Properties

Label 32.0.614...625.1
Degree $32$
Signature $[0, 16]$
Discriminant $6.143\times 10^{48}$
Root discriminant $33.47$
Ramified primes $5, 67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_8.A_4$ (as 32T402)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 + 38*x^30 - 130*x^29 + 699*x^28 - 2191*x^27 + 8745*x^26 - 25571*x^25 + 83386*x^24 - 222738*x^23 + 612551*x^22 - 1445800*x^21 + 3366820*x^20 - 6839883*x^19 + 13440516*x^18 - 23161945*x^17 + 38310126*x^16 - 55684805*x^15 + 77474162*x^14 - 94829103*x^13 + 110710129*x^12 - 113727761*x^11 + 110357640*x^10 - 94103904*x^9 + 74479117*x^8 - 51698016*x^7 + 32429638*x^6 - 17496051*x^5 + 7973462*x^4 - 3036857*x^3 + 868340*x^2 - 166271*x + 16531)
 
gp: K = bnfinit(x^32 - 4*x^31 + 38*x^30 - 130*x^29 + 699*x^28 - 2191*x^27 + 8745*x^26 - 25571*x^25 + 83386*x^24 - 222738*x^23 + 612551*x^22 - 1445800*x^21 + 3366820*x^20 - 6839883*x^19 + 13440516*x^18 - 23161945*x^17 + 38310126*x^16 - 55684805*x^15 + 77474162*x^14 - 94829103*x^13 + 110710129*x^12 - 113727761*x^11 + 110357640*x^10 - 94103904*x^9 + 74479117*x^8 - 51698016*x^7 + 32429638*x^6 - 17496051*x^5 + 7973462*x^4 - 3036857*x^3 + 868340*x^2 - 166271*x + 16531, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16531, -166271, 868340, -3036857, 7973462, -17496051, 32429638, -51698016, 74479117, -94103904, 110357640, -113727761, 110710129, -94829103, 77474162, -55684805, 38310126, -23161945, 13440516, -6839883, 3366820, -1445800, 612551, -222738, 83386, -25571, 8745, -2191, 699, -130, 38, -4, 1]);
 

\( x^{32} - 4 x^{31} + 38 x^{30} - 130 x^{29} + 699 x^{28} - 2191 x^{27} + 8745 x^{26} - 25571 x^{25} + 83386 x^{24} - 222738 x^{23} + 612551 x^{22} - 1445800 x^{21} + 3366820 x^{20} - 6839883 x^{19} + 13440516 x^{18} - 23161945 x^{17} + 38310126 x^{16} - 55684805 x^{15} + 77474162 x^{14} - 94829103 x^{13} + 110710129 x^{12} - 113727761 x^{11} + 110357640 x^{10} - 94103904 x^{9} + 74479117 x^{8} - 51698016 x^{7} + 32429638 x^{6} - 17496051 x^{5} + 7973462 x^{4} - 3036857 x^{3} + 868340 x^{2} - 166271 x + 16531 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6142666889587199870339155304469168186187744140625\)\(\medspace = 5^{28}\cdot 67^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.47$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 67$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{11111} a^{30} + \frac{3099}{11111} a^{29} - \frac{1480}{11111} a^{28} - \frac{568}{11111} a^{27} - \frac{1279}{11111} a^{26} + \frac{1337}{11111} a^{25} + \frac{4661}{11111} a^{24} + \frac{1889}{11111} a^{23} + \frac{4889}{11111} a^{22} + \frac{2663}{11111} a^{21} - \frac{2921}{11111} a^{20} + \frac{3509}{11111} a^{19} + \frac{4063}{11111} a^{18} + \frac{1573}{11111} a^{17} - \frac{654}{11111} a^{16} + \frac{2569}{11111} a^{15} - \frac{2627}{11111} a^{14} - \frac{5052}{11111} a^{13} - \frac{500}{11111} a^{12} + \frac{505}{11111} a^{11} - \frac{5033}{11111} a^{10} + \frac{3346}{11111} a^{9} - \frac{3744}{11111} a^{8} + \frac{3427}{11111} a^{7} - \frac{1000}{11111} a^{6} + \frac{3973}{11111} a^{5} + \frac{2816}{11111} a^{4} + \frac{321}{11111} a^{3} + \frac{2226}{11111} a^{2} - \frac{2484}{11111} a + \frac{6}{41}$, $\frac{1}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{31} + \frac{3022990281389662504175997725649858443703869232954787494965687508347926560554915482484469953525854025}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{30} + \frac{7106484644496517793254208315641515904197035912219992654675777574602218890089330402607108977069994443654}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{29} + \frac{38016723411765531537711550620960027090440836420093624528826153173922055885896281272138328054417058537628}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{28} - \frac{49203851486993140657777241962781368645347942578868208347882238430606444289699632631891686492379727824370}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{27} - \frac{11969196746175437006458031360893710904199152370723149608033007148314166667252908764239680897391727280361}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{26} - \frac{8119262074019553535978273787971235247610442022800343867590361288446202246191495595592717283186704726758}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{25} - \frac{24905875991746503330988758503506640263672041825632188111254761865164199399267050491406844676314775184383}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{24} + \frac{39502192456203462675904192298728936581405866173991126381650498891021579160751770288813121459645017027412}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{23} - \frac{44306203900832344067069861385261565067604235473016378484098438861320109759727422130259930243330048631817}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{22} + \frac{6870129459914979256127350141295314067540016928794905719370450632328633674300301682990088695906368172252}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{21} - \frac{31839608345435224072033481556502007243340709811063298097374924281688635088165864030650759145239497787184}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{20} - \frac{42332684839109427449315175217610460772616118995451492484822986765931059825571078323503369611732829980089}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{19} - \frac{6475355618642551615971139092851229639177739826980259392718208261624216696115141647843327042155147234893}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{18} - \frac{18220687304147082318535431313849888615218461017389885006264164472494084585928219799292084835566131116067}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{17} - \frac{5864477062508982966555259846382673015718466410184518148734252289278621188394620557558538971886747146060}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{16} + \frac{24254285373058930536309656964490995820934782786735649341190136335343771934526549580180338950762486489636}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{15} - \frac{42922125196527742555485714517964673729964952737993908950184064491240471560417284655708654257299016040940}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{14} - \frac{27996514559118209380055911601190607990291131900855770478623242877988148332238407718869709101876623226471}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{13} - \frac{51434193082379725960724632212261486472190626481420074891748428626299181835960953155877203205302346416629}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{12} + \frac{35313265557883914998263258963766380436701858906905028728974864449893275661833278598566078624414889234601}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{11} + \frac{1838576918460368947105734953293528895570775194933546710427660972216644239559648419011735373963294520532}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{10} + \frac{12966659366751088479851004960438569663182008009600247847214952380965299906992763638386775142133141032402}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{9} + \frac{18991772054802530739129985140438092798068725804462824161784960828904529915016293516491775741893917381469}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{8} - \frac{902797328035264285171390739504572171399596128177723428368875015772500316048445850109322199580440804910}{2559422513371445398872125376891010855322485572532487330134700345165605555382506046748905218725377365769} a^{7} - \frac{46519521634636219697149301524039691039362026931554331710903243174050274874207120372141279608316828265113}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{6} - \frac{50959278661458914436321168609044675211242628320162922817407832873578738375571162670300732655599426108438}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{5} + \frac{14297565224010523140256095210099164551862291013665732512033191269774762722307557130330486231877922964219}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{4} + \frac{47588035336785513321945196485908902013585750073566677144802170276438361661966542778626931898837390369550}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{3} + \frac{3939446001663970934855514299267887434914695046320127867887476088820082727819773343408392278141609351484}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a^{2} + \frac{46389375688435024152840378234458504983241570929944064513704595778937169278427132485637135698720827150104}{104936323048229261353757140452531445068221908473831980535522714151789827770682747916705113967740471996529} a + \frac{146047651475834222482966933697544159989344956073059705170475355065939199013978848908981308126690576358}{387218904237008344478808636356204594347682319091630924485323668456789032364142981242454295083913180799}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{90291092229261887443385456616074916398978755267412463416561909877613940161435669771329}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{31} - \frac{326293861374252267718960501123735594187278074872001671062473218909249456142379348799725}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{30} + \frac{3305154096143990392604649798762942607867306264720207923477624652218036895218836003202534}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{29} - \frac{10461690802345099154530269581539479045321507098775569027949299742209818807011522328606725}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{28} + \frac{59076540449768798365339443556406147821189119998109577865019001278374041805237391000487993}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{27} - \frac{175020865780655680054521017559555208821739954355861526584956218623243415861830902558917403}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{26} + \frac{722054428148362810852832897517949444384106217547533358114802914764800217807440121482780103}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{25} - \frac{2030102867700341092154519098787788542561405180585632607518354735676306694452163049517591225}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{24} + \frac{6745538009756878158670077952511831026407870567468592795902440264488931742230875893875663363}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{23} - \frac{17507456506922919028605150368419511115381553393756873992029625268648309339120729287570929712}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{22} + \frac{48550964685839112673543639217140380073995718013091683010443195043120534890322897081827171836}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{21} - \frac{2726887888613371311358439032071704143516050344997100249618906216885775007281681106714092074}{1926471398139242548692299969926083464153523628756370639078055861330368926975524629054759} a^{20} + \frac{260842161685592046478217690638601576196459598024569296311552414516854444441961965979294902467}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{19} - \frac{12607148598487985715691601977749226198458987208740649592266176275690648007890158250641956570}{1926471398139242548692299969926083464153523628756370639078055861330368926975524629054759} a^{18} + \frac{1014038302805579516883748690454047666626744080361581545680800514963866383412354709181118128334}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{17} - \frac{1699840212878636981318756132668485384640593490633787573152854743388825786644931185549283217772}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{16} + \frac{2802796859392509866194196150951587189128843535895512389770489831999055839979632407181712131291}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{15} - \frac{3945514531436039917140596518512799971134658899455870720061861706664154040159572196753446606073}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{14} + \frac{5471411309429574305726100198981996873782371108304937591515796293751668712357740845213176992967}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{13} - \frac{6448418347151372248884875151093679389592854692923362773680691665323768383649750950175658841641}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{12} + \frac{7504193913797034248152721199072468804507080997237263521587185760349461083497808181229392761477}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{11} - \frac{7367404887694245182183355247759212899870499318961047997562356619769340060480154655320472262050}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{10} + \frac{7114652527498930046212196958911145559339685665261738767429766698825685263160161798308668654987}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{9} - \frac{5743225305234312339309698853333507080650223695207073057414408368042493066144871539076000396105}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{8} + \frac{4500273156231488719635591244227838501192296891953780224759379956536807694684089138747815724966}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{7} - \frac{2923324478271884015395228983314938090778539961640820594981063836806908283906316601681829979214}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{6} + \frac{1793317501876108688802041042857819091330637363687747341190229034398474359440555154595714596640}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{5} - \frac{882720900799482757398234079484700737952316904690387446099147908964017412130318991753413929884}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{4} + \frac{375941787770800027486369121930360849013952961878983102335331883844167432202116229709525004752}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{3} - \frac{127182923862640809174451022277550069287811252883267530116690283917743722115071570374504254120}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a^{2} + \frac{28383098606015499999371450964778404350196797663676067794692006283517482652058428111959088483}{78985327323708944496384298766969422030294468779011196202200290314545126005996509791245119} a - \frac{13695001339095496795758407553150380970912238336373525155267460422275494561694364526388153}{291458772412210127292930991760034767639462984424395557941698488245553970501832139451089} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 879491729100.8733 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 879491729100.8733 \cdot 1}{10\sqrt{6142666889587199870339155304469168186187744140625}}\approx 0.209378059774320$ (assuming GRH)

Galois group

$C_8.A_4$ (as 32T402):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.112225.1, 8.0.12594450625.1, 16.0.99137616590976806640625.1

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 $24{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R $24{,}\,{\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ $24{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }$ $24{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $24{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ $24{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ $24{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
67Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.335.6t1.b.a$1$ $ 5 \cdot 67 $ 6.6.2518890125.1 $C_6$ (as 6T1) $0$ $1$
1.335.6t1.b.b$1$ $ 5 \cdot 67 $ 6.6.2518890125.1 $C_6$ (as 6T1) $0$ $1$
1.67.3t1.a.a$1$ $ 67 $ 3.3.4489.1 $C_3$ (as 3T1) $0$ $1$
1.67.3t1.a.b$1$ $ 67 $ 3.3.4489.1 $C_3$ (as 3T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.335.12t1.a.a$1$ $ 5 \cdot 67 $ 12.0.793100932727814453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.335.12t1.a.b$1$ $ 5 \cdot 67 $ 12.0.793100932727814453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.335.12t1.a.c$1$ $ 5 \cdot 67 $ 12.0.793100932727814453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.335.12t1.a.d$1$ $ 5 \cdot 67 $ 12.0.793100932727814453125.1 $C_{12}$ (as 12T1) $0$ $-1$
2.112225.48.a.a$2$ $ 5^{2} \cdot 67^{2}$ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.112225.48.a.b$2$ $ 5^{2} \cdot 67^{2}$ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.112225.48.a.c$2$ $ 5^{2} \cdot 67^{2}$ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
2.112225.48.a.d$2$ $ 5^{2} \cdot 67^{2}$ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.a$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.b$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.c$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.d$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.e$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.f$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.g$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 2.1675.32t402.a.h$2$ $ 5^{2} \cdot 67 $ 32.0.6142666889587199870339155304469168186187744140625.1 $C_8.A_4$ (as 32T402) $0$ $0$
* 3.112225.4t4.a.a$3$ $ 5^{2} \cdot 67^{2}$ 4.0.112225.1 $A_4$ (as 4T4) $1$ $-1$
* 3.22445.6t6.a.a$3$ $ 5 \cdot 67^{2}$ 6.2.100755605.1 $A_4\times C_2$ (as 6T6) $1$ $-1$
* 3.561125.12t29.a.a$3$ $ 5^{3} \cdot 67^{2}$ 12.8.793100932727814453125.1 $C_4\times A_4$ (as 12T29) $0$ $1$
* 3.561125.12t29.a.b$3$ $ 5^{3} \cdot 67^{2}$ 12.8.793100932727814453125.1 $C_4\times A_4$ (as 12T29) $0$ $1$

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 *.