185179; 449:1187087; 450:1189923;
451:1192855; 452:1193515; 453:1195512; 454:1196767; 455:1198516; 456:1206435; 457:1207795; 458:1210319; 459:1213163; 460:1219447;
461:1222552; 462:1225348; 463:1228159; 464:1228943; 465:1229503; 466:1230955; 467:1235119; 468:1236180; 469:1237079; 470:1238039;
471:1239608; 472:1240584; 473:1247028; 474:1250783; 475:1252808; 476:1262248; 477:1270355; 478:1270852; 479:1271796; 480:1272603;
481:1273583; 482:1278427; 483:1280703; 484:1281352; 485:1282019; 486:1284376; 487:1284468; 488:1284616; 489:1286180; 490:1286651;
491:1287111; 492:1287688; 493:1289707; 494:1289720; 495:1290612; 496:1293263; 497:1295827; 498:1302767; 499:1308968; 500:1312468;
501:1319523; 502:1319639; 503:1320451; 504:1323896; 505:1326235; 506:1331083; 507:1334008; 508:1335988; 509:1336523; 510:1341204;
511:1341287; 512:1345391; 513:1346707; 514:1346788; 515:1346983; 516:1347752; 517:1348447; 518:1350619; 519:1353571; 520:1353583;
521:1353608; 522:1355176; 523:1356499; 524:1357991; 525:1360963; 526:1361883; 527:1363207; 528:1368559; 529:1370324; 530:1371839;
531:1375979; 532:1376931; 533:1378244; 534:1379215; 535:1380548; 536:1381055; 537:1381784; 538:1382232; 539:1382459; 540:1383247;
541:1383880; 542:1384139; 543:1393363; 544:1398520; 545:1400163; 546:1408996; 547:1413635; 548:1418299; 549:1420296; 550:1420719;
551:1422436; 552:1423603; 553:1426843; 554:1427091; 555:1428735; 556:1429431; 557:1431348; 558:1436072; 559:1438887; 560:1439112;
561:1440647; 562:1444807; 563:1448040; 564:1449464; 565:1450115; 566:1452376; 567:1464504; 568:1468839; 569:1468920; 570:1470043;
571:1473508; 572:1476987; 573:1482267; 574:1484083; 575:1484135; 576:1484547; 577:1484772; 578:1484984; 579:1485051; 580:1485163;
581:1485219; 582:1488815; 583:1492660; 584:1510139; 585:1512323; 586:1512559; 587:1513720; 588:1515707; 589:1519519; 590:1520223;
591:1522139; 592:1532147; 593:1533860; 594:1534291; 595:1536472; 596:1542399; 597:1543579; 598:1547687; 599:1552379; 600:1552468;
601:1553767; 602:1554567; 603:1561723; 604:1563143; 605:1567487; 606:1567495; 607:1567799; 608:1569795; 609:1574263; 610:1578247;
611:1579080; 612:1582087; 613:1582887; 614:1588996; 615:1593635; 616:1597591; 617:1602463; 618:1602739; 619:1602779; 620:1605039;
621:1605471; 622:1606472; 623:1609415; 624:1613647; 625:1615735; 626:1621587; 627:1625783; 628:1627451; 629:1631211; 630:1631267;
631:1633188; 632:1633487; 633:1635683; 634:1637720; 635:1638651; 636:1640599; 637:1641867; 638:1642207; 639:1642927; 640:1648751;
641:1650023; 642:1655083; 643:1656596; 644:1657443; 645:1662863; 646:1663023; 647:1668623; 648:1670648; 649:1671163; 650:1672148;
651:1674340; 652:1675156; 653:1676536; 654:1681611; 655:1687352; 656:1688471; 657:1689431; 658:1691095; 659:1691508; 660:1696859;
661:1698851; 662:1699412; 663:1707691; 664:1707695; 665:1709439; 666:1711527; 667:1713679; 668:1721931; 669:1722191; 670:1723803;
671:1728435; 672:1728659; 673:1731643; 674:1733543; 675:1736619; 676:1739035; 677:1739203; 678:1743963; 679:1744163; 680:1745128;
681:1745988; 682:1751335; 683:1752727; 684:1753864; 685:1754455; 686:1758440; 687:1758655; 688:1760263; 689:1760763; 690:1763383;
691:1763555; 692:1765384; 693:1767135; 694:1770499; 695:1772583; 696:1772639; 697:1774007; 698:1780303; 699:1785595; 700:1785859;
701:1790599; 702:1794903; 703:1798987; 704:1799783; 705:1802631; 706:1805155; 707:1805943; 708:1806815; 709:1810079; 710:1810183;
711:1810756; 712:1810927; 713:1812903; 714:1813607; 715:1814264; 716:1818203; 717:1822820; 718:1825060; 719:1825160; 720:1831139;
721:1832071; 722:1833887; 723:1836199; 724:1836499; 725:1836516; 726:1837659; 727:1838419; 728:1839787; 729:1840319; 730:1846255;
731:1846499; 732:1846799; 733:1847795; 734:1848619; 735:1854935; 736:1856408; 737:1856568; 738:1857128; 739:1858967; 740:1859987;
741:1860251; 742:1861563; 743:1861647; 744:1862036; 745:1862603; 746:1863167; 747:1863912; 748:1864067; 749:1865032; 750:1867247;
751:1870915; 752:1872447; 753:1876207; 754:1877448; 755:1878184; 756:1880072; 757:1884103; 758:1887419; 759:1889231; 760:1892355;
761:1895143; 762:1898403; 763:1899812; 764:1904563; 765:1905047; 766:1905107; 767:1908867; 768:1909268; 769:1909623; 770:1914104;
771:1914616; 772:1915172; 773:1915639; 774:1916740; 775:1920040; 776:1920948; 777:1923771; 778:1924967; 779:1925671; 780:1926143;
781:1926835; 782:1932099; 783:1937876; 784:1938308; 785:1941415; 786:1943319; 787:1945531; 788:1947268; 789:1948952; 790:1951923;
791:1954771; 792:1955955; 793:1957395; 794:1958308; 795:1958507; 796:1961167; 797:1962659; 798:1963627; 799:1965255; 800:1967827;
801:1968043; 802:1969167; 803:1975348; 804:1979771; 805:1980955; 806:1981803; 807:1983431; 808:1984667; 809:1986099; 810:1991683;
811:1993091; 812:1996091; 813:1998516.

5.2.1. Real Base Objects

Recently we started a search for analogous positive discriminants and discovered
the existence of 119 pentadic quantum class groups G52(K)
with abelianizations of diamond type (5,5)
over real base objects K = Q(D1/2) with 5-irregular discriminants D
extending to the upper bound 107 > D > 0.
They are given in the following table.
(Output format is Counter:Discriminant separated by semicolons.)


1: 244641; 2: 871733; 3: 995353; 4: 1129841; 5: 1167541;
6: 1277996; 7: 1305577; 8: 1378696; 9: 1455745; 10: 1469297;
11: 1497941; 12: 1510889; 13: 1677921; 14: 1777441; 15: 1787441;
16: 1791949; 17: 1915448; 18: 2041741; 19: 2138961; 20: 2177256;
21: 2183893; 22: 2238988; 23: 2353628; 24: 2368533; 25: 2594248;
26: 2646556; 27: 2723021; 28: 3092161; 29: 3148873; 30: 3323973;
31: 3431756; 32: 3535729; 33: 3542541; 34: 3545804; 35: 3567349;
36: 3622873; 37: 3715529; 38: 3812377; 39: 3845857; 40: 3905485;
41: 4019221; 42: 4268364; 43: 4389753; 44: 4405580; 45: 4410012;
46: 4652661; 47: 4686744; 48: 4731713; 49: 4888193; 50: 4925181;
51: 4954652; 52: 4992481; 53: 5026556; 54: 5041489; 55: 5340152;
56: 5377116; 57: 5425345; 58: 5539513; 59: 5571960; 60: 5653101;
61: 5741429; 62: 5814485; 63: 5927829; 64: 6048037; 65: 6429692;
66: 6563820; 67: 6586517; 68: 6658437; 69: 6661532; 70: 6854392;
71: 6959845; 72: 6963065; 73: 6969592; 74: 7012373; 75: 7148136;
76: 7216401; 77: 7232201; 78: 7276812; 79: 7306081; 80: 7561397;
81: 7564297; 82: 7783980; 83: 7814344; 84: 7835928; 85: 7893596;
86: 7965233; 87: 7979605; 88: 7982885; 89: 8008513; 90: 8117697;
91: 8182353; 92: 8266952; 93: 8410933; 94: 8476833; 95: 8486249;
96: 8588504; 97: 8633589; 98: 8641337; 99: 8664653; 100: 8784316;
101: 8897737; 102: 9082201; 103: 9098641; 104: 9145765; 105: 9171793;
106: 9198633; 107: 9217388; 108: 9318872; 109: 9385141; 110: 9424781;
111: 9479596; 112: 9638081; 113: 9697533; 114: 9704497; 115: 9781909;
116: 9818961; 117: 9820229; 118: 9897404; 119: 9924277.

5.2.2. Extended Real Base Objects

Hoping to find an example of the exceptional first excited state of TKT a.3* ↑ (200000)
with elementary abelian distinguished 5-class group Cl5(N1) of type (5,5,5,5,5),
we extended table 5.2.1 to a total of
270 pentadic quantum class groups G52(K)
with abelianizations of diamond type (5,5)
over real base objects K = Q(D1/2) with 5-irregular discriminants D
up to the upper bound 2*107 > D > 0.
(Output format is Counter:Discriminant separated by semicolons.)


120:10008677;
121:10049233; 122:10103217; 123:10107064; 124:10172165; 125:10188856; 126:10243165; 127:10486805; 128:10488389; 129:10514760; 130:10583833;
131:10638001; 132:10660229; 133:10847801; 134:10870081; 135:10891921; 136:10942621; 137:10976941; 138:11015149; 139:11056953; 140:11072396;
141:11179317; 142:11229197; 143:11255081; 144:11313161; 145:11545953; 146:11576168; 147:11717013; 148:11717917; 149:11847224; 150:12038956;
151:12136312; 152:12154204; 153:12304472; 154:12378669; 155:12384056; 156:12460121; 157:12514261; 158:12562849; 159:12570801; 160:12633877;
161:12919021; 162:13075457; 163:13217601; 164:13283261; 165:13344445; 166:13357661; 167:13371429; 168:13393393; 169:13403921; 170:13410001;
171:13477681; 172:13605033; 173:13611445; 174:13617553; 175:13689868; 176:13855145; 177:13990273; 178:14008161; 179:14010936; 180:14028221;
181:14174744; 182:14211685; 183:14276681; 184:14406748; 185:14458369; 186:14538981; 187:14624097; 188:14646444; 189:14707865; 190:14731689;
191:14737301; 192:14752001; 193:14825569; 194:14867701; 195:14870593; 196:14941592; 197:14963612; 198:14988513; 199:15062001; 200:15092857;
201:15119932; 202:15147736; 203:15411532; 204:15428561; 205:15480485; 206:15614413; 207:15621429; 208:15915373; 209:15923441; 210:15936056;
211:15976981; 212:16021820; 213:16038001; 214:16069649; 215:16115281; 216:16208213; 217:16434245; 218:16474956; 219:16494873; 220:16499549;
221:16547217; 222:16587849; 223:16654605; 224:16708337; 225:16986557; 226:17279837; 227:17287997; 228:17330561; 229:17405592; 230:17431597;
231:17535473; 232:17537797; 233:17577545; 234:17716744; 235:17744504; 236:17818888; 237:17838797; 238:18022141; 239:18052457; 240:18070649;
241:18109761; 242:18160609; 243:18221329; 244:18250008; 245:18327201; 246:18417196; 247:18434456; 248:18494312; 249:18537469; 250:18723997;
251:18764161; 252:18834493; 253:18876697; 254:18898449; 255:18931073; 256:19055605; 257:19115293; 258:19197057; 259:19211073; 260:19268728;
261:19282972; 262:19361153; 263:19527121; 264:19555789; 265:19565501; 266:19573057; 267:19590917; 268:19621905; 269:19766877; 270:19924748.


5.3. Recall of Triadic TTTs


The triadic TTT (transfer target type) τ = (str(Cl3(N1)),…,str(Cl3(N4)))
is well known by our theory of nearly homocyclic and exotic 3-class groups.
Its first component τ(1) is not able to distinguish between the transfer kernel types (TKT) a.2 (1000) and a.3 (2000),
neither in the GS (ground state) nor in the ES (excited states).

The following table shows the minimal representatives of all known GS and ES of TKTs for vertices of coclass graph G(3,1)
among the 2576 triadic quantum class groups G32(K) with abelianizations of diamond type (3,3)
over real base objects K = Q(D1/2) with 3-irregular discriminants 0 < D < 107,
computed from January to March 2010 with the aid of PARI/GP V2.3.4.
The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 4 | Cl3(Ni) ≅ (3,3,3) }.

No. Discriminant 3-Class Group of Cohomology Transfer Kernel Quantum 3-Class
D K L1 L2 L3 L4 N1 N2 N3 N4 F31(K) ε1 Type Type (TKT) Group, G32(K)
Coclass 1 (GS)
1 32009 (3,3) 3 3 3 3 (9,3) (3,3) (3,3) (3,3) (3,3) 0 (δααα) a.3 (2000) < 81,8 >
4 72329 (3,3) 3 3 3 3 (9,3) (3,3) (3,3) (3,3) (3,3) 0 (δααα) a.2 (1000) < 81,10 >
9 142079 (3,3) 3 3 3 3 (3,3,3) (3,3) (3,3) (3,3) (3,3) 1 (δααα) a.3* (2000) < 81,7 >
Coclass 1 (ES 1)
3 62501 (3,3) 9 3 3 3 (9,9) (3,3) (3,3) (3,3) (9,9) 0 (αααα) a.1 ↑ (0000) < 729,99…101 >
58 494236 (3,3) 9 3 3 3 (27,9) (3,3) (3,3) (3,3) (9,9) 0 (δααα) a.3 ↑ (2000) < 729,97…98 >
110 790085 (3,3) 9 3 3 3 (27,9) (3,3) (3,3) (3,3) (9,9) 0 (δααα) a.2 ↑ (1000) < 729,96 >
Coclass 1 (ES 2)
559 2905160 (3,3) 27 3 3 3 (27,27) (3,3) (3,3) (3,3) (27,27) 0 (αααα) a.1 ↑2 (0000) < 6561,# >

5.4. Introduction of Pentadic TTTs


The pentadic TTT (transfer target type) is defined by τ = (str(Cl5(N1)),…,str(Cl5(N6))).
Frequently, the TTT admits an identification of TKTs (transfer kernel types).
The reason is that, according to Theorem 2, for any Ni with 1 ≤ i ≤ 6,
an elementary abelian structure (5,…,5) of 5-rank at most 4 implies a coarse TKT (Taussky type) (A) and
an exceptional structure (25,5,…,5) of 5-rank at most 3 implies a coarse TKT (Taussky type) (B).

5.4.1. TTTs for Quantum Groups of Coclass 1

The first component τ(1) of the TTT admits the separation of the GS (ground state) of TKT (transfer kernel type)
a.2 (100000), where τ(1) = str(Cl5(N1)) = (5,5,5) and
a.3 (200000), where τ(1) = str(Cl5(N1)) = (25,5).
However, the first component τ(1) of the TTT is not able to distinguish the ES (excited states) of these TKTs.

The following table shows the minimal representatives of all known GS and ES of TKTs for vertices on the coclass graphs G(5,1) and G(5,2)
among the 270 pentadic quantum class groups G52(K) with abelianizations of diamond type (5,5)
over real base objects K = Q(D1/2) with 5-irregular discriminants 0 < D < 2*107,
computed in December 2011 with the aid of MAGMA V2.18-2 and valuable advice by C. Fieker .
The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 6 | Cl5(Ni) ≅ (5,5,5,5,5) }.
Additionally, we need another invariant ε2 = #{ 1 ≤ i ≤ 6 | Cl5(Ni) ≅ (5,5,5) }.
No. Discriminant 5-Class Group of Cohomology Transfer Kernel Quantum 5-Class
D K L1 L2 L3 L4 L5 L6 N1 N2 N3 N4 N5 N6 F51(K) ε1 ε2 Type Type (TKT) Group, G52(K)
Coclass 1 (GS)
1 244641 (5,5) 5 5 5 5 5 5 (25,5) (5,5) (5,5) (5,5) (5,5) (5,5) (5,5) 0 0 (δααααα) a.3 (200000) < 625,9…10 >
5 1167541 (5,5) 5 5 5 5 5 5 (5,5,5) (5,5) (5,5) (5,5) (5,5) (5,5) (5,5) 0 1 (δααααα) a.2 (100000) < 625,8 >
Coclass 1 (ES 1)
4 1129841 (5,5) (5,5) 5 5 5 5 5 (5,5,5,5) (5,5) (5,5) (5,5) (5,5) (5,5) (5,5,5,5) 0 0 (αααααα) a.1 ↑ (000000) < 15625,636…642 > ↓2
38 3812377 (5,5) (5,5) 5 5 5 5 5 (25,5,5,5) (5,5) (5,5) (5,5) (5,5) (5,5) (5,5,5,5) 0 0 (δααααα) a.2 ↑ (100000) or a.3 ↑ (200000) < 15625,631…635 >
Coclass 2 (GS)
51 4954652 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (25,5) (25,5) (25,5) (5,5,5) 0 0 (δδδδδδ) (BBBBBB) < 3125,9|12 > or < 15625,680 >
240 18070649 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (5,5,5) (25,5) (25,5) (5,5,5) 0 1 (δδδδδδ) (612435) < 3125,8|13 >
127 10486805 (5,5) 5 5 5 5 5 5 (5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (5,5,5) 0 2 (δδδδδδ) (AABBBB) < 3125,11 > or < 15625,674 >
Coclass 2 (ES 1)
79 7306081 (5,5) (5,5) 5 5 5 5 5 (5,5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (5,5,5,5) 0 1 δδδδδ) ↑ (022222) < 15625,564 >

Statistical evaluation:

Pentadic Quantum Class Groups of Coclass 1:

There occur 8 cases of TKT a.1 ↑ (000000) for
no. 4: D = 1129841, 126: 10243165, 188: 14646444, 233: 17577545,
255: 18931073, 259: 19211073, 263: 19527121, and 265: 19565501;
2 cases of ES 1 of TKT a.2 ↑ (100000) or a.3 ↑ (200000) for
38: 3812377 and 268: 19621905;
unfortunately no example of the exceptional ES 1 of TKT a.3* ↑ (200000)
with distinguished 5-class group Cl5(N1) of type (5,5,5,5,5);
and 34 cases (13%) of the GS of TKT a.2 (100000) for
5: 1167541; 7: 1305577; 8: 1378696; 9: 1455745; 10: 1469297;
17: 1915448; 27: 2723021; 29: 3148873; 30: 3323973; 35: 3567349;
59: 5571960; 63: 5927829; 87: 7979605; 91: 8182353; 96: 8588504;
103: 9098641; 112: 9638081; 115: 9781909; 116: 9818961; 125: 10188856;
136: 10942621; 147: 11717013; 148: 11717917; 156: 12460121; 159: 12570801;
160: 12633877; 196: 14941592; 213: 16038001; 222: 16587849; 225: 16986557;
226: 17279837; 237: 17838797; 248: 18494312; 249: 18537469.
The remaining 214 cases (79.3%) of the GS of TKT a.3 (200000), starting with 1: 244641, are clearly dominating.

Pentadic Quantum Class Groups of Coclass 2:

There occur 6 cases of coarse TKT (BBBBBB) for
51: 4954652, 76: 7216401, 158: 12562849, 217: 16434245, 247: 18434456, and 257: 19115293;
a single case of TKT (612435) for 240: 18070649;
2 cases of coarse TKT (AABBBB) for
127: 10486805 and 252: 18834493;
and 3 cases of TKT ↑ (022222) for
79: 7306081, 145: 11545953, and 197: 14963612.

5.4.2. TTTs for Quantum Groups of Coclass 2


The possibilities for coclass 2 are more extensive than those for coclass 1.

The following table shows the minimal representatives of all known GS and ES of TKTs for vertices on the coclass graph G(5,2)
among the 813 pentadic quantum class groups G52(K) with abelianizations of diamond type (5,5)
over complex base objects K = Q(D1/2) with 5-irregular discriminants -2*106 < D < 0,
computed in December 2011 with the aid of MAGMA V2.18-2 and valuable advice by C. Fieker .
The invariant ε1 denotes the cardinality #{ 1 ≤ i ≤ 6 | Cl5(Ni) ≅ (5,5,5,5,5) }.
Additionally, we need another invariant ε2 = #{ 1 ≤ i ≤ 6 | Cl5(Ni) ≅ (5,5,5) }.

For (first and second) excited states (ES 1 and ES 2) of coclass 2 as well as of coclass 1,
one of the 5-class groups Cl5(Li) of the non-Galois absolute quintic subfields Li of Ni (unramified over K) is of 5-rank 2,
which shows impressively that the rank equation for p = 3,
r3(Li) = r3(K)-1,
by G. Gras and F. Gerth III, conjectured by T. Callahan in 1974,
generalizes to a double inequality for p ≥ 5,
rp(K)-1 ≤ rp(Li) ≤ (rp(K)-1)*(p-1)/2,
as predicted, and partially proved, by R. Bölling and F. Lemmermeyer .

No. Discriminant 5-Class Group of Transfer Kernel Quantum 5-Class
|D| K L1 L2 L3 L4 L5 L6 N1 N2 N3 N4 N5 N6 F51(K) ε1 ε2 Type (TKT) Group, G52(K)
Coclass 2 (GS)
1 11199 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (25,5) (25,5) (25,5) (5,5,5) 0 0 (512643) < 3125,12 >
2 12451 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (5,5,5) (25,5) (25,5) (5,5,5) 0 1 (612435) < 3125,8|13 >
3 17944 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (25,5) (25,5) (25,5) (5,5,5) 0 0 (312564) < 3125,9 >
4 30263 (5,5) 5 5 5 5 5 5 (5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (25,5,5) 0 2 (126543) < 15625,674 >
6 37363 (5,5) 5 5 5 5 5 5 (5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (5,5,5) 0 2 (125364) < 3125,11 >
9 42871 (5,5) 5 5 5 5 5 5 (25,5) (25,5) (25,5) (25,5) (25,5) (25,5) (25,5,5) 0 0 (214365) < 15625,680 >
31 89751 (5,5) 5 5 5 5 5 5 (5,5,5) (5,5,5) (5,5,5) (5,5,5) (5,5,5) (5,5,5) (5,5,5) 0 6 (123456) < 3125,14 >
Coclass 2 (ES 1)
14 62632 (5,5) (5,5) 5 5 5 5 5 (25,5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (5,5,5,5,5) 0 1 (BABBBB) < 15625,564 > ↓
19 67031 (5,5) (5,5) 5 5 5 5 5 (5,5,5,5,5) (25,5) (25,5) (25,5) (25,5) (25,5) (5,5,5,5,5) 1 0 (BBBBBB) < 15625,557|558 > ↓
20 67063 (5,5) (5,5) 5 5 5 5 5 (5,5,5,5,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) (5,5,5,5,5) 1 1 (BABBBB) < 15625,564 > ↓
98 280847 (5,5) (5,5) 5 5 5 5 5 (25,5,5,5) (25,5) (25,5) (25,5) (25,5) (25,5) (5,5,5,5,5) 0 0 (BBBBBB) < 15625,557|558 > ↓
Coclass 2 (ES 2)
77 181752 (5,5) (25,5) 5 5 5 5 5 (25,25,25,5) (5,5,5) (25,5) (25,5) (25,5) (25,5) ? 0 1 (BABBBB) < 15625,564 > ↓3

Statistical evaluation:

Pentadic Quantum Class Groups of Coclass 2:

0. Ground States (GS).

Ground states appear exclusively with sporadic, and mostly terminal, top vertices of G(5,2).
We are pleased to present the solution of a problem posed in 1970 by O. Taussky :
there occur 4 cases of the very special TKT (123456) with 6 fixed points for
no. 31: D = -89751, 87: -235796, 362: -1006931, and 812: -1996091;
142 cases (17.5%) of TKT (612435) (5-cycle), starting with 2: -12451,
which was attempted but not analyzed completely in 1982 by Heider and Schmithals ;
and 241 cases (29.6%) of coarse TKT (AABBBB), starting with 4: -30263.
The remaining 253 cases (31.1%) of cTKT (BBBBBB), starting with 1: -11199 are slightly dominating.

1. First Excited States (ES).

There occur 6 cases of ES 1 of TKT ↑ (BBBBBB) with exotic 5-class group of type (5,5,5,5,5) for
19: -67031, 36: -99031, 173: -469871, 257: -702067, 732: -1846799 and 798: -1963627;
31 other cases of ES 1 of TKT ↑ (BBBBBB) starting with 98: -280847;
29 cases of ES 1 of TKT ↑ (BABBBB) with exotic 5-class group of type (5,5,5,5,5), starting with 20: -67063;
and 103 other cases (12.7%) of ES 1 of TKT ↑ (BABBBB), starting with 14: -62632 and clearly dominating.

2. Second Excited State (ES).

There occur 4 cases of ES 2 of TKT 2 (BABBBB) for
77: -181752, 375: -1030195, 435: -1157995, and 565: -1450115.
The first of these occurrences takes place very soon in comparison with the considerable delay of the triadic analog ,
where the ES 2 of TKT E.14 ↑2 (2441) occurs for no. 463: D = -262744.

Pentadic Quantum Class Groups of Coclass 4:

It is astonishing that no pentadic quantum class group of coclass 4 appeared among the 357 cases investigated,
and, unfortunately, the extension to 813 cases turned out to be in vain, too,
whereas the 29th example D = -27156 of TKT F.11 ↑↑ (4221) already was of coclass 4 among the 2020 triadic analogs .

*
Web master's e-mail address:
contact@algebra.at
*

Navigation 2012
Back to Algebra