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Properties

Label 33.33.2677496390...3841.1
Degree $33$
Signature $[33, 0]$
Discriminant $3^{44}\cdot 67^{32}$
Root discriminant $255.21$
Ramified primes $3, 67$
Class number Not computed
Class group Not computed
Galois group $C_{33}$ (as 33T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![703500067, 80878788432, 1020301549110, 2783095185303, -1812694476681, -14010626855421, -5285639627900, 26552533093695, 19258042021938, -24919584215523, -24622684612068, 11906847973017, 16594056412925, -2289097051344, -6509647282689, -286240447964, 1560447894561, 238566048363, -236603093887, -53976965718, 23212311336, 6740683807, -1484472234, -525017025, 61127718, 26459037, -1554534, -861687, 22110, 17487, -134, -201, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 201*x^31 - 134*x^30 + 17487*x^29 + 22110*x^28 - 861687*x^27 - 1554534*x^26 + 26459037*x^25 + 61127718*x^24 - 525017025*x^23 - 1484472234*x^22 + 6740683807*x^21 + 23212311336*x^20 - 53976965718*x^19 - 236603093887*x^18 + 238566048363*x^17 + 1560447894561*x^16 - 286240447964*x^15 - 6509647282689*x^14 - 2289097051344*x^13 + 16594056412925*x^12 + 11906847973017*x^11 - 24622684612068*x^10 - 24919584215523*x^9 + 19258042021938*x^8 + 26552533093695*x^7 - 5285639627900*x^6 - 14010626855421*x^5 - 1812694476681*x^4 + 2783095185303*x^3 + 1020301549110*x^2 + 80878788432*x + 703500067)
 
gp: K = bnfinit(x^33 - 201*x^31 - 134*x^30 + 17487*x^29 + 22110*x^28 - 861687*x^27 - 1554534*x^26 + 26459037*x^25 + 61127718*x^24 - 525017025*x^23 - 1484472234*x^22 + 6740683807*x^21 + 23212311336*x^20 - 53976965718*x^19 - 236603093887*x^18 + 238566048363*x^17 + 1560447894561*x^16 - 286240447964*x^15 - 6509647282689*x^14 - 2289097051344*x^13 + 16594056412925*x^12 + 11906847973017*x^11 - 24622684612068*x^10 - 24919584215523*x^9 + 19258042021938*x^8 + 26552533093695*x^7 - 5285639627900*x^6 - 14010626855421*x^5 - 1812694476681*x^4 + 2783095185303*x^3 + 1020301549110*x^2 + 80878788432*x + 703500067, 1)
 

Normalized defining polynomial

\( x^{33} - 201 x^{31} - 134 x^{30} + 17487 x^{29} + 22110 x^{28} - 861687 x^{27} - 1554534 x^{26} + 26459037 x^{25} + 61127718 x^{24} - 525017025 x^{23} - 1484472234 x^{22} + 6740683807 x^{21} + 23212311336 x^{20} - 53976965718 x^{19} - 236603093887 x^{18} + 238566048363 x^{17} + 1560447894561 x^{16} - 286240447964 x^{15} - 6509647282689 x^{14} - 2289097051344 x^{13} + 16594056412925 x^{12} + 11906847973017 x^{11} - 24622684612068 x^{10} - 24919584215523 x^{9} + 19258042021938 x^{8} + 26552533093695 x^{7} - 5285639627900 x^{6} - 14010626855421 x^{5} - 1812694476681 x^{4} + 2783095185303 x^{3} + 1020301549110 x^{2} + 80878788432 x + 703500067 \)

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

Invariants

Degree:  $33$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[33, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26774963908351534896674972007892455806354317687496033867399094865996622071193841=3^{44}\cdot 67^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.21$
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:  $3, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(603=3^{2}\cdot 67\)
Dirichlet character group:    $\lbrace$$\chi_{603}(1,·)$, $\chi_{603}(397,·)$, $\chi_{603}(400,·)$, $\chi_{603}(529,·)$, $\chi_{603}(274,·)$, $\chi_{603}(157,·)$, $\chi_{603}(160,·)$, $\chi_{603}(418,·)$, $\chi_{603}(550,·)$, $\chi_{603}(424,·)$, $\chi_{603}(553,·)$, $\chi_{603}(304,·)$, $\chi_{603}(49,·)$, $\chi_{603}(565,·)$, $\chi_{603}(442,·)$, $\chi_{603}(64,·)$, $\chi_{603}(583,·)$, $\chi_{603}(457,·)$, $\chi_{603}(205,·)$, $\chi_{603}(592,·)$, $\chi_{603}(82,·)$, $\chi_{603}(595,·)$, $\chi_{603}(88,·)$, $\chi_{603}(91,·)$, $\chi_{603}(220,·)$, $\chi_{603}(478,·)$, $\chi_{603}(226,·)$, $\chi_{603}(238,·)$, $\chi_{603}(211,·)$, $\chi_{603}(502,·)$, $\chi_{603}(169,·)$, $\chi_{603}(121,·)$, $\chi_{603}(508,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{29} a^{22} - \frac{12}{29} a^{21} - \frac{11}{29} a^{20} + \frac{4}{29} a^{19} - \frac{7}{29} a^{18} - \frac{6}{29} a^{17} - \frac{8}{29} a^{16} - \frac{12}{29} a^{15} + \frac{7}{29} a^{14} + \frac{5}{29} a^{13} - \frac{12}{29} a^{12} - \frac{5}{29} a^{11} - \frac{4}{29} a^{10} - \frac{11}{29} a^{9} + \frac{12}{29} a^{8} - \frac{2}{29} a^{7} - \frac{10}{29} a^{6} - \frac{4}{29} a^{5} - \frac{8}{29} a^{4} - \frac{9}{29} a^{3} + \frac{11}{29} a^{2} - \frac{6}{29} a$, $\frac{1}{29} a^{23} - \frac{10}{29} a^{21} - \frac{12}{29} a^{20} + \frac{12}{29} a^{19} - \frac{3}{29} a^{18} + \frac{7}{29} a^{17} + \frac{8}{29} a^{16} + \frac{8}{29} a^{15} + \frac{2}{29} a^{14} - \frac{10}{29} a^{13} - \frac{4}{29} a^{12} - \frac{6}{29} a^{11} - \frac{1}{29} a^{10} - \frac{4}{29} a^{9} - \frac{3}{29} a^{8} - \frac{5}{29} a^{7} - \frac{8}{29} a^{6} + \frac{2}{29} a^{5} + \frac{11}{29} a^{4} - \frac{10}{29} a^{3} + \frac{10}{29} a^{2} - \frac{14}{29} a$, $\frac{1}{29} a^{24} + \frac{13}{29} a^{21} - \frac{11}{29} a^{20} + \frac{8}{29} a^{19} - \frac{5}{29} a^{18} + \frac{6}{29} a^{17} - \frac{14}{29} a^{16} - \frac{2}{29} a^{15} + \frac{2}{29} a^{14} - \frac{12}{29} a^{13} - \frac{10}{29} a^{12} + \frac{7}{29} a^{11} + \frac{14}{29} a^{10} + \frac{3}{29} a^{9} - \frac{1}{29} a^{8} + \frac{1}{29} a^{7} - \frac{11}{29} a^{6} - \frac{3}{29} a^{4} + \frac{7}{29} a^{3} + \frac{9}{29} a^{2} - \frac{2}{29} a$, $\frac{1}{29} a^{25} + \frac{6}{29} a^{20} + \frac{1}{29} a^{19} + \frac{10}{29} a^{18} + \frac{6}{29} a^{17} - \frac{14}{29} a^{16} + \frac{13}{29} a^{15} + \frac{13}{29} a^{14} + \frac{12}{29} a^{13} - \frac{11}{29} a^{12} - \frac{8}{29} a^{11} - \frac{3}{29} a^{10} - \frac{3}{29} a^{9} - \frac{10}{29} a^{8} - \frac{14}{29} a^{7} + \frac{14}{29} a^{6} - \frac{9}{29} a^{5} - \frac{5}{29} a^{4} + \frac{10}{29} a^{3} - \frac{9}{29} a$, $\frac{1}{29} a^{26} + \frac{6}{29} a^{21} + \frac{1}{29} a^{20} + \frac{10}{29} a^{19} + \frac{6}{29} a^{18} - \frac{14}{29} a^{17} + \frac{13}{29} a^{16} + \frac{13}{29} a^{15} + \frac{12}{29} a^{14} - \frac{11}{29} a^{13} - \frac{8}{29} a^{12} - \frac{3}{29} a^{11} - \frac{3}{29} a^{10} - \frac{10}{29} a^{9} - \frac{14}{29} a^{8} + \frac{14}{29} a^{7} - \frac{9}{29} a^{6} - \frac{5}{29} a^{5} + \frac{10}{29} a^{4} - \frac{9}{29} a^{2}$, $\frac{1}{29} a^{27} - \frac{14}{29} a^{21} - \frac{11}{29} a^{20} + \frac{11}{29} a^{19} - \frac{1}{29} a^{18} - \frac{9}{29} a^{17} + \frac{3}{29} a^{16} - \frac{3}{29} a^{15} + \frac{5}{29} a^{14} - \frac{9}{29} a^{13} + \frac{11}{29} a^{12} - \frac{2}{29} a^{11} + \frac{14}{29} a^{10} - \frac{6}{29} a^{9} + \frac{3}{29} a^{7} - \frac{3}{29} a^{6} + \frac{5}{29} a^{5} - \frac{10}{29} a^{4} - \frac{13}{29} a^{3} - \frac{8}{29} a^{2} + \frac{7}{29} a$, $\frac{1}{29} a^{28} - \frac{5}{29} a^{21} + \frac{2}{29} a^{20} - \frac{3}{29} a^{19} + \frac{9}{29} a^{18} + \frac{6}{29} a^{17} + \frac{1}{29} a^{16} + \frac{11}{29} a^{15} + \frac{2}{29} a^{14} - \frac{6}{29} a^{13} + \frac{4}{29} a^{12} + \frac{2}{29} a^{11} - \frac{4}{29} a^{10} - \frac{9}{29} a^{9} - \frac{3}{29} a^{8} - \frac{2}{29} a^{7} + \frac{10}{29} a^{6} - \frac{8}{29} a^{5} - \frac{9}{29} a^{4} + \frac{11}{29} a^{3} - \frac{13}{29} a^{2} + \frac{3}{29} a$, $\frac{1}{29} a^{29} - \frac{1}{29} a$, $\frac{1}{1073} a^{30} - \frac{6}{1073} a^{27} - \frac{14}{1073} a^{26} - \frac{1}{1073} a^{25} + \frac{13}{1073} a^{24} - \frac{11}{1073} a^{23} + \frac{10}{1073} a^{22} + \frac{507}{1073} a^{21} + \frac{331}{1073} a^{20} - \frac{50}{1073} a^{19} - \frac{277}{1073} a^{18} + \frac{11}{1073} a^{17} + \frac{479}{1073} a^{16} - \frac{469}{1073} a^{15} - \frac{369}{1073} a^{14} + \frac{142}{1073} a^{13} + \frac{170}{1073} a^{12} + \frac{82}{1073} a^{11} - \frac{89}{1073} a^{10} - \frac{196}{1073} a^{9} - \frac{495}{1073} a^{8} - \frac{36}{1073} a^{7} + \frac{62}{1073} a^{6} - \frac{535}{1073} a^{5} + \frac{207}{1073} a^{4} + \frac{324}{1073} a^{3} - \frac{3}{37} a^{2} + \frac{267}{1073} a - \frac{1}{37}$, $\frac{1}{8370473} a^{31} - \frac{600}{8370473} a^{30} + \frac{1701}{226229} a^{29} - \frac{70935}{8370473} a^{28} + \frac{18164}{8370473} a^{27} - \frac{2479}{226229} a^{26} + \frac{46826}{8370473} a^{25} + \frac{58641}{8370473} a^{24} - \frac{20326}{8370473} a^{23} + \frac{93334}{8370473} a^{22} - \frac{3784311}{8370473} a^{21} - \frac{2955927}{8370473} a^{20} - \frac{443100}{8370473} a^{19} - \frac{3278933}{8370473} a^{18} - \frac{2629495}{8370473} a^{17} + \frac{662772}{8370473} a^{16} + \frac{78956}{288637} a^{15} + \frac{1211144}{8370473} a^{14} + \frac{926772}{8370473} a^{13} - \frac{3971563}{8370473} a^{12} + \frac{911712}{8370473} a^{11} - \frac{3327745}{8370473} a^{10} + \frac{131}{226229} a^{9} + \frac{956415}{8370473} a^{8} - \frac{2199559}{8370473} a^{7} + \frac{3433975}{8370473} a^{6} - \frac{2592987}{8370473} a^{5} + \frac{61347}{226229} a^{4} - \frac{966455}{8370473} a^{3} - \frac{33447}{8370473} a^{2} + \frac{3039975}{8370473} a - \frac{52865}{288637}$, $\frac{1}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{32} - \frac{862620682884021915344933728141776620338083354360089373519666125296595321481779842638944002530438622693620954105544227135495106546455047252918243988830840354630974686112965912}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{31} - \frac{6701033377532132423490962326049401745445610951437898189278216525657857218106660604965750396316918464875917481362442335425027487666004533819062945235511653870313139145696845321962}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{30} - \frac{1404669715559628339281165798524373776678985301553780235916393254848261259902300943885948122955298069712481273590558789925404313903058018077639368363038816912509336710899955864164}{82044016197127780692469780884764070727861426497646053480495674323758543886115453407440071115102659329199139284767372868644451075906240528583628365307183705942166013743083303284083} a^{29} - \frac{9603748123852022028293950599085053795647421472270872608814855177893508345161775544459072251564069107553897550441863422138841234682221160153967508663880270044575021582531307470145}{596482171811550621791199217783825270967424965618021307736576659272731035280136674773010246755746361069042391016281710855820468632939964924026919736963038294552504262078092123876171} a^{28} + \frac{376091280167328540403152885024064266238555188251547091228336606096090953697600653358516381762703644928889082504990904615056186144798395577111475974964169721243880604895489825158532}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{27} + \frac{260252456526252887662307057538019967670181262792061851626117068953321027916871482818116165021570811565129616392993201200531025308113848713716294278669061842614361402475986543088408}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{26} + \frac{310707383290052412508950198353408690518767372670770550508968482339844200161367427767771618065524785404965276380526322246618991213280674695101883167218973376200778167027899454093389}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{25} - \frac{4235969187415074994411521774661478339929274040437191904779759519723453183487191380589753655126357266224817729094855858618791125896923370213294644937850252279621171676180841147963}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{24} + \frac{319034065823354490915848048761580432011267816623709893183459977233297165137758644092525686603560226971951219076706098481854217456632627848949461150004729097973117097516970205554007}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{23} - \frac{225285068593281735345793081865748899693938848232158375936958310603637009014477767815054211883322249982982013630526380050481560315655905997882190807225132438972326460045909223812330}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{22} - \frac{4903890862464811584196987853913465738362100016844817665707960149606854730165870522321686131863305674355865497544742673484977755378208989095379710816149956771351147568562249523866860}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{21} - \frac{6735932913665681157238970925776817723214814626527240365082324185580096127166363025793157500472383152195865914618872700355468483994330643044317438698928097421467451999959345472747428}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{20} + \frac{2871889835466074812961920989970886728072442665033779022217737222027660754226159780814391123920395299159148785276896841241242594917778787298739399162050545566837060868294804859228937}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{19} + \frac{8316683769170368861739761657366646510234253271566385018706657661397667940224166705118734020429144528606067863190177633089274887890350616970776057556090993261312979389916255547124500}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{18} - \frac{9480900124831806384380032241855991903238413751625229057382205846215284393069836564023353716824466566295609597463748548026941045396376377809209725009251756121953441215456316346597135}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{17} - \frac{6621676468982616860128149999793814738552206919513996743980804872668579649103743374460963557373744093949431461811159781063049591564003160378739446870195393713774251623546731568558897}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{16} - \frac{10238721053025193922569737679075917743100883927470576146359986985671171156696821660048124313808905624725722327615804432035303765263949505106740170762330005786309683829544970766509503}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{15} + \frac{5006898202270739234061477712182281017097686304320660369578837899069114343093643262682551064711360836012060129884558037981380692242460079424156974344196374407526727734637840233731413}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{14} + \frac{9549351106245773336792859228029199538205516840512549045155844089659120393422495326533773937338687330453679291226533911991136087567083282507937070233542355471662060621547249321187839}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{13} + \frac{1308863452773377978091354956540241160310054449357237190029472667775675166255766434784551619966044161802879603866673854587360567373347588446877424394224105872451896185352756880638772}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{12} - \frac{9748972467570231201233266844545654922124738970537728253562836443210198958875710225946468361777960015590512016470498229920080904500205735894527003848280423972701605404712396541591651}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{11} + \frac{4005043220220098575351604125086115671308380410287993322887858071009785527754924697367777622225300564186770891916951084322130916940263257022443828030285897103221899845206384838765397}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{10} - \frac{243995887938741627029951212543271762184285548528266252385797491420379397992338033926808136923981367927373508138389011433809101494177061974884163866977711886156430181206233301200515}{761028977828530103664633484758673621579128404409199599525977117003139596736726102296599280343538460674295464400083562126391632393750989730655035526470083341325608886099634778738563} a^{9} - \frac{7326810055617989575340138968218932227786962097186769181572638082652806158668983509047636856172840169337347433354018975012613162269214730095152224009366267062501921178292448364757146}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{8} - \frac{4464541818950060840445348275226406307564561195687958843888353623982396104905844177721249807798279611441016332387665063462918131938837363906353176485665132151332001426704296174236840}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{7} - \frac{22703687399371944533366424403453632117899648061534465462245009715524586948695733445324771972093086228706970369045609390559895134370151735745621633496354845077894348506369792719575}{82044016197127780692469780884764070727861426497646053480495674323758543886115453407440071115102659329199139284767372868644451075906240528583628365307183705942166013743083303284083} a^{6} - \frac{10776427518167001748332109371294027236305009586370601839685097460577255976707529753786069213159816462993995796739538809059557357375984278077735280950581169696823073489435038409124274}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{5} - \frac{7206202051194358453051988120354844126435994776344751077580334855898605129820764560214189278363378642648422362806080579844429651621529651189400694315694642146802151428823341750213022}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{4} - \frac{4360674282092452871327259450383541328531186387853684542186172599566447704259724248935812895593637349008814194865164758433744560051597220094280944130187676067754453177468608239226902}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{3} - \frac{2086542377588899208783713678636744125687701272759163308135132728379244138781332500178049152983094010003869057287656423667772350391732300293488387478930669239595990658603419019585600}{22069840357027373006274371058001535025794723727866788386253336393091048305365056966601379129962615359554568467602423301665357339418778702188996030267632416898442657696889408583418327} a^{2} + \frac{21989805998210471177781161438202343224613858224868856310748405497630740191348526282442196457250407458286470107607207860775218854629623465058095836593843934207359448035361019834778}{82044016197127780692469780884764070727861426497646053480495674323758543886115453407440071115102659329199139284767372868644451075906240528583628365307183705942166013743083303284083} a + \frac{220485544961951818868821614273717160494988786642258018876306597348377812880526383920632069413885200782316887128357703840619806794319310217558622272154293931149999346347948398}{2101889357632191940388802920876058490451069835885424047698027494767957479753102591761789273159365923827489965724995959682799776820857322031587999874250718347402315266149918327}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $32$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{33}$ (as 33T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 33
The 33 conjugacy class representatives for $C_{33}$
Character table for $C_{33}$ is not computed

Intermediate fields

3.3.363609.1, 11.11.1822837804551761449.1

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 $33$ R $33$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{3}$ $33$ $33$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{33}$ $33$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ $33$ $33$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

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
67Data not computed