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Properties

Label 33.33.4310183904...4961.1
Degree $33$
Signature $[33, 0]$
Discriminant $331^{32}$
Root discriminant $277.63$
Ramified prime $331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11929082177, 32809035298, 78469286329, -284076623751, -121808299470, 900256719370, -54894829630, -1552342739498, 367945862575, 1698429690588, -487547125457, -1269500051881, 343137082097, 670475622740, -146537401384, -252575482403, 39220743072, 67632545787, -6539712577, -12788859303, 645059522, 1696328846, -30572191, -156418516, -311795, 9866153, 118107, -413016, -5947, 10849, 127, -160, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 160*x^31 + 127*x^30 + 10849*x^29 - 5947*x^28 - 413016*x^27 + 118107*x^26 + 9866153*x^25 - 311795*x^24 - 156418516*x^23 - 30572191*x^22 + 1696328846*x^21 + 645059522*x^20 - 12788859303*x^19 - 6539712577*x^18 + 67632545787*x^17 + 39220743072*x^16 - 252575482403*x^15 - 146537401384*x^14 + 670475622740*x^13 + 343137082097*x^12 - 1269500051881*x^11 - 487547125457*x^10 + 1698429690588*x^9 + 367945862575*x^8 - 1552342739498*x^7 - 54894829630*x^6 + 900256719370*x^5 - 121808299470*x^4 - 284076623751*x^3 + 78469286329*x^2 + 32809035298*x - 11929082177)
 
gp: K = bnfinit(x^33 - x^32 - 160*x^31 + 127*x^30 + 10849*x^29 - 5947*x^28 - 413016*x^27 + 118107*x^26 + 9866153*x^25 - 311795*x^24 - 156418516*x^23 - 30572191*x^22 + 1696328846*x^21 + 645059522*x^20 - 12788859303*x^19 - 6539712577*x^18 + 67632545787*x^17 + 39220743072*x^16 - 252575482403*x^15 - 146537401384*x^14 + 670475622740*x^13 + 343137082097*x^12 - 1269500051881*x^11 - 487547125457*x^10 + 1698429690588*x^9 + 367945862575*x^8 - 1552342739498*x^7 - 54894829630*x^6 + 900256719370*x^5 - 121808299470*x^4 - 284076623751*x^3 + 78469286329*x^2 + 32809035298*x - 11929082177, 1)
 

Normalized defining polynomial

\( x^{33} - x^{32} - 160 x^{31} + 127 x^{30} + 10849 x^{29} - 5947 x^{28} - 413016 x^{27} + 118107 x^{26} + 9866153 x^{25} - 311795 x^{24} - 156418516 x^{23} - 30572191 x^{22} + 1696328846 x^{21} + 645059522 x^{20} - 12788859303 x^{19} - 6539712577 x^{18} + 67632545787 x^{17} + 39220743072 x^{16} - 252575482403 x^{15} - 146537401384 x^{14} + 670475622740 x^{13} + 343137082097 x^{12} - 1269500051881 x^{11} - 487547125457 x^{10} + 1698429690588 x^{9} + 367945862575 x^{8} - 1552342739498 x^{7} - 54894829630 x^{6} + 900256719370 x^{5} - 121808299470 x^{4} - 284076623751 x^{3} + 78469286329 x^{2} + 32809035298 x - 11929082177 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(431018390484252150311278335994398797372126151977884846814848506899060252737944961=331^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.63$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $331$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(331\)
Dirichlet character group:    $\lbrace$$\chi_{331}(1,·)$, $\chi_{331}(259,·)$, $\chi_{331}(132,·)$, $\chi_{331}(270,·)$, $\chi_{331}(146,·)$, $\chi_{331}(131,·)$, $\chi_{331}(280,·)$, $\chi_{331}(283,·)$, $\chi_{331}(284,·)$, $\chi_{331}(31,·)$, $\chi_{331}(163,·)$, $\chi_{331}(293,·)$, $\chi_{331}(167,·)$, $\chi_{331}(169,·)$, $\chi_{331}(299,·)$, $\chi_{331}(180,·)$, $\chi_{331}(308,·)$, $\chi_{331}(223,·)$, $\chi_{331}(318,·)$, $\chi_{331}(198,·)$, $\chi_{331}(74,·)$, $\chi_{331}(79,·)$, $\chi_{331}(80,·)$, $\chi_{331}(212,·)$, $\chi_{331}(85,·)$, $\chi_{331}(88,·)$, $\chi_{331}(89,·)$, $\chi_{331}(219,·)$, $\chi_{331}(95,·)$, $\chi_{331}(274,·)$, $\chi_{331}(111,·)$, $\chi_{331}(297,·)$, $\chi_{331}(120,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{14} - \frac{1}{8} a^{7} + \frac{1}{8}$, $\frac{1}{8} a^{22} - \frac{1}{8} a^{15} - \frac{1}{8} a^{8} + \frac{1}{8} a$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{16} - \frac{1}{8} a^{9} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{24} - \frac{1}{16} a^{22} - \frac{1}{16} a^{21} - \frac{1}{8} a^{18} - \frac{1}{16} a^{17} - \frac{1}{8} a^{16} - \frac{1}{16} a^{15} + \frac{1}{16} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{8} - \frac{3}{16} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{7}{16} a^{3} - \frac{3}{8} a^{2} - \frac{3}{16} a - \frac{5}{16}$, $\frac{1}{16} a^{25} - \frac{1}{16} a^{23} - \frac{1}{16} a^{22} - \frac{1}{8} a^{19} - \frac{1}{16} a^{18} - \frac{1}{8} a^{17} - \frac{1}{16} a^{16} + \frac{1}{16} a^{15} - \frac{1}{4} a^{12} - \frac{1}{16} a^{11} - \frac{3}{16} a^{9} - \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} - \frac{7}{16} a^{4} - \frac{3}{8} a^{3} - \frac{3}{16} a^{2} - \frac{5}{16} a - \frac{1}{4}$, $\frac{1}{16} a^{26} - \frac{1}{16} a^{23} - \frac{1}{16} a^{22} - \frac{1}{16} a^{21} - \frac{1}{8} a^{20} - \frac{1}{16} a^{19} - \frac{1}{8} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{15} + \frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{3}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{5}{16} a^{2} + \frac{1}{16} a + \frac{3}{16}$, $\frac{1}{16} a^{27} - \frac{1}{16} a^{23} - \frac{1}{16} a^{21} - \frac{1}{16} a^{20} - \frac{1}{8} a^{17} + \frac{1}{16} a^{16} - \frac{1}{8} a^{15} - \frac{1}{16} a^{14} + \frac{3}{16} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{16} a^{9} - \frac{3}{16} a^{7} + \frac{5}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} + \frac{7}{16} a^{2} - \frac{3}{8} a - \frac{3}{16}$, $\frac{1}{5792} a^{28} - \frac{131}{5792} a^{27} + \frac{3}{724} a^{26} + \frac{21}{2896} a^{25} - \frac{69}{5792} a^{24} - \frac{205}{5792} a^{23} - \frac{151}{5792} a^{22} - \frac{4}{181} a^{21} - \frac{217}{5792} a^{20} - \frac{75}{1448} a^{19} + \frac{1}{181} a^{18} - \frac{177}{5792} a^{17} + \frac{239}{5792} a^{16} - \frac{197}{5792} a^{15} + \frac{45}{724} a^{14} + \frac{1443}{5792} a^{13} - \frac{85}{1448} a^{12} + \frac{317}{2896} a^{11} - \frac{639}{5792} a^{10} - \frac{403}{5792} a^{9} + \frac{275}{5792} a^{8} + \frac{20}{181} a^{7} + \frac{2281}{5792} a^{6} + \frac{57}{724} a^{5} - \frac{1}{1448} a^{4} + \frac{2693}{5792} a^{3} - \frac{127}{5792} a^{2} + \frac{1017}{5792} a - \frac{969}{5792}$, $\frac{1}{5792} a^{29} - \frac{123}{5792} a^{27} - \frac{9}{724} a^{26} + \frac{3}{5792} a^{25} + \frac{21}{724} a^{24} + \frac{9}{362} a^{23} - \frac{361}{5792} a^{22} - \frac{333}{5792} a^{21} - \frac{129}{5792} a^{20} - \frac{267}{2896} a^{19} - \frac{691}{5792} a^{18} + \frac{55}{1448} a^{17} - \frac{191}{2896} a^{16} - \frac{107}{5792} a^{15} - \frac{629}{5792} a^{14} - \frac{633}{5792} a^{13} - \frac{13}{724} a^{12} - \frac{483}{5792} a^{11} - \frac{213}{1448} a^{10} + \frac{355}{1448} a^{9} + \frac{1189}{5792} a^{8} + \frac{1413}{5792} a^{7} + \frac{2789}{5792} a^{6} - \frac{1085}{2896} a^{5} + \frac{2531}{5792} a^{4} - \frac{41}{362} a^{3} - \frac{751}{2896} a^{2} + \frac{607}{2896} a + \frac{1933}{5792}$, $\frac{1}{5792} a^{30} + \frac{105}{5792} a^{27} + \frac{59}{5792} a^{26} - \frac{3}{181} a^{25} - \frac{17}{5792} a^{24} - \frac{59}{1448} a^{23} - \frac{41}{2896} a^{22} + \frac{55}{5792} a^{21} - \frac{75}{5792} a^{20} - \frac{667}{5792} a^{19} + \frac{87}{2896} a^{18} - \frac{71}{5792} a^{17} - \frac{197}{2896} a^{16} - \frac{61}{1448} a^{15} - \frac{517}{5792} a^{14} - \frac{1081}{5792} a^{13} + \frac{1137}{5792} a^{12} + \frac{3}{724} a^{11} - \frac{71}{5792} a^{10} + \frac{4}{181} a^{9} - \frac{481}{2896} a^{8} + \frac{421}{5792} a^{7} - \frac{709}{5792} a^{6} + \frac{1423}{5792} a^{5} + \frac{693}{2896} a^{4} - \frac{769}{5792} a^{3} + \frac{2245}{5792} a^{2} + \frac{131}{724} a - \frac{1175}{5792}$, $\frac{1}{5792} a^{31} + \frac{29}{2896} a^{27} - \frac{41}{2896} a^{26} - \frac{83}{5792} a^{25} + \frac{131}{5792} a^{24} + \frac{85}{5792} a^{23} - \frac{9}{2896} a^{22} + \frac{333}{5792} a^{21} + \frac{199}{2896} a^{20} + \frac{45}{1448} a^{19} + \frac{189}{5792} a^{18} - \frac{271}{5792} a^{17} + \frac{363}{5792} a^{16} - \frac{13}{724} a^{15} + \frac{215}{5792} a^{14} + \frac{469}{2896} a^{13} + \frac{667}{2896} a^{12} + \frac{1415}{5792} a^{11} + \frac{253}{5792} a^{10} + \frac{447}{5792} a^{9} + \frac{253}{2896} a^{8} + \frac{147}{5792} a^{7} + \frac{1143}{2896} a^{6} + \frac{297}{724} a^{5} + \frac{1823}{5792} a^{4} + \frac{15}{2896} a^{3} + \frac{1713}{5792} a^{2} + \frac{20}{181} a - \frac{1063}{5792}$, $\frac{1}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{32} - \frac{166857052534244019064537945070193662973032697322934012424023711580927746967856506388928321474047522911573017477487625629889681}{3068824740457925414763930127935600464277782149015462094237341504656407990646735339745218788204223354106471601387050518218840376352} a^{31} - \frac{67947909676207260983053617922969422244677663065893198296580504883442107451921438741266748825531831047312812650832278647855359}{3068824740457925414763930127935600464277782149015462094237341504656407990646735339745218788204223354106471601387050518218840376352} a^{30} + \frac{39881142992725189040232003731208679834594558186648439270088404171566963917191305203972606897896018358024833195481147850729585}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{29} - \frac{3130690420027675129274385406520569437932702594704092173934699482070382283820606993402742666328912576998867668161889311429469}{1534412370228962707381965063967800232138891074507731047118670752328203995323367669872609394102111677053235800693525259109420188176} a^{28} - \frac{135291346973935006603905274514163642555592354284321256097989045189408172209154475854991824849153936970708882228997417762699652367}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{27} + \frac{143285514071505853980044298636106637299490784671866769179563795910119557152343053278394005988820804137960189884499438509946110139}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{26} + \frac{35920426504103726879079675301519560075079136369758941673589097090086523779841087439423119116208328480622911240175677445276624627}{1534412370228962707381965063967800232138891074507731047118670752328203995323367669872609394102111677053235800693525259109420188176} a^{25} - \frac{121397550943702912775544043163454298647229977615539385173594783912416963028889483938360962687988963518834627939373927987148318083}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{24} - \frac{84800471549418776413480423774744123429344327312551376767898593213547007563922249930171233330311910620039818584968940287401294345}{1534412370228962707381965063967800232138891074507731047118670752328203995323367669872609394102111677053235800693525259109420188176} a^{23} + \frac{9244472756202725571389955723812725849996356373074069121898035783274134257739405429448893044153580848563046198694261928705627891}{191801546278620338422745632995975029017361384313466380889833844041025499415420958734076174262763959631654475086690657388677523522} a^{22} - \frac{379434433483205317631491205014707982241216490146375634693770598232162941568477059204210735500007145054839536168841889252972587903}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{21} + \frac{348302559309398322956858453859029452429508907906313687410577655567211702695043420652165702540942927544117253923980646518131157889}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{20} - \frac{651391902699785073079518137411450122114822141764110413879087741059755514479763974742854078064531656748133744775154528168920174763}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{19} - \frac{60369110631264874515308286735939593186371411144687266461465924262662434060766400911078782571509717763494487159282504362223229585}{767206185114481353690982531983900116069445537253865523559335376164101997661683834936304697051055838526617900346762629554710094088} a^{18} - \frac{694922853606211150344749174680522293674883513456330158385356130880556832245307045150253424329656093610576320809258090610973228417}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{17} - \frac{23339316729455510785599967800678593892769972271553972805308087931468366293548133039098504241517128612475225803329152225557678459}{383603092557240676845491265991950058034722768626932761779667688082050998830841917468152348525527919263308950173381314777355047044} a^{16} + \frac{39498539884253338256649710123202314585755283703469723917531516386536033767776287892625919752579216841966838491979779951967841611}{767206185114481353690982531983900116069445537253865523559335376164101997661683834936304697051055838526617900346762629554710094088} a^{15} - \frac{137294643503344416765771960106648125360560363958000113655598365851458034271754323927737448775444426662597230402124873926556499867}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{14} + \frac{392430545691461935846353094233730401268143446019787111039242461251164993797480460705770674853279673590525157830731856214145325911}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{13} + \frac{557106841710180136228964181863092550677081717111052564653623996669785989152115507953238077592374524025511187445899010623537803577}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{12} - \frac{12349740312082892623914460384056135069937994832694150229848456163065632741002013669152313109564626085746973646042621781841789345}{1534412370228962707381965063967800232138891074507731047118670752328203995323367669872609394102111677053235800693525259109420188176} a^{11} + \frac{1346091525235358576998688294698629979789818372109987625000893878287136846108197127249890905610994822383151670815068510930127280283}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{10} + \frac{32269801288833108540644563480101710773189598128533899595329052946171773903219568959841174200386875305573530999003604476668371881}{767206185114481353690982531983900116069445537253865523559335376164101997661683834936304697051055838526617900346762629554710094088} a^{9} + \frac{206546104073067069509671344673375633840892292999915319914541339613237683367208584323721451719087887349118014758592807389969168493}{1534412370228962707381965063967800232138891074507731047118670752328203995323367669872609394102111677053235800693525259109420188176} a^{8} - \frac{248073579334872874885586444085688711541835229450921564509201294559014669375154206645998373885339687506716603726154809237382593941}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{7} + \frac{2568197605012534573026916082040419613294061010137409351557468672531408281736843268021172484421825337277583397269121720242559254151}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{6} + \frac{482302328827597240763638055053412536791034292548344316141676551055131838021042279662631194544138434811910444646887249684331096103}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{5} - \frac{1821853277140147504676929193734821264336789326981369436697710578209518600556934014028690223851354044612781596796663744853111691265}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{4} - \frac{2789055168123612669765970510525901980829046938066860817381826134677317528739758709791745748083424995578699083104344855907063730437}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a^{3} - \frac{556780071622873386435008787443240168762819920358009215959446880585923073164933907537462696911745164559583586556288368797533455915}{3068824740457925414763930127935600464277782149015462094237341504656407990646735339745218788204223354106471601387050518218840376352} a^{2} + \frac{2733494329005887270091947844121231024939128750169184147772542631032945981425435528604468628502734968249028555863907510651986611939}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704} a - \frac{903648549198294686089883344962100574009063860384800423659476021625655767602530776453387698829122709429648492588821317446381583021}{6137649480915850829527860255871200928555564298030924188474683009312815981293470679490437576408446708212943202774101036437680752704}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $32$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2249458248958823000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

3.3.109561.1, 11.11.15786284949774657045043801.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{11}$ $33$ $33$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{3}$ $33$ $33$ $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{11}$ $33$ $33$ $33$ $33$ $33$ $33$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
331Data not computed