↑

Properties

Label 28.0.30734458946...5625.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $88.31$
Ramified primes $3, 5, 29$
Class number $118784$ (GRH)
Class group $[8, 8, 8, 232]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![83521, -1267554, 19656335, -8234460, 118266095, -204723830, 548816761, -583060481, 740257288, -455634883, 445333440, -193045342, 188495906, -53967794, 54491661, -10760627, 11890509, -1581476, 1880966, -181133, 224851, -15433, 19003, -1036, 1159, -40, 43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 43*x^26 - 40*x^25 + 1159*x^24 - 1036*x^23 + 19003*x^22 - 15433*x^21 + 224851*x^20 - 181133*x^19 + 1880966*x^18 - 1581476*x^17 + 11890509*x^16 - 10760627*x^15 + 54491661*x^14 - 53967794*x^13 + 188495906*x^12 - 193045342*x^11 + 445333440*x^10 - 455634883*x^9 + 740257288*x^8 - 583060481*x^7 + 548816761*x^6 - 204723830*x^5 + 118266095*x^4 - 8234460*x^3 + 19656335*x^2 - 1267554*x + 83521)
 
gp: K = bnfinit(x^28 - x^27 + 43*x^26 - 40*x^25 + 1159*x^24 - 1036*x^23 + 19003*x^22 - 15433*x^21 + 224851*x^20 - 181133*x^19 + 1880966*x^18 - 1581476*x^17 + 11890509*x^16 - 10760627*x^15 + 54491661*x^14 - 53967794*x^13 + 188495906*x^12 - 193045342*x^11 + 445333440*x^10 - 455634883*x^9 + 740257288*x^8 - 583060481*x^7 + 548816761*x^6 - 204723830*x^5 + 118266095*x^4 - 8234460*x^3 + 19656335*x^2 - 1267554*x + 83521, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 43 x^{26} - 40 x^{25} + 1159 x^{24} - 1036 x^{23} + 19003 x^{22} - 15433 x^{21} + 224851 x^{20} - 181133 x^{19} + 1880966 x^{18} - 1581476 x^{17} + 11890509 x^{16} - 10760627 x^{15} + 54491661 x^{14} - 53967794 x^{13} + 188495906 x^{12} - 193045342 x^{11} + 445333440 x^{10} - 455634883 x^{9} + 740257288 x^{8} - 583060481 x^{7} + 548816761 x^{6} - 204723830 x^{5} + 118266095 x^{4} - 8234460 x^{3} + 19656335 x^{2} - 1267554 x + 83521 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3073445894692486733947614496733715330677488092041015625=3^{14}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.31$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(4,·)$, $\chi_{435}(326,·)$, $\chi_{435}(289,·)$, $\chi_{435}(136,·)$, $\chi_{435}(371,·)$, $\chi_{435}(226,·)$, $\chi_{435}(109,·)$, $\chi_{435}(16,·)$, $\chi_{435}(401,·)$, $\chi_{435}(146,·)$, $\chi_{435}(149,·)$, $\chi_{435}(281,·)$, $\chi_{435}(154,·)$, $\chi_{435}(286,·)$, $\chi_{435}(161,·)$, $\chi_{435}(34,·)$, $\chi_{435}(419,·)$, $\chi_{435}(209,·)$, $\chi_{435}(64,·)$, $\chi_{435}(299,·)$, $\chi_{435}(274,·)$, $\chi_{435}(431,·)$, $\chi_{435}(434,·)$, $\chi_{435}(179,·)$, $\chi_{435}(181,·)$, $\chi_{435}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{17} a^{17} + \frac{5}{17} a^{16} + \frac{2}{17} a^{15} - \frac{6}{17} a^{14} + \frac{6}{17} a^{13} + \frac{4}{17} a^{12} + \frac{7}{17} a^{11} - \frac{1}{17} a^{10} - \frac{1}{17} a^{9} + \frac{3}{17} a^{8} + \frac{1}{17} a^{6} - \frac{3}{17} a^{5} - \frac{8}{17} a^{4} - \frac{5}{17} a^{3} - \frac{1}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{6}{17} a^{16} + \frac{1}{17} a^{15} + \frac{2}{17} a^{14} + \frac{8}{17} a^{13} + \frac{4}{17} a^{12} - \frac{2}{17} a^{11} + \frac{4}{17} a^{10} + \frac{8}{17} a^{9} + \frac{2}{17} a^{8} + \frac{1}{17} a^{7} - \frac{8}{17} a^{6} + \frac{7}{17} a^{5} + \frac{1}{17} a^{4} + \frac{7}{17} a^{3} + \frac{4}{17} a^{2} + \frac{5}{17} a$, $\frac{1}{17} a^{19} - \frac{3}{17} a^{16} - \frac{3}{17} a^{15} + \frac{6}{17} a^{14} + \frac{6}{17} a^{13} + \frac{5}{17} a^{12} - \frac{5}{17} a^{11} + \frac{2}{17} a^{10} - \frac{4}{17} a^{9} + \frac{2}{17} a^{8} - \frac{8}{17} a^{7} - \frac{4}{17} a^{6} - \frac{7}{17} a^{4} + \frac{8}{17} a^{3} - \frac{1}{17} a^{2} - \frac{6}{17} a$, $\frac{1}{17} a^{20} - \frac{5}{17} a^{16} - \frac{5}{17} a^{15} + \frac{5}{17} a^{14} + \frac{6}{17} a^{13} + \frac{7}{17} a^{12} + \frac{6}{17} a^{11} - \frac{7}{17} a^{10} - \frac{1}{17} a^{9} + \frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{3}{17} a^{6} + \frac{1}{17} a^{5} + \frac{1}{17} a^{4} + \frac{1}{17} a^{3} + \frac{8}{17} a^{2} - \frac{3}{17} a$, $\frac{1}{17} a^{21} + \frac{3}{17} a^{16} - \frac{2}{17} a^{15} - \frac{7}{17} a^{14} + \frac{3}{17} a^{13} - \frac{8}{17} a^{12} - \frac{6}{17} a^{11} - \frac{6}{17} a^{10} - \frac{4}{17} a^{9} - \frac{6}{17} a^{8} + \frac{3}{17} a^{7} + \frac{6}{17} a^{6} + \frac{3}{17} a^{5} - \frac{5}{17} a^{4} - \frac{8}{17} a^{2} - \frac{5}{17} a$, $\frac{1}{17} a^{22} + \frac{4}{17} a^{15} + \frac{4}{17} a^{14} + \frac{8}{17} a^{13} - \frac{1}{17} a^{12} + \frac{7}{17} a^{11} - \frac{1}{17} a^{10} - \frac{3}{17} a^{9} - \frac{6}{17} a^{8} + \frac{6}{17} a^{7} + \frac{4}{17} a^{5} + \frac{7}{17} a^{4} + \frac{7}{17} a^{3} - \frac{2}{17} a^{2} + \frac{3}{17} a$, $\frac{1}{17} a^{23} + \frac{4}{17} a^{16} + \frac{4}{17} a^{15} + \frac{8}{17} a^{14} - \frac{1}{17} a^{13} + \frac{7}{17} a^{12} - \frac{1}{17} a^{11} - \frac{3}{17} a^{10} - \frac{6}{17} a^{9} + \frac{6}{17} a^{8} + \frac{4}{17} a^{6} + \frac{7}{17} a^{5} + \frac{7}{17} a^{4} - \frac{2}{17} a^{3} + \frac{3}{17} a^{2}$, $\frac{1}{1003} a^{24} - \frac{15}{1003} a^{23} + \frac{25}{1003} a^{22} - \frac{3}{1003} a^{21} + \frac{29}{1003} a^{20} + \frac{19}{1003} a^{19} - \frac{10}{1003} a^{18} - \frac{21}{1003} a^{17} + \frac{331}{1003} a^{16} - \frac{480}{1003} a^{15} - \frac{138}{1003} a^{14} - \frac{188}{1003} a^{13} - \frac{16}{59} a^{12} - \frac{500}{1003} a^{11} + \frac{226}{1003} a^{10} + \frac{281}{1003} a^{9} - \frac{352}{1003} a^{8} - \frac{371}{1003} a^{7} + \frac{352}{1003} a^{6} + \frac{248}{1003} a^{5} - \frac{426}{1003} a^{4} - \frac{440}{1003} a^{3} + \frac{382}{1003} a^{2} + \frac{1}{59} a + \frac{12}{59}$, $\frac{1}{1003} a^{25} - \frac{23}{1003} a^{23} + \frac{18}{1003} a^{22} - \frac{16}{1003} a^{21} - \frac{18}{1003} a^{20} - \frac{20}{1003} a^{19} + \frac{6}{1003} a^{18} + \frac{16}{1003} a^{17} + \frac{355}{1003} a^{16} + \frac{23}{59} a^{15} - \frac{16}{1003} a^{14} - \frac{260}{1003} a^{13} - \frac{37}{1003} a^{12} + \frac{396}{1003} a^{11} - \frac{105}{1003} a^{10} - \frac{90}{1003} a^{9} - \frac{164}{1003} a^{8} + \frac{97}{1003} a^{7} - \frac{431}{1003} a^{6} - \frac{128}{1003} a^{5} - \frac{281}{1003} a^{4} + \frac{390}{1003} a^{3} + \frac{201}{1003} a^{2} + \frac{27}{59} a + \frac{3}{59}$, $\frac{1}{70897743630713} a^{26} + \frac{5137429766}{70897743630713} a^{25} - \frac{5137429724}{70897743630713} a^{24} - \frac{1468104312015}{70897743630713} a^{23} - \frac{1311026835214}{70897743630713} a^{22} - \frac{1321935902207}{70897743630713} a^{21} + \frac{1653760220453}{70897743630713} a^{20} - \frac{781493432858}{70897743630713} a^{19} - \frac{81810839380}{70897743630713} a^{18} + \frac{491173640447}{70897743630713} a^{17} + \frac{211102488155}{1201656671707} a^{16} - \frac{18933377310}{70685686571} a^{15} - \frac{30923118300931}{70897743630713} a^{14} + \frac{20098505203047}{70897743630713} a^{13} + \frac{13722835338156}{70897743630713} a^{12} + \frac{4566907110986}{70897743630713} a^{11} + \frac{10384969607439}{70897743630713} a^{10} - \frac{32416420371454}{70897743630713} a^{9} - \frac{31548364792799}{70897743630713} a^{8} - \frac{28964080610984}{70897743630713} a^{7} + \frac{35219295571493}{70897743630713} a^{6} + \frac{1563775045097}{70897743630713} a^{5} + \frac{5050935223830}{70897743630713} a^{4} - \frac{6045253072746}{70897743630713} a^{3} + \frac{35190035318250}{70897743630713} a^{2} + \frac{320110954861}{4170455507689} a - \frac{72683231182}{245320912217}$, $\frac{1}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{27} + \frac{1368524431225929431051005298715358778292541768569321585635558623788507513020287289}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{26} + \frac{49269865172229601517890320074022843117222051783256491833922292075608427717129892507237650385}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{25} - \frac{2844217168711317009720589439888128138577783556662535678290201732428660314585575589806940264}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{24} + \frac{1861971194126779856178466680928375310066639616937405865022126735494226085594176308782065629606}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{23} + \frac{5675135292132896137940657383675758479155669724089017298280048005061026720780820530479554099656}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{22} + \frac{6641483978574268207836630657303020953983765469352587644487416597865990074192021233449251602606}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{21} - \frac{4815415547308651270008378222583355559839600985363831293889619372352801960171470800726143177720}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{20} - \frac{6913781515729200661831471040665013470682626455194614289709952041842849921680314946211419210638}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{19} - \frac{4807923461834573563511423178601166210555695086360308592135603487976566316628949508530969900040}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{18} + \frac{5369800331174993678198854146856751906165503171438081423862526656238368645860726479656336521258}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{17} + \frac{4359515472358383779714484130612151414058329265681222053056945834748172711711618944972815545957}{15153346087528942918303048509544252180206552699840635785035317203906259268886010469052918178651} a^{16} - \frac{33024719087587908079723975836853843434090901746694169787780722985297679545697466836593766620066}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{15} + \frac{85243498640754236841601644435742317226727604588990053275286674640739241145266755141413572234328}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{14} - \frac{100093744105022916430078205175389958701220324072760975133370652801697007731944734556811961855524}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{13} - \frac{38392199769505875273229597020602930470322441604999121062644409041770469256193013282475691408965}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{12} + \frac{203148503508220008454947554655344100738168204772200350231305825888268473883560974547068517082}{839110369667726480818084119421017221705248846570979831744626685558327060492059211641366804681} a^{11} - \frac{60106499124788812091694044530585813821173138397526999694698710341186223967865567851014272021580}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{10} - \frac{66581707340182618150268472374095723822975937838472865095362284534163477638456474232180434543334}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{9} + \frac{416908142649388405897330393241400392604533853721608639062474180831668272831369769148887170192}{839110369667726480818084119421017221705248846570979831744626685558327060492059211641366804681} a^{8} + \frac{7331899629741052192741679718199354563086342820637732985301648470575663869205392812512914805522}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{7} - \frac{49994447061288863866836929020588853128690990261395047878051195868247192540658429480634253000277}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{6} + \frac{98116167176233511026819339524405541148644598451255343890554378816386349633487485382253721937023}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{5} - \frac{37229756899688985054321994869170875247840025184541986876706041100038482293195069888922038878232}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{4} + \frac{8884540096434184572424585996630416535532412507395460785257639764931819544248897656054929127554}{257606883487992029611151824662252287063511395897290808345600392466406407571062177973899609037067} a^{3} - \frac{2759385690263818750685691482172875356138862431170365698143245249144537545999670001949317848320}{15153346087528942918303048509544252180206552699840635785035317203906259268886010469052918178651} a^{2} - \frac{55001269097420323348006818815589475158255841075858025578295656388597602101232135235788047730}{891373299266408406959002853502603069423914864696507987355018659053309368758000615826642245803} a - \frac{2983159647907773759575154377342826193893267234935539144828964143691489924055466673505889538}{52433723486259318056411932558976651142583227335088705138530509356077021691647095048626014459}$

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

Class group and class number

$C_{8}\times C_{8}\times C_{8}\times C_{232}$, which has order $118784$ (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:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{880800517087610968387752804623189218042914691383995149653178490400910092649769}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{27} + \frac{884766994589138604816745212562241170475095736013825962744963323948424693374837}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{26} - \frac{37875017172464543348369071269943772847676091556198232151951924807305854376350290}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{25} + \frac{35387270362165703855178215326193788140991270474298827087558327929177456875627213}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{24} - \frac{1020848499875761792322444256355324049908237529608840016136688242046243966023400482}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{23} + \frac{916457868548835624270548565683348432453822235695876805284958872064657670362121750}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{22} - \frac{16737535731701947785542150847786469646138765500869275131734109811564550637437125535}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{21} + \frac{13651430681339882791755938005371851048316209740409912446889047270338429012732881802}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{20} - \frac{198032624020438271696805331031230881251271179892538469781423223795571644586700210152}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{19} + \frac{160154858196564207379080978014087415323515632350405515856305627153920026295654564969}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{18} - \frac{1656516883459234166286359328790383704515522975951763147435592583619906657860580191680}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{17} + \frac{82184217035841389555011630812310558714322878805361378543202407504972980440657019228}{61769483696216509076992449954029563285587889331130742739860456858877730450103426003} a^{16} - \frac{10470776643356763689702765375027500052333354937492812068439242737690822439417083040123}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{15} + \frac{9497369060628693902147444245080714554006874315380780038469406951686963729381726301609}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{14} - \frac{47978916114764658273221883373004616268999364756950899368535854023372589631493697244305}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{13} + \frac{47573150675679174326882995570211628060439444603724451159829807679210089349347975349494}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{12} - \frac{540469152474917980625089090657033436097024047075576838886098976240750230046816908305}{3420460009236744802309028173350171256856658366870432008396181650165867809940580593} a^{11} + \frac{169952629976532186631726752385225813196339546474095042325278240780338134119019343478195}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{10} - \frac{391724324721925935149789823810173349636521975644114780693377615016207586311354201861589}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{9} + \frac{1303612193991661230474586130090895051617004354732693388941888548670689494029148465369}{3420460009236744802309028173350171256856658366870432008396181650165867809940580593} a^{8} - \frac{650027200797473366118867894807274303821408966187264235699146088434698381938060300208602}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{7} + \frac{509743297490074307915533318027945890179389494398348604139372199543290752173380462894168}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{6} - \frac{477991515341630073632462877498809098857285719866086086763467117546073694733955511718159}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{5} + \frac{172073095625464718739928087906440116583663198991610018260932573353368592392923609105245}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{4} - \frac{96857927326056803531378039016846564271071789302772092662584492045470787668623810297410}{1050081222835680654308871649218502575854994118629222626577627766600921417651758242051} a^{3} + \frac{69222131742676639691146439148056774814517287513653824955399107835593812462948020424}{61769483696216509076992449954029563285587889331130742739860456858877730450103426003} a^{2} - \frac{55163722481609436230029236732407726809830182524317828932275647038379368848640850752}{3633499040953912298646614703178209605034581725360631925874144521110454732359025059} a + \frac{209219313325961810523027192297359788921652424481913155249699058640111381300453301}{213735237703171311685094982539894682649093042668272466227890854182967925432883827} \) (order $6$)
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:  \( 73869644668.60387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-435}) \), \(\Q(\sqrt{-3}, \sqrt{145})\), 7.7.594823321.1, 14.14.801611618199890796015625.1, 14.0.773792930870360792667.1, 14.0.1753124609003161170886171875.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5Data not computed
29Data not computed