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Properties

Label 32.0.22940818697...0625.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{32}\cdot 5^{28}\cdot 7^{16}$
Root discriminant $32.45$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8.A_4$ (as 32T402)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3131, -7709, 25063, -54894, 79930, -97500, 157773, -215882, 132237, -50246, 366311, -946897, 1021466, -315448, -441722, 581095, -229142, -149755, 359728, -419617, 387026, -307078, 211471, -127199, 67152, -30713, 12498, -4335, 1315, -336, 68, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131)
 
gp: K = bnfinit(x^32 - 11*x^31 + 68*x^30 - 336*x^29 + 1315*x^28 - 4335*x^27 + 12498*x^26 - 30713*x^25 + 67152*x^24 - 127199*x^23 + 211471*x^22 - 307078*x^21 + 387026*x^20 - 419617*x^19 + 359728*x^18 - 149755*x^17 - 229142*x^16 + 581095*x^15 - 441722*x^14 - 315448*x^13 + 1021466*x^12 - 946897*x^11 + 366311*x^10 - 50246*x^9 + 132237*x^8 - 215882*x^7 + 157773*x^6 - 97500*x^5 + 79930*x^4 - 54894*x^3 + 25063*x^2 - 7709*x + 3131, 1)
 

Normalized defining polynomial

\( x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2294081869728198830572049522437155246734619140625=3^{32}\cdot 5^{28}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.45$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{101} a^{28} + \frac{35}{101} a^{27} - \frac{33}{101} a^{26} + \frac{20}{101} a^{25} - \frac{7}{101} a^{24} - \frac{10}{101} a^{23} + \frac{3}{101} a^{22} - \frac{7}{101} a^{21} + \frac{21}{101} a^{20} - \frac{11}{101} a^{19} + \frac{7}{101} a^{18} + \frac{24}{101} a^{17} - \frac{24}{101} a^{16} - \frac{22}{101} a^{15} - \frac{48}{101} a^{14} - \frac{21}{101} a^{13} + \frac{31}{101} a^{12} - \frac{36}{101} a^{11} - \frac{20}{101} a^{10} - \frac{47}{101} a^{9} - \frac{19}{101} a^{8} + \frac{20}{101} a^{7} - \frac{9}{101} a^{6} - \frac{48}{101} a^{5} - \frac{6}{101} a^{4} + \frac{27}{101} a^{3} + \frac{8}{101} a^{2} - \frac{13}{101} a$, $\frac{1}{25351} a^{29} + \frac{32}{25351} a^{28} - \frac{4683}{25351} a^{27} + \frac{2947}{25351} a^{26} - \frac{976}{25351} a^{25} - \frac{5443}{25351} a^{24} + \frac{9729}{25351} a^{23} - \frac{420}{25351} a^{22} - \frac{8038}{25351} a^{21} - \frac{10073}{25351} a^{20} - \frac{1374}{25351} a^{19} - \frac{6057}{25351} a^{18} - \frac{3429}{25351} a^{17} + \frac{1363}{25351} a^{16} - \frac{2002}{25351} a^{15} - \frac{7351}{25351} a^{14} + \frac{6255}{25351} a^{13} + \frac{780}{25351} a^{12} - \frac{7891}{25351} a^{11} - \frac{4330}{25351} a^{10} - \frac{1797}{25351} a^{9} + \frac{8763}{25351} a^{8} + \frac{234}{25351} a^{7} - \frac{4566}{25351} a^{6} - \frac{2993}{25351} a^{5} + \frac{6711}{25351} a^{4} + \frac{634}{25351} a^{3} + \frac{11881}{25351} a^{2} + \frac{6806}{25351} a - \frac{40}{251}$, $\frac{1}{19292111} a^{30} - \frac{347}{19292111} a^{29} - \frac{7524}{19292111} a^{28} + \frac{531087}{19292111} a^{27} - \frac{3477791}{19292111} a^{26} - \frac{5736847}{19292111} a^{25} - \frac{780993}{19292111} a^{24} + \frac{8818866}{19292111} a^{23} - \frac{3902508}{19292111} a^{22} + \frac{4897996}{19292111} a^{21} + \frac{3200088}{19292111} a^{20} - \frac{6178728}{19292111} a^{19} - \frac{8315588}{19292111} a^{18} + \frac{1016822}{19292111} a^{17} + \frac{2300653}{19292111} a^{16} - \frac{1988007}{19292111} a^{15} + \frac{9241980}{19292111} a^{14} - \frac{8877291}{19292111} a^{13} - \frac{8406795}{19292111} a^{12} + \frac{852106}{19292111} a^{11} + \frac{5256183}{19292111} a^{10} + \frac{8999432}{19292111} a^{9} + \frac{7226077}{19292111} a^{8} + \frac{6278132}{19292111} a^{7} - \frac{6797945}{19292111} a^{6} + \frac{8680847}{19292111} a^{5} + \frac{190053}{19292111} a^{4} + \frac{7906505}{19292111} a^{3} - \frac{4269691}{19292111} a^{2} - \frac{6709703}{19292111} a - \frac{41817}{191011}$, $\frac{1}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{31} + \frac{3025478061973959547589865419216628872475589057706319352516239975546521386116811243313}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{30} + \frac{1484012568092064139352246242950802633165807965493801943658483516107523771351321801555590}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{29} - \frac{485062661136990034625296148867811159350629656804639735909870527796539534764928988137668006}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{28} - \frac{413560337412748469468826209472596293226812811878364183995323530782786039066782889413926206}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{27} + \frac{2604116400287630145580519286324359293476243658013543060155399060350788209529743117844363757}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{26} + \frac{12437424018398591756544438974947675369175595392588163064626457835926669998015981679653148945}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{25} - \frac{45642884897715281443459780114656041180201569334971718400806810028402759766515085236886675387}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{24} + \frac{2678589785232056265014545008442328081946882686395028362541985996598096997827156476052413406}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{23} - \frac{2019903127891199514977146903548043698614729356903987762472995285230153436552167188690211907}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{22} + \frac{28386988812817098092320636314832249500300170721299859081117655059900423760791472056042509832}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{21} - \frac{14450266322925004006210183782088547242948898117521073394267158915656686017416429663216443256}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{20} - \frac{62280282343605637121557189465270076653419934202193399679181705315788360607517218021345135651}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{19} + \frac{20416736089254884010790129890208301333343838788096693898814887818516099573418530408614967712}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{18} - \frac{61633872592275153488992288090944320204186423670189902942136905288927361219224023771764231641}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{17} - \frac{48484487106730215402305848034601162213684352607737691301826337330884506370361983470980334653}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{16} - \frac{47444648057485751317313601968334553916554501983866413802484590953488440491274525428488937534}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{15} - \frac{33749757739098110500889060575221009826225339657303943783735843683743608673127808989191798112}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{14} - \frac{40481075243038187178582373240196446758447235403545711249784521218425016951380948535425217563}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{13} - \frac{36843499369615161858909432245074506825513635927694417696815251826292838766374462136337663720}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{12} - \frac{11797846801227561008422823442077304269430681682695528517302702124612606424204542100488013656}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{11} + \frac{10730576975757125102142828543926403803961492503376967961366347185717123694372690948404523892}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{10} + \frac{56950807015566692259262716914552922277996106752169916253798670148738655973497204997817628529}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{9} - \frac{36929909120464706212675398653592158994851254770709922483064183707514457453382854650270386096}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{8} - \frac{18015304585112911641972087974367876208688867705556761748720025325941730914153650966694320209}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{7} - \frac{3003193913583388064933877218082412842130401208703575347349656050348758210225897428367307195}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{6} + \frac{15205402129196595507281307833609477416538161445443147633256557295905453129819027707646620065}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{5} - \frac{27729176930488948309571214778802118440492216870594065372101663434484202828672408344548663658}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{4} + \frac{29327862832311151620103403239515263045127555876250323662867785878489322203392214593313386614}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{3} - \frac{4907379012919130990978227992397577648794834667129989215726124687624352661157346735811645868}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a^{2} - \frac{68144143239118605157142206190680646527111659186154852540249001679153453130159870768161909945}{143863033309802086607455221385447194823075373346926711213759131281965350269006073102088573999} a + \frac{164257539148729213088941098730806287923870102535021128783474819750736728326076399976626230}{1424386468413882045618368528568784107159162112345809021918407240415498517514911614872164099}$

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:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{6638407240373683378630867815201395558285438932852502041753695589293418306971}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{31} - \frac{83360160096908894117006724134204811125762338095037488824399616851135042247073}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{30} + \frac{552586014163738577155325274058813465697021226866645545690642768270585828377127}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{29} - \frac{2804666642179450335549806174053850230883742957875856976185287871731565471456526}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{28} + \frac{11448376057233387831714100426239885927797227045726692325427247350711119768648762}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{27} - \frac{38750301851901452722084707016343656282044360782463035469104270631518054084347649}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{26} + \frac{114170354710479830755093491839812819515607372992189798390448294068544665228544452}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{25} - \frac{289781024804178125867948634924493148332501126369397401545487471366754020067608848}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{24} + \frac{642294565831738938401279843876035384621982698559680210429712279484438072942824730}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{23} - \frac{1254643786551414847989052661021681817905378418949421619315407380772118301660628402}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{22} + \frac{2119308879716823472559849162376310767995881815904308276858058405453685192621204068}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{21} - \frac{3147697461597465605827034390643392983115068424290894060040244414995736709618880289}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{20} + \frac{4044401234852695421880787416418196281248982391636883170032454679841276132159812162}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{19} - \frac{4464045586583621418007736119964451771989326114767631768491293495241784993241203536}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{18} + \frac{4004323544016001996908612230217484383514809114810692254535415898424143975324824488}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{17} - \frac{2004683475728018368574031666377328534245129839642259818000104859960660679573890136}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{16} - \frac{1888906683192466235594012371985261427063627709181935555341632451433730519214407584}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{15} + \frac{6147209007970713554223697174773821893593305027167001966564936272839200752533150253}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{14} - \frac{5839198610682105903044455661450306513902137529572679820326119883996285846661576201}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{13} - \frac{2253866069737533778572774913210273797380936431286312886877130995780313755236445307}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{12} + \frac{11056486303856033150293046022056592504630520843935380664945908986247760761649475200}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{11} - \frac{10905253226140676702017327308033055010532233874258641053349183755102273542016509251}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{10} + \frac{3985653179434570741793226675990179085650249929689165464680093821044730224390711367}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{9} - \frac{418220658061310907142377654321005795757674696030982683732573291628270701602078679}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{8} + \frac{1574132625456407462857469230118220107450785261564550312371024038606391021213825259}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{7} - \frac{2295789050320103733137733529913993719794609059227436965884441595334184141780030029}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{6} + \frac{1488203124856075108444096916970177017747200691082406759687608266401704060065732827}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{5} - \frac{1052713753858510674562695324286078925244442381986209996397154848513096067361582265}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{4} + \frac{921264150530854391006387252236104268354459447580578950407124966545463249105098373}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{3} - \frac{583911646184010750699507249489113873344240992705018065478318298849750464246397581}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a^{2} + \frac{221824581777955967679290896885505956341350187897596320724267372048361108090422140}{52706400691464760340439956855014258785409963498694716620815628009662672815456109} a - \frac{1038474931296181961991593062358694559628567681565259383280398696825689577356089}{521845551400641191489504523316972859261484787115789273473422059501610621935209} \) (order $10$)
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:  \( 584489557946.5273 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.A_4$ (as 32T402):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 28 conjugacy class representatives for $C_8.A_4$
Character table for $C_8.A_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.99225.1, 8.0.9845600625.1, Deg 16

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 $24{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ $24{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ $24{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $24{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
7Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3e2_5_7.6t1.4c1$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{6} - 3 x^{5} - 42 x^{4} + 19 x^{3} + 483 x^{2} + 732 x + 251$ $C_6$ (as 6T1) $0$ $1$
1.3e2_5_7.6t1.4c2$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{6} - 3 x^{5} - 42 x^{4} + 19 x^{3} + 483 x^{2} + 732 x + 251$ $C_6$ (as 6T1) $0$ $1$
1.3e2_7.3t1.1c1$1$ $ 3^{2} \cdot 7 $ $x^{3} - 21 x - 35$ $C_3$ (as 3T1) $0$ $1$
1.3e2_7.3t1.1c2$1$ $ 3^{2} \cdot 7 $ $x^{3} - 21 x - 35$ $C_3$ (as 3T1) $0$ $1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
1.3e2_5_7.12t1.1c1$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
1.3e2_5_7.12t1.1c2$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
1.3e2_5_7.12t1.1c3$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
1.3e2_5_7.12t1.1c4$1$ $ 3^{2} \cdot 5 \cdot 7 $ $x^{12} + 21 x^{10} - 35 x^{9} + 441 x^{8} + 2205 x^{7} + 10486 x^{6} + 30870 x^{5} + 143031 x^{4} + 281260 x^{3} + 540225 x^{2} + 900375 x + 1500625$ $C_{12}$ (as 12T1) $0$ $-1$
2.3e4_5e2_7e2.48.1c1$2$ $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
2.3e4_5e2_7e2.48.1c2$2$ $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
2.3e4_5e2_7e2.48.1c3$2$ $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
2.3e4_5e2_7e2.48.1c4$2$ $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c1$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c2$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c3$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c4$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c5$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c6$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c7$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 2.3e2_5e2_7.32.1c8$2$ $ 3^{2} \cdot 5^{2} \cdot 7 $ $x^{32} - 11 x^{31} + 68 x^{30} - 336 x^{29} + 1315 x^{28} - 4335 x^{27} + 12498 x^{26} - 30713 x^{25} + 67152 x^{24} - 127199 x^{23} + 211471 x^{22} - 307078 x^{21} + 387026 x^{20} - 419617 x^{19} + 359728 x^{18} - 149755 x^{17} - 229142 x^{16} + 581095 x^{15} - 441722 x^{14} - 315448 x^{13} + 1021466 x^{12} - 946897 x^{11} + 366311 x^{10} - 50246 x^{9} + 132237 x^{8} - 215882 x^{7} + 157773 x^{6} - 97500 x^{5} + 79930 x^{4} - 54894 x^{3} + 25063 x^{2} - 7709 x + 3131$ $C_8.A_4$ (as 32T402) $0$ $0$
* 3.3e4_5e2_7e2.4t4.1c1$3$ $ 3^{4} \cdot 5^{2} \cdot 7^{2}$ $x^{4} - x^{3} + 7 x + 14$ $A_4$ (as 4T4) $1$ $-1$
* 3.3e4_5_7e2.6t6.1c1$3$ $ 3^{4} \cdot 5 \cdot 7^{2}$ $x^{6} - 3 x^{5} + 12 x^{4} - 19 x^{3} + 15 x^{2} - 6 x - 1$ $A_4\times C_2$ (as 6T6) $1$ $-1$
* 3.3e4_5e3_7e2.12t29.1c1$3$ $ 3^{4} \cdot 5^{3} \cdot 7^{2}$ $x^{12} - 3 x^{11} - 39 x^{10} + 125 x^{9} + 240 x^{8} - 753 x^{7} + 114 x^{6} + 462 x^{5} - 750 x^{4} + 2115 x^{3} - 3129 x^{2} + 1617 x + 251$ C4*A4 $0$ $1$
* 3.3e4_5e3_7e2.12t29.1c2$3$ $ 3^{4} \cdot 5^{3} \cdot 7^{2}$ $x^{12} - 3 x^{11} - 39 x^{10} + 125 x^{9} + 240 x^{8} - 753 x^{7} + 114 x^{6} + 462 x^{5} - 750 x^{4} + 2115 x^{3} - 3129 x^{2} + 1617 x + 251$ C4*A4 $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.