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Properties

Label 27.27.5659106580...6209.9
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $296.70$
Ramified primes $3, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1335980438165647, -2609849224374930, -9665204246538732, 7574956000765869, 15167762481168378, -5379727037407434, -7789317328108479, 1787389461378522, 1968336232189875, -343364658706698, -288687216994956, 42243715332927, 26759487401136, -3503200886262, -1642096841094, 200583185334, 68210967663, -7986694239, -1923801079, 219630546, 36210411, -4074096, -434322, 48483, 2994, -333, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 333*x^25 + 2994*x^24 + 48483*x^23 - 434322*x^22 - 4074096*x^21 + 36210411*x^20 + 219630546*x^19 - 1923801079*x^18 - 7986694239*x^17 + 68210967663*x^16 + 200583185334*x^15 - 1642096841094*x^14 - 3503200886262*x^13 + 26759487401136*x^12 + 42243715332927*x^11 - 288687216994956*x^10 - 343364658706698*x^9 + 1968336232189875*x^8 + 1787389461378522*x^7 - 7789317328108479*x^6 - 5379727037407434*x^5 + 15167762481168378*x^4 + 7574956000765869*x^3 - 9665204246538732*x^2 - 2609849224374930*x + 1335980438165647)
 
gp: K = bnfinit(x^27 - 9*x^26 - 333*x^25 + 2994*x^24 + 48483*x^23 - 434322*x^22 - 4074096*x^21 + 36210411*x^20 + 219630546*x^19 - 1923801079*x^18 - 7986694239*x^17 + 68210967663*x^16 + 200583185334*x^15 - 1642096841094*x^14 - 3503200886262*x^13 + 26759487401136*x^12 + 42243715332927*x^11 - 288687216994956*x^10 - 343364658706698*x^9 + 1968336232189875*x^8 + 1787389461378522*x^7 - 7789317328108479*x^6 - 5379727037407434*x^5 + 15167762481168378*x^4 + 7574956000765869*x^3 - 9665204246538732*x^2 - 2609849224374930*x + 1335980438165647, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 333 x^{25} + 2994 x^{24} + 48483 x^{23} - 434322 x^{22} - 4074096 x^{21} + 36210411 x^{20} + 219630546 x^{19} - 1923801079 x^{18} - 7986694239 x^{17} + 68210967663 x^{16} + 200583185334 x^{15} - 1642096841094 x^{14} - 3503200886262 x^{13} + 26759487401136 x^{12} + 42243715332927 x^{11} - 288687216994956 x^{10} - 343364658706698 x^{9} + 1968336232189875 x^{8} + 1787389461378522 x^{7} - 7789317328108479 x^{6} - 5379727037407434 x^{5} + 15167762481168378 x^{4} + 7574956000765869 x^{3} - 9665204246538732 x^{2} - 2609849224374930 x + 1335980438165647 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.70$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2457=3^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2457}(256,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(835,·)$, $\chi_{2457}(1990,·)$, $\chi_{2457}(1537,·)$, $\chi_{2457}(841,·)$, $\chi_{2457}(2122,·)$, $\chi_{2457}(718,·)$, $\chi_{2457}(16,·)$, $\chi_{2457}(1873,·)$, $\chi_{2457}(1171,·)$, $\chi_{2457}(22,·)$, $\chi_{2457}(1303,·)$, $\chi_{2457}(2011,·)$, $\chi_{2457}(1054,·)$, $\chi_{2457}(352,·)$, $\chi_{2457}(484,·)$, $\chi_{2457}(1894,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1192,·)$, $\chi_{2457}(235,·)$, $\chi_{2457}(1075,·)$, $\chi_{2457}(2356,·)$, $\chi_{2457}(373,·)$, $\chi_{2457}(1654,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1660,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} - \frac{3}{13} a^{8} - \frac{3}{13} a^{7} + \frac{1}{13} a^{6} + \frac{5}{13} a^{4} - \frac{6}{13} a^{3} - \frac{4}{13} a^{2} - \frac{6}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{10} + \frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} + \frac{4}{13} a^{3} - \frac{5}{13} a^{2} - \frac{4}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{8} + \frac{6}{13} a^{7} + \frac{4}{13} a^{6} - \frac{4}{13} a^{5} - \frac{1}{13} a^{4} + \frac{1}{13} a^{3} - \frac{4}{13} a - \frac{1}{13}$, $\frac{1}{13} a^{12} + \frac{4}{13} a^{8} + \frac{2}{13} a^{7} + \frac{1}{13} a^{6} - \frac{1}{13} a^{5} - \frac{4}{13} a^{3} + \frac{2}{13} a^{2} - \frac{5}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{13} + \frac{1}{13} a^{8} - \frac{5}{13} a^{6} + \frac{2}{13} a^{4} - \frac{2}{13} a^{2} + \frac{3}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{14} + \frac{3}{13} a^{8} - \frac{2}{13} a^{7} - \frac{1}{13} a^{6} + \frac{2}{13} a^{5} - \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{6}{13} a^{2} + \frac{2}{13} a - \frac{1}{13}$, $\frac{1}{221} a^{15} + \frac{2}{221} a^{14} + \frac{4}{221} a^{13} + \frac{2}{221} a^{11} - \frac{7}{221} a^{10} + \frac{3}{221} a^{9} + \frac{4}{221} a^{8} - \frac{67}{221} a^{7} - \frac{98}{221} a^{6} - \frac{70}{221} a^{5} + \frac{54}{221} a^{4} + \frac{67}{221} a^{3} - \frac{48}{221} a^{2} + \frac{48}{221} a - \frac{2}{221}$, $\frac{1}{221} a^{16} - \frac{8}{221} a^{13} + \frac{2}{221} a^{12} + \frac{6}{221} a^{11} - \frac{2}{221} a^{9} + \frac{44}{221} a^{8} + \frac{53}{221} a^{7} - \frac{6}{17} a^{6} + \frac{41}{221} a^{5} + \frac{10}{221} a^{4} - \frac{12}{221} a^{3} + \frac{8}{221} a^{2} - \frac{98}{221} a - \frac{64}{221}$, $\frac{1}{221} a^{17} - \frac{8}{221} a^{14} + \frac{2}{221} a^{13} + \frac{6}{221} a^{12} - \frac{2}{221} a^{10} - \frac{7}{221} a^{9} - \frac{15}{221} a^{8} + \frac{75}{221} a^{7} - \frac{10}{221} a^{6} + \frac{10}{221} a^{5} - \frac{46}{221} a^{4} + \frac{93}{221} a^{3} + \frac{106}{221} a^{2} + \frac{21}{221} a - \frac{3}{13}$, $\frac{1}{2873} a^{18} - \frac{6}{2873} a^{17} + \frac{3}{2873} a^{16} - \frac{6}{2873} a^{15} + \frac{3}{2873} a^{14} - \frac{22}{2873} a^{13} + \frac{89}{2873} a^{12} + \frac{37}{2873} a^{11} + \frac{110}{2873} a^{10} - \frac{41}{2873} a^{9} - \frac{18}{2873} a^{8} + \frac{96}{221} a^{7} + \frac{167}{2873} a^{6} - \frac{922}{2873} a^{5} + \frac{813}{2873} a^{4} - \frac{1017}{2873} a^{3} + \frac{418}{2873} a^{2} - \frac{324}{2873} a + \frac{1028}{2873}$, $\frac{1}{2873} a^{19} + \frac{6}{2873} a^{17} - \frac{1}{2873} a^{16} + \frac{6}{2873} a^{15} - \frac{1}{169} a^{14} + \frac{74}{2873} a^{13} - \frac{105}{2873} a^{12} - \frac{110}{2873} a^{11} + \frac{47}{2873} a^{10} + \frac{48}{2873} a^{9} + \frac{581}{2873} a^{8} - \frac{15}{2873} a^{7} + \frac{639}{2873} a^{6} + \frac{11}{221} a^{5} + \frac{22}{221} a^{4} + \frac{1154}{2873} a^{3} + \frac{79}{221} a^{2} - \frac{1371}{2873} a + \frac{1397}{2873}$, $\frac{1}{2873} a^{20} - \frac{4}{2873} a^{17} + \frac{1}{2873} a^{16} + \frac{6}{2873} a^{15} - \frac{100}{2873} a^{14} + \frac{14}{2873} a^{13} + \frac{32}{2873} a^{12} + \frac{98}{2873} a^{11} - \frac{1}{2873} a^{10} - \frac{70}{2873} a^{9} + \frac{535}{2873} a^{8} + \frac{1289}{2873} a^{7} + \frac{1117}{2873} a^{6} + \frac{683}{2873} a^{5} + \frac{1255}{2873} a^{4} + \frac{44}{2873} a^{3} - \frac{1097}{2873} a^{2} - \frac{3}{221} a - \frac{1007}{2873}$, $\frac{1}{2873} a^{21} + \frac{3}{2873} a^{17} + \frac{5}{2873} a^{16} + \frac{6}{2873} a^{15} + \frac{6}{221} a^{14} - \frac{43}{2873} a^{13} - \frac{79}{2873} a^{12} + \frac{108}{2873} a^{11} + \frac{71}{2873} a^{10} - \frac{58}{2873} a^{9} + \frac{1217}{2873} a^{8} + \frac{1312}{2873} a^{7} + \frac{1078}{2873} a^{6} - \frac{1198}{2873} a^{5} - \frac{955}{2873} a^{4} + \frac{815}{2873} a^{3} + \frac{15}{169} a^{2} + \frac{674}{2873} a + \frac{1369}{2873}$, $\frac{1}{2873} a^{22} - \frac{3}{2873} a^{17} - \frac{3}{2873} a^{16} + \frac{5}{2873} a^{15} - \frac{2}{221} a^{14} + \frac{1}{221} a^{13} - \frac{94}{2873} a^{12} - \frac{1}{2873} a^{11} + \frac{80}{2873} a^{10} - \frac{77}{2873} a^{9} - \frac{376}{2873} a^{8} + \frac{376}{2873} a^{7} - \frac{1361}{2873} a^{6} - \frac{35}{2873} a^{5} + \frac{1067}{2873} a^{4} + \frac{1200}{2873} a^{3} + \frac{1253}{2873} a^{2} - \frac{142}{2873} a + \frac{1297}{2873}$, $\frac{1}{2873} a^{23} + \frac{5}{2873} a^{17} + \frac{1}{2873} a^{16} - \frac{5}{2873} a^{15} - \frac{108}{2873} a^{14} - \frac{69}{2873} a^{13} - \frac{46}{2873} a^{12} - \frac{30}{2873} a^{11} - \frac{72}{2873} a^{10} - \frac{96}{2873} a^{9} + \frac{179}{2873} a^{8} + \frac{368}{2873} a^{7} + \frac{492}{2873} a^{6} - \frac{503}{2873} a^{5} + \frac{662}{2873} a^{4} - \frac{589}{2873} a^{3} - \frac{318}{2873} a^{2} + \frac{71}{221} a - \frac{1024}{2873}$, $\frac{1}{2873} a^{24} + \frac{5}{2873} a^{17} + \frac{6}{2873} a^{16} + \frac{59}{2873} a^{14} - \frac{105}{2873} a^{13} + \frac{84}{2873} a^{12} + \frac{55}{2873} a^{11} - \frac{35}{2873} a^{10} + \frac{5}{169} a^{9} + \frac{1420}{2873} a^{8} + \frac{609}{2873} a^{7} - \frac{363}{2873} a^{6} - \frac{708}{2873} a^{5} - \frac{24}{221} a^{4} + \frac{633}{2873} a^{3} + \frac{1381}{2873} a^{2} - \frac{743}{2873} a - \frac{1435}{2873}$, $\frac{1}{565981} a^{25} + \frac{84}{565981} a^{24} + \frac{5}{565981} a^{23} - \frac{64}{565981} a^{22} + \frac{3}{43537} a^{21} - \frac{33}{565981} a^{20} + \frac{21}{565981} a^{19} + \frac{19}{565981} a^{18} - \frac{1159}{565981} a^{17} - \frac{56}{43537} a^{16} - \frac{6}{43537} a^{15} - \frac{8343}{565981} a^{14} - \frac{16604}{565981} a^{13} - \frac{17042}{565981} a^{12} - \frac{16777}{565981} a^{11} - \frac{9675}{565981} a^{10} + \frac{19600}{565981} a^{9} - \frac{66115}{565981} a^{8} - \frac{225494}{565981} a^{7} - \frac{180373}{565981} a^{6} - \frac{220319}{565981} a^{5} + \frac{1653}{565981} a^{4} + \frac{228662}{565981} a^{3} - \frac{124226}{565981} a^{2} + \frac{258845}{565981} a - \frac{255416}{565981}$, $\frac{1}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{26} - \frac{1413343968866318407862382009172404587296804736866189454520156243146073855250218116313316390287266107658670237513941824951665602173908614262656}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{25} + \frac{3562966870912499480501871095240821577772209962967796895334127839214232079835082191589885954627770368151908066411666443768798499823967852879284671}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{24} - \frac{2338059270400768186246765465983020557414037500525560286761266365572505499542419190570197652388855008533106672771081274604999146973720088461974989}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{23} + \frac{8293485863191147358058064015940099993282639265838992823527511541560012558812246061272004681560595580234136325647614117826140415959406471716644182}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{22} - \frac{8085837036050903252932876968799468495454964379571277420761851530242532628964711082988085878993896965688355481155216306853545038461876033094863014}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{21} + \frac{984866345389855190105166187533014450351409750558143110419841754403124549599726583123288248641571676376233513251914077968038194738848230284822157}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{20} - \frac{4315510555304553055631555381834862749763070092959856312150898053400898276431574269275810371736876038054941858308296444849687894572343727601867955}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{19} - \frac{8889779300912383081298956865633245125670520695184591853288263762226370752607979978795538248039100810488382635343089932788909190673470419427368}{56154324529804668649646431313949808695679498034045637661571646444653840421644727610702623125352150567632994839853554664959830639973438701812552601} a^{18} - \frac{77097028227448354685567360516209334579471570850359204423853366980811751780037798002578136299331191567546662368918500938360905934639801313317077122}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{17} - \frac{6010904323135606295747427733318134291190300761753760796581602252293698850440445885616285776281610200222931339696202360461077963287719421794329364}{3986957041616131474124896623290436417393244360417240273971586897570422669936775660359886241900002690301942633629602381212147975438114147828691234671} a^{16} + \frac{19522219387048983516964799410987856304256759299067008433471259571308859494072646280033225699402626300243510612539242313127413493255400433230978324}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{15} + \frac{1262593280066131750303269104067171817238323015990265466012773352085493094696445958904392012909796619284309099804412550092704203369338193750758586357}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{14} + \frac{522414767900286240738786015386799731104836518592998510146213971968606140902287157581178398327171873993061697271591215020518344800510620427921079}{306689003201240882624992047945418185953326489262864636459352838274647897687444281566145095530769437715534048740738644708626767341393395986822402667} a^{13} - \frac{1316954099877850543015167642421238056004058762007877328591823983518260159656932470570419649825726759514242090977989579771103020468369825887589550040}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{12} - \frac{341387265169630930775643003693326952483536842948246083630621830849477353857761651682182151170330303587660950466568491168015034988286917110433759302}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{11} - \frac{514579787350650397940124887602938031851538560382971320661026464558416354993934794832387917598582700524024845998968363994133972031807279466011731405}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{10} - \frac{981553061455843920057524224879583608475271362236790886019683794798873625054722565885709432290306404471649291750538649244240531390826099994065892779}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{9} - \frac{1511461085203330802610430163198708368699078680742917818569982665336497412039104198621058540839306018090989699368303630282168831867448441840715816159}{3986957041616131474124896623290436417393244360417240273971586897570422669936775660359886241900002690301942633629602381212147975438114147828691234671} a^{8} - \frac{807386525535151121041011701777044496709157650323455103912875406412049943332096752548285581515573180294262084767500239951455157089916854252470798537}{3048849502412335833154332711927980789771304510907301385978272333436205571128122563804618890864707939642662013952048879750466098864440230692528591219} a^{7} - \frac{231537674151575946725056753812898696597963048969308344604196136735186568753475487951067929664749732976902719914430000845746800146431833185460436136}{730006218887460692445403607081347513043833474442593289600431403780499925481381458939134100629577957379228932918096210644477798319654703123563183813} a^{6} + \frac{8651193714877378475036395343155831882421055800788427697133098775020111259377722597812935915941987443413473050383921944421830865826081827012873542427}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{5} + \frac{13779665758594242704249208835188732551606752693713459563420777831694476382758701270422647882198595062513065752797213488511294482737270579080262380041}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{4} + \frac{22799530882144797537179092368889983619853962593508802450227374154717432280063793225852658437982887475894050009720713673614368531987089853151488581800}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{3} + \frac{17047245742977044856491850607393139256547189362574137502239727629523227114325057740048848464985062955042809522773361945537574732126941436464032250540}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a^{2} - \frac{24184753454841031647797153219976939715790417102873923303560362642048421191632211554677742641571845203874834704582803854492916688303409949264441917076}{51830441541009709163623656102775673426112176685424123561630629668415494709178083584678521144700034973925254237184830955757923680695483921772986050723} a + \frac{323926285232775679975064192028723903739028822178983343800872572869015625690780310076047543053708744629415298114624039309035432967027221064411096}{900130972734230200302593843503511113494245961088277792355649079877311868657683673167859556879874176793130620120957103137457211244950312112901583}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $26$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.3969.1, \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1, 9.9.17820338848416865911969.4, 9.9.17820338848416865911969.2, 9.9.151470380950257681.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. 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
3Data not computed
7Data not computed
$13$13.9.6.3$x^{9} - 52 x^{6} + 676 x^{3} - 79092$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
13.9.6.3$x^{9} - 52 x^{6} + 676 x^{3} - 79092$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
13.9.6.3$x^{9} - 52 x^{6} + 676 x^{3} - 79092$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$