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Properties

Label 32.0.72742794030...0625.2
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $69.09$
Ramified primes $3, 5, 17$
Class number $17520$ (GRH)
Class group $[2, 8760]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18800896, 33890176, 7184704, -3964352, 24696960, 28090720, 11656076, -1571294, -24556589, -21617481, 15851367, 21128450, -464629, -8705985, -1392802, 904191, 320975, -33646, 175483, -49605, -21524, 14365, 7117, -9994, 761, 1509, -94, -275, 125, 22, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 11*x^30 + 22*x^29 + 125*x^28 - 275*x^27 - 94*x^26 + 1509*x^25 + 761*x^24 - 9994*x^23 + 7117*x^22 + 14365*x^21 - 21524*x^20 - 49605*x^19 + 175483*x^18 - 33646*x^17 + 320975*x^16 + 904191*x^15 - 1392802*x^14 - 8705985*x^13 - 464629*x^12 + 21128450*x^11 + 15851367*x^10 - 21617481*x^9 - 24556589*x^8 - 1571294*x^7 + 11656076*x^6 + 28090720*x^5 + 24696960*x^4 - 3964352*x^3 + 7184704*x^2 + 33890176*x + 18800896)
 
gp: K = bnfinit(x^32 - x^31 - 11*x^30 + 22*x^29 + 125*x^28 - 275*x^27 - 94*x^26 + 1509*x^25 + 761*x^24 - 9994*x^23 + 7117*x^22 + 14365*x^21 - 21524*x^20 - 49605*x^19 + 175483*x^18 - 33646*x^17 + 320975*x^16 + 904191*x^15 - 1392802*x^14 - 8705985*x^13 - 464629*x^12 + 21128450*x^11 + 15851367*x^10 - 21617481*x^9 - 24556589*x^8 - 1571294*x^7 + 11656076*x^6 + 28090720*x^5 + 24696960*x^4 - 3964352*x^3 + 7184704*x^2 + 33890176*x + 18800896, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 11 x^{30} + 22 x^{29} + 125 x^{28} - 275 x^{27} - 94 x^{26} + 1509 x^{25} + 761 x^{24} - 9994 x^{23} + 7117 x^{22} + 14365 x^{21} - 21524 x^{20} - 49605 x^{19} + 175483 x^{18} - 33646 x^{17} + 320975 x^{16} + 904191 x^{15} - 1392802 x^{14} - 8705985 x^{13} - 464629 x^{12} + 21128450 x^{11} + 15851367 x^{10} - 21617481 x^{9} - 24556589 x^{8} - 1571294 x^{7} + 11656076 x^{6} + 28090720 x^{5} + 24696960 x^{4} - 3964352 x^{3} + 7184704 x^{2} + 33890176 x + 18800896 \)

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:  \(72742794030721358896825924818930661340866148471832275390625=3^{16}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.09$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(128,·)$, $\chi_{255}(1,·)$, $\chi_{255}(2,·)$, $\chi_{255}(4,·)$, $\chi_{255}(8,·)$, $\chi_{255}(16,·)$, $\chi_{255}(149,·)$, $\chi_{255}(154,·)$, $\chi_{255}(32,·)$, $\chi_{255}(166,·)$, $\chi_{255}(169,·)$, $\chi_{255}(43,·)$, $\chi_{255}(172,·)$, $\chi_{255}(178,·)$, $\chi_{255}(53,·)$, $\chi_{255}(191,·)$, $\chi_{255}(64,·)$, $\chi_{255}(202,·)$, $\chi_{255}(77,·)$, $\chi_{255}(83,·)$, $\chi_{255}(212,·)$, $\chi_{255}(86,·)$, $\chi_{255}(89,·)$, $\chi_{255}(223,·)$, $\chi_{255}(101,·)$, $\chi_{255}(106,·)$, $\chi_{255}(239,·)$, $\chi_{255}(247,·)$, $\chi_{255}(251,·)$, $\chi_{255}(253,·)$, $\chi_{255}(254,·)$, $\chi_{255}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{9} - \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{32} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{19} - \frac{1}{16} a^{13} - \frac{1}{8} a^{10} + \frac{1}{32} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} + \frac{1}{32} a^{14} + \frac{3}{32} a^{11} + \frac{5}{64} a^{8} - \frac{5}{64} a^{5} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{21} - \frac{1}{64} a^{18} - \frac{1}{32} a^{15} - \frac{1}{32} a^{12} + \frac{5}{64} a^{9} + \frac{3}{64} a^{6} + \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{64} a^{22} - \frac{1}{64} a^{19} - \frac{1}{32} a^{13} - \frac{7}{64} a^{10} - \frac{13}{64} a^{7} + \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{64} a^{23} - \frac{1}{64} a^{17} - \frac{1}{64} a^{11} - \frac{1}{8} a^{8} + \frac{1}{64} a^{5} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{24} + \frac{1}{128} a^{18} - \frac{5}{128} a^{12} + \frac{3}{128} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{256} a^{25} - \frac{1}{256} a^{24} + \frac{1}{256} a^{19} - \frac{1}{256} a^{18} - \frac{1}{16} a^{14} - \frac{5}{256} a^{13} - \frac{11}{256} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{61}{256} a^{7} - \frac{51}{256} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{512} a^{26} - \frac{1}{512} a^{24} - \frac{1}{128} a^{23} + \frac{1}{512} a^{20} - \frac{1}{512} a^{18} - \frac{1}{128} a^{17} - \frac{1}{32} a^{15} + \frac{11}{512} a^{14} - \frac{1}{32} a^{13} - \frac{27}{512} a^{12} - \frac{7}{128} a^{11} + \frac{3}{32} a^{10} - \frac{1}{16} a^{9} + \frac{19}{512} a^{8} + \frac{1}{32} a^{7} + \frac{93}{512} a^{6} - \frac{23}{128} a^{5} - \frac{3}{32} a^{4} - \frac{5}{32} a^{3} + \frac{3}{16} a^{2} + \frac{1}{8}$, $\frac{1}{512} a^{27} - \frac{1}{512} a^{25} + \frac{1}{512} a^{21} - \frac{1}{512} a^{19} + \frac{11}{512} a^{15} - \frac{1}{32} a^{14} - \frac{27}{512} a^{13} + \frac{1}{32} a^{12} + \frac{3}{32} a^{11} - \frac{45}{512} a^{9} + \frac{1}{32} a^{8} - \frac{35}{512} a^{7} + \frac{7}{32} a^{6} - \frac{3}{32} a^{5} - \frac{1}{4} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{4602368} a^{28} - \frac{59}{575296} a^{27} - \frac{535}{2301184} a^{26} - \frac{1935}{1150592} a^{25} + \frac{6209}{4602368} a^{24} + \frac{1033}{1150592} a^{23} - \frac{7943}{4602368} a^{22} + \frac{933}{143824} a^{21} + \frac{14453}{2301184} a^{20} - \frac{7201}{1150592} a^{19} + \frac{52921}{4602368} a^{18} - \frac{10665}{1150592} a^{17} - \frac{3205}{4602368} a^{16} + \frac{165}{5696} a^{15} - \frac{64237}{2301184} a^{14} - \frac{5441}{1150592} a^{13} - \frac{32773}{4602368} a^{12} - \frac{4997}{1150592} a^{11} - \frac{448517}{4602368} a^{10} - \frac{831}{35956} a^{9} + \frac{189631}{2301184} a^{8} - \frac{117807}{1150592} a^{7} + \frac{1149899}{4602368} a^{6} - \frac{154587}{1150592} a^{5} + \frac{10737}{287648} a^{4} - \frac{112377}{287648} a^{3} - \frac{47331}{143824} a^{2} - \frac{6003}{17978} a + \frac{15675}{71912}$, $\frac{1}{9204736} a^{29} - \frac{1}{9204736} a^{28} - \frac{3823}{4602368} a^{27} + \frac{663}{9204736} a^{26} - \frac{7775}{9204736} a^{25} + \frac{3573}{4602368} a^{24} - \frac{39351}{9204736} a^{23} + \frac{28127}{9204736} a^{22} + \frac{34121}{4602368} a^{21} + \frac{3543}{9204736} a^{20} - \frac{102175}{9204736} a^{19} + \frac{50229}{4602368} a^{18} - \frac{68573}{9204736} a^{17} + \frac{62005}{9204736} a^{16} - \frac{56965}{4602368} a^{15} + \frac{331501}{9204736} a^{14} - \frac{395717}{9204736} a^{13} - \frac{280649}{4602368} a^{12} - \frac{118789}{9204736} a^{11} + \frac{1070173}{9204736} a^{10} - \frac{445909}{4602368} a^{9} + \frac{62117}{9204736} a^{8} - \frac{1502845}{9204736} a^{7} - \frac{108321}{4602368} a^{6} + \frac{268883}{1150592} a^{5} - \frac{26167}{575296} a^{4} - \frac{503}{17978} a^{3} - \frac{13891}{35956} a^{2} + \frac{32163}{143824} a - \frac{7511}{71912}$, $\frac{1}{15127730412585778567476224} a^{30} - \frac{115517553030792851}{15127730412585778567476224} a^{29} - \frac{449907427870601859}{7563865206292889283738112} a^{28} - \frac{14766938582716358224671}{15127730412585778567476224} a^{27} - \frac{14470041742338218754237}{15127730412585778567476224} a^{26} - \frac{11913163529448032096865}{7563865206292889283738112} a^{25} - \frac{41317223986620867349445}{15127730412585778567476224} a^{24} - \frac{5444188611100756101419}{15127730412585778567476224} a^{23} - \frac{50451461053937917432555}{7563865206292889283738112} a^{22} + \frac{27206274912104532507057}{15127730412585778567476224} a^{21} - \frac{14940561023826820618013}{15127730412585778567476224} a^{20} + \frac{93000313313639389078607}{7563865206292889283738112} a^{19} - \frac{59055744679313530096283}{15127730412585778567476224} a^{18} + \frac{220684528495119405956279}{15127730412585778567476224} a^{17} - \frac{60414489124465316861553}{7563865206292889283738112} a^{16} + \frac{469690557900121309043963}{15127730412585778567476224} a^{15} + \frac{13514274148497304966765}{285428875709165633348608} a^{14} + \frac{191505561591301429837445}{7563865206292889283738112} a^{13} - \frac{95026359597023039955743}{15127730412585778567476224} a^{12} - \frac{1392381119774916658693697}{15127730412585778567476224} a^{11} - \frac{552794972019793834513041}{7563865206292889283738112} a^{10} + \frac{387441535890979479468307}{15127730412585778567476224} a^{9} - \frac{1055075380983359865377463}{15127730412585778567476224} a^{8} + \frac{35221807052502033161447}{175903842006811378691584} a^{7} + \frac{1547195637208216499088511}{7563865206292889283738112} a^{6} + \frac{11399855746332200372809}{1890966301573222320934528} a^{5} + \frac{47978190725290881131895}{472741575393305580233632} a^{4} - \frac{113816599311456539345229}{472741575393305580233632} a^{3} - \frac{10814825509885562502523}{59092696924163197529204} a^{2} + \frac{45362979919647550020407}{118185393848326395058408} a - \frac{176550136022501900963}{436108464384968247448}$, $\frac{1}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{31} + \frac{39762103271343174356927959331420333512453021905}{18979063356084915356238107225894089988917626616825913366800548861670076416} a^{30} + \frac{571201665957475744368027877260190846062117329563975007550145496989}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{29} - \frac{11697511646716407593514285642944080589766785286613788006856286408485}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{28} + \frac{6606799234575229590626422584623749317149993759026652379433789639515909}{9489531678042457678119053612947044994458813308412956683400274430835038208} a^{27} - \frac{31824376370337194966337910735059111494294595056908202422818705500619043}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{26} - \frac{182079310336063109218951242652470149572682112209825013554575396699304729}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{25} - \frac{77851082597375813838783684152228886515511017064017892305498671864201207}{37958126712169830712476214451788179977835253233651826733601097723340152832} a^{24} - \frac{281456021768221208239447750250431616166431958417781413291660979501043939}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{23} + \frac{356748032549832967647479721757525992085077113041296747048350572896091451}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{22} - \frac{13574634961586608375869632297001063479624143669391492051970044163469}{2188545128699828800304209781583728089127955098803726172370911999731328} a^{21} + \frac{199165845512811057341398647763688983896730958711674223672788340909772861}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{20} + \frac{476102145755452776477064768838442172735251511969791191702521499602120081}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{19} - \frac{17547369864445790517469738517460892044779812823512265689306756346916387}{37958126712169830712476214451788179977835253233651826733601097723340152832} a^{18} - \frac{2323861603004373068287178578585906340686493310862599103150023951598222801}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{17} + \frac{1127656828321771204951412401891150552774774939568386946803975704462815481}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{16} - \frac{34251382537418804647685834206839944824599183309103385463284670728029249}{9489531678042457678119053612947044994458813308412956683400274430835038208} a^{15} - \frac{6002356785762918048258423952247049869930667923672924574968192277623544369}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{14} + \frac{3727309894426644517352180600422397003405749009288849118984489983956887077}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{13} + \frac{1861270717198263022126393802384633019561709507970631828301247182544083771}{37958126712169830712476214451788179977835253233651826733601097723340152832} a^{12} - \frac{95877979987752429484429554383055141940907169692490005070302388632053559}{794934590830781795025679883807082303200738287615745062483792622478327808} a^{11} + \frac{6290439808775868861839162590573693773310368916364488379674826358459706225}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{10} + \frac{206109929190574454630644486298785548959239222406309053655212536698757427}{2372382919510614419529763403236761248614703327103239170850068607708759552} a^{9} - \frac{16331625852922975346880227632632606500641049742117102060171121813068919593}{151832506848679322849904857807152719911341012934607306934404390893360611328} a^{8} - \frac{13535320671924391777336021104497508774667255160535371752162590710549093647}{75916253424339661424952428903576359955670506467303653467202195446680305664} a^{7} - \frac{2412400221659919710346333873068797545929527694303819462250456141379551619}{37958126712169830712476214451788179977835253233651826733601097723340152832} a^{6} - \frac{795533777512728142493461058874876934798197594635644909654515716766820135}{4744765839021228839059526806473522497229406654206478341700137215417519104} a^{5} + \frac{230574519216947761098270354230333094160073161554866317544183321969600295}{1186191459755307209764881701618380624307351663551619585425034303854379776} a^{4} - \frac{112819668074912336678543877401094749979549332760859096074091571177845719}{296547864938826802441220425404595156076837915887904896356258575963594944} a^{3} + \frac{128860726817266234986529289749243890430383645192735743376928718156988993}{2372382919510614419529763403236761248614703327103239170850068607708759552} a^{2} - \frac{11593826038429242258362214166432278587400768518085253569284613492098329}{1186191459755307209764881701618380624307351663551619585425034303854379776} a - \frac{763841603087058958092226462229509421918693129373246240705039614568503}{2188545128699828800304209781583728089127955098803726172370911999731328}$

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

Class group and class number

$C_{2}\times C_{8760}$, which has order $17520$ (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{657086522963876357752357180103109886947710301498087}{6347527028096214749375865632248157525209282966626646065152} a^{31} + \frac{198089833044375900276681898849353788890920495165213}{793440878512026843671983204031019690651160370828330758144} a^{30} + \frac{5363382593499191810303096248093222056522670407729941}{6347527028096214749375865632248157525209282966626646065152} a^{29} - \frac{22818877709758050576101964062486822541536586328821565}{6347527028096214749375865632248157525209282966626646065152} a^{28} - \frac{3310380077898263986615432312827880846869363665947561}{396720439256013421835991602015509845325580185414165379072} a^{27} + \frac{267065549892646616430612952841071623073530236695687317}{6347527028096214749375865632248157525209282966626646065152} a^{26} - \frac{283663254484817028983535480287806892646310861457247185}{6347527028096214749375865632248157525209282966626646065152} a^{25} - \frac{181611215872741068212982100463467493145586263494406775}{1586881757024053687343966408062039381302320741656661516288} a^{24} + \frac{665676030874397632896851649074786011495659762787581077}{6347527028096214749375865632248157525209282966626646065152} a^{23} + \frac{5999274116639353411020096427508481906036965650452084771}{6347527028096214749375865632248157525209282966626646065152} a^{22} - \frac{416315378919535389760936937583857518873018227173509035}{198360219628006710917995801007754922662790092707082689536} a^{21} + \frac{6131388654539596390310570038264851636676033517177795829}{6347527028096214749375865632248157525209282966626646065152} a^{20} + \frac{11972223987899263176939981710738875158438883021043971785}{6347527028096214749375865632248157525209282966626646065152} a^{19} + \frac{71514265471474608727055043927354494589038356210726247}{36904226907536132263813172280512543751216761433875849216} a^{18} - \frac{136478844484578459002108091640650400141396406366242407209}{6347527028096214749375865632248157525209282966626646065152} a^{17} + \frac{204088198438157544272183400055290465898720148333897138545}{6347527028096214749375865632248157525209282966626646065152} a^{16} - \frac{26702058225341597338373521164907956468616132482566892571}{396720439256013421835991602015509845325580185414165379072} a^{15} - \frac{89450815413451995210665969474640547227101244914794227657}{6347527028096214749375865632248157525209282966626646065152} a^{14} + \frac{1293670923127189052844457564503760334190033352184173924733}{6347527028096214749375865632248157525209282966626646065152} a^{13} + \frac{989546943239048462290957791304324296086599395293743898811}{1586881757024053687343966408062039381302320741656661516288} a^{12} - \frac{30563578416100012887058842044333931948855628139689095327}{33233125801550862562177306975121243587483156893333225472} a^{11} - \frac{7985772305795335088180152042813006516916885706955332895191}{6347527028096214749375865632248157525209282966626646065152} a^{10} + \frac{2281944244564557596299810076070253948014320884112310551}{4613028363442016532976646535064067968902095179234481152} a^{9} + \frac{13735179644498090285589039168058206461045310963745758621183}{6347527028096214749375865632248157525209282966626646065152} a^{8} - \frac{1329707116919508401717574919856793835487207635189658661767}{3173763514048107374687932816124078762604641483313323032576} a^{7} - \frac{259227216163361422178976367673902113071789234701513720203}{1586881757024053687343966408062039381302320741656661516288} a^{6} - \frac{200275781746886197810466864932980626664366187488776646815}{198360219628006710917995801007754922662790092707082689536} a^{5} - \frac{81420677440999307558399358071012524529243017075301129141}{49590054907001677729498950251938730665697523176770672384} a^{4} + \frac{3621934332213683209926021211996821954024648058174490383}{12397513726750419432374737562984682666424380794192668096} a^{3} + \frac{93278827355879248489973154198656796700011628993432195801}{99180109814003355458997900503877461331395046353541344768} a^{2} - \frac{66508149048790161345910539109211099058420310060015743873}{49590054907001677729498950251938730665697523176770672384} a - \frac{62125854269166979955366737349386131713195075201589439}{91494566249080586216787731092137879457006500326145152} \) (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:  \( 162037216252175.56 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{85})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{-15}, \sqrt{-51})\), \(\Q(\sqrt{-15}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{-51})\), 4.4.122825.1, 4.0.1105425.1, 4.4.4913.1, 4.0.44217.1, 8.0.4228250625.1, 8.0.1221964430625.4, 8.0.1955143089.1, 8.8.15085980625.1, 8.0.1221964430625.3, 8.0.1221964430625.1, 8.0.1221964430625.2, 8.0.6411541765625.2, 8.8.519334883015625.1, 8.0.6411541765625.1, 8.8.519334883015625.2, 16.0.1493197069712680437890625.1, 16.0.269708720716852904093994140625.2, 16.0.269708720716852904093994140625.1, 16.0.41107867812353742431640625.1, 16.16.269708720716852904093994140625.1, 16.0.269708720716852904093994140625.4, 16.0.269708720716852904093994140625.3

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{16}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
17Data not computed