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Properties

Label 33.33.1988802753...8081.1
Degree $33$
Signature $[33, 0]$
Discriminant $463^{32}$
Root discriminant $384.42$
Ramified prime $463$
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![509248489, 6767585596, 934259347, -250097496007, -598044769496, 1343595105723, 5769638127754, 2241066960907, -14284279230133, -21152129527475, -1491679119790, 18101684222486, 11512222917539, -4114024022131, -6282532031836, -536588718261, 1554506047014, 424789196171, -213351348451, -90626504470, 17186085873, 10645175728, -797096725, -781070288, 18323632, 37089947, -33149, -1132606, -7901, 21325, 159, -224, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - x^32 - 224*x^31 + 159*x^30 + 21325*x^29 - 7901*x^28 - 1132606*x^27 - 33149*x^26 + 37089947*x^25 + 18323632*x^24 - 781070288*x^23 - 797096725*x^22 + 10645175728*x^21 + 17186085873*x^20 - 90626504470*x^19 - 213351348451*x^18 + 424789196171*x^17 + 1554506047014*x^16 - 536588718261*x^15 - 6282532031836*x^14 - 4114024022131*x^13 + 11512222917539*x^12 + 18101684222486*x^11 - 1491679119790*x^10 - 21152129527475*x^9 - 14284279230133*x^8 + 2241066960907*x^7 + 5769638127754*x^6 + 1343595105723*x^5 - 598044769496*x^4 - 250097496007*x^3 + 934259347*x^2 + 6767585596*x + 509248489)
 
gp: K = bnfinit(x^33 - x^32 - 224*x^31 + 159*x^30 + 21325*x^29 - 7901*x^28 - 1132606*x^27 - 33149*x^26 + 37089947*x^25 + 18323632*x^24 - 781070288*x^23 - 797096725*x^22 + 10645175728*x^21 + 17186085873*x^20 - 90626504470*x^19 - 213351348451*x^18 + 424789196171*x^17 + 1554506047014*x^16 - 536588718261*x^15 - 6282532031836*x^14 - 4114024022131*x^13 + 11512222917539*x^12 + 18101684222486*x^11 - 1491679119790*x^10 - 21152129527475*x^9 - 14284279230133*x^8 + 2241066960907*x^7 + 5769638127754*x^6 + 1343595105723*x^5 - 598044769496*x^4 - 250097496007*x^3 + 934259347*x^2 + 6767585596*x + 509248489, 1)
 

Normalized defining polynomial

\( x^{33} - x^{32} - 224 x^{31} + 159 x^{30} + 21325 x^{29} - 7901 x^{28} - 1132606 x^{27} - 33149 x^{26} + 37089947 x^{25} + 18323632 x^{24} - 781070288 x^{23} - 797096725 x^{22} + 10645175728 x^{21} + 17186085873 x^{20} - 90626504470 x^{19} - 213351348451 x^{18} + 424789196171 x^{17} + 1554506047014 x^{16} - 536588718261 x^{15} - 6282532031836 x^{14} - 4114024022131 x^{13} + 11512222917539 x^{12} + 18101684222486 x^{11} - 1491679119790 x^{10} - 21152129527475 x^{9} - 14284279230133 x^{8} + 2241066960907 x^{7} + 5769638127754 x^{6} + 1343595105723 x^{5} - 598044769496 x^{4} - 250097496007 x^{3} + 934259347 x^{2} + 6767585596 x + 509248489 \)

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:  \(19888027534781987791463720075885348940229889728076773633013606286908499586225227118081=463^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $384.42$
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:  $463$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(463\)
Dirichlet character group:    $\lbrace$$\chi_{463}(128,·)$, $\chi_{463}(1,·)$, $\chi_{463}(132,·)$, $\chi_{463}(133,·)$, $\chi_{463}(134,·)$, $\chi_{463}(15,·)$, $\chi_{463}(21,·)$, $\chi_{463}(158,·)$, $\chi_{463}(36,·)$, $\chi_{463}(293,·)$, $\chi_{463}(39,·)$, $\chi_{463}(425,·)$, $\chi_{463}(179,·)$, $\chi_{463}(55,·)$, $\chi_{463}(441,·)$, $\chi_{463}(315,·)$, $\chi_{463}(194,·)$, $\chi_{463}(68,·)$, $\chi_{463}(457,·)$, $\chi_{463}(77,·)$, $\chi_{463}(337,·)$, $\chi_{463}(228,·)$, $\chi_{463}(143,·)$, $\chi_{463}(94,·)$, $\chi_{463}(95,·)$, $\chi_{463}(225,·)$, $\chi_{463}(356,·)$, $\chi_{463}(229,·)$, $\chi_{463}(362,·)$, $\chi_{463}(370,·)$, $\chi_{463}(373,·)$, $\chi_{463}(247,·)$, $\chi_{463}(122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{233} a^{31} - \frac{63}{233} a^{30} + \frac{51}{233} a^{29} - \frac{27}{233} a^{28} - \frac{14}{233} a^{27} - \frac{99}{233} a^{26} - \frac{106}{233} a^{25} - \frac{65}{233} a^{24} + \frac{114}{233} a^{23} - \frac{46}{233} a^{22} - \frac{69}{233} a^{21} + \frac{52}{233} a^{20} - \frac{20}{233} a^{19} + \frac{41}{233} a^{18} - \frac{10}{233} a^{17} + \frac{78}{233} a^{16} + \frac{89}{233} a^{15} + \frac{89}{233} a^{14} + \frac{1}{233} a^{13} - \frac{49}{233} a^{12} - \frac{78}{233} a^{11} - \frac{28}{233} a^{10} + \frac{33}{233} a^{9} + \frac{105}{233} a^{8} + \frac{14}{233} a^{7} + \frac{83}{233} a^{6} - \frac{74}{233} a^{5} + \frac{77}{233} a^{4} + \frac{87}{233} a^{3} - \frac{1}{233} a^{2} + \frac{69}{233} a - \frac{2}{233}$, $\frac{1}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{32} + \frac{76118267690298803437986561908555620720202586251932806948643218985102368636916620156534412719830907138436492101807446505781227165958930779510088502889971773427592042416406857167357784360836}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{31} + \frac{26213956441450000495483333666490148023118870386771371918219518336253041845170563270988423894805348584040648027941100559581281388780231914983765400036057844311123406529702976892008197665495367}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{30} + \frac{12223810892546425600421117482835555586691931275328763064218178589011956277383839967276643895282332961614237156330128685840445043899095825521139136303279863813086538144040232958926563801589169}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{29} - \frac{13580916630915253594386284607876456292402591590112486368476063479494331047836686642078086825451060269343257581519149002996657507023709294139846303675848214144253078473515054936356494935817957}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{28} + \frac{49593985830123017856106691091610997232058729505658554348898712616830409103583953513229617607123918140031140173678117405357749941969978367943417484031684662210059461630961202049112527330591518}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{27} - \frac{30477093336222170998113656509117149100904142681324581462844700461630975699069863975421513533807517773628795739689505201694358415469474415881928818380145202321040165293286349068695229354680008}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{26} - \frac{3954098489561051147851871521437130719993990801783675296202531298220982971176215001661697309529619595593263848045316317720699406792530223700743348236315062848700286873872346500369748266011495}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{25} - \frac{32762746860370025842146420169898042141853157831873280011074502569460198250147370470460114675925381013618856735851563789831139410882183829702616741887116411519329744420775903231120857883355378}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{24} - \frac{23333854791460167082857841150840174618788387601960036783945712489691967980570729036737807302259381680758045818920335585465537659020663270308549242132658012459668088452806389694444948857722767}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{23} - \frac{16653047556652216727696979373021304384199867716565157531143023953367447409197632102931165137059474170156345475851192459562839569131782005350801403864646109777503448993285217532671107892405808}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{22} - \frac{10642926355593177663227498641759629825085113896130686230456653978495892506981850969294192271845048723433427263590578819578548995687637277414305460219391442585980084346633453962792172512978935}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{21} + \frac{46497219152061241225758258594286298823284636117225445930822119234481212053899038081926373393185508445274076809519097815877314038483406567892743722414720066229660292149247051449862788817566602}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{20} + \frac{24854710546668351560598126938580866649038785873842432593680974807028947079971042124999781394834739322459465864289520682461906846247633686911353059710516468800651867925479176689559607671622862}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{19} + \frac{48264043546487619160935448991939453031142812675192473956196321905297787186339719460613259199430709109502289619928597402010981655365361980780767713487602602361431662617077349037102138131040027}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{18} - \frac{36363915841690055234124026101141842210843351950946743014443313168972473076320472647358269308786179322179410233299201553171823547538011642174982218956759106299470568377480425962819584662235850}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{17} - \frac{37498659618879745413295784828412227811344156428126575616209052555766308858865494934712317198754679382237306055534238969466899930641430900373166799870330692813782946385726990579814719530928342}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{16} + \frac{48510752960573233119640316643045391639487390456845374459457589564419467808325239409880570692946358522758772711593218746862232826009455298515295220537798970499341925827967508201430717429441376}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{15} + \frac{44547501738630872900910569740798556167623345568145283709523576915536580206356106101555604871979263879565146968650272104638468399351622839556071893163280635904963414005750849220691913658266867}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{14} - \frac{9950002883335992340343135589802102583993040081451131652813549503832634193626824747275778572111774713138430115594788989274535952232615090579081691449978848841977270003313826545215077541811264}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{13} - \frac{32840806150548229745689944278842217186937775390096443574981449101928926908775846834231363252227049048200658936416869874937167992554445702447629902908674832331996346283985966724946289752334407}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{12} - \frac{16552659087290099908775903162440212010390735506256540913495998126524547398251793446489422545116080749681860415056082298378338249038432460891078327559062362612749883682077422793409589769196555}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{11} + \frac{39567520638386798634625198260854618657135919385940519813384998090623653795399324910900975332079181264028398987427416168761325097215418074873305786506340627467303451778027803504684181181768821}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{10} - \frac{17263610983975768262525659683605478145936226370150905740295662955150563543194838705951061359989810428705807900272097521771687125155716228667136632230776063674325585504424373751847251388528106}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{9} - \frac{24604997676121621900150999770296592861770273509509025192227243091304628928253017019964178740961413784346308112273226443888402052508241994046313911376130725343835113438546294277174022730852439}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{8} + \frac{27194646003634718642624429828171833924608542508167897620565983761163892262792314206274368298481776231163269233843743307939034876991802568919083097500302337931865530984147470085357173850290668}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{7} - \frac{30165987248790023289635123296937830827630431635282941403978665353391477644536653890246790076324230904699405617298215270126466399082821081889122280401694651286796683468539839439231422231131374}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{6} - \frac{45665131342152433273806216271857069619731903304249495486333016632452985527528769383150046038773095122934243415333063528631494491087052759867681414982961542286912312367099974850127007317895791}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{5} - \frac{4359369383035764462379193083209538387169988140979162046731135207614470327635248505446049875994148755838312021657768703486952786925151615058008151212914222673583475670969539629202693935020922}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{4} + \frac{25608093385334411725709296111225242653222073740854392923204042291675118465295118213275741569198127886139884137801733939025294073578695013347517461428631306923646735220997437779542019581943963}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{3} - \frac{26640449981613890257177734253437059447578620607852755419877624445033429956712198800141302951003115728706369819699339948018881795259183312209906976003546307088044193937708757916008413710849301}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a^{2} + \frac{6307014182200729166662934763069849917383974316617043923011586650651828191819677809940137016159415186983034017821379851789403564162076859069225762217182631464846416846159760162231566526227238}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519} a - \frac{30477548723053068956044440888123140652287774451708571124955068251703882823848117082086533281249503293159618319248928976132694834001946864944485991945072068265750453759794616150526672807336901}{100303661339122511105382554421261025935149951202330912113024590674851359285521057403430213388981335609319080378871187751493654207295028312260971369611542902198605984643712657538306355444859519}$

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

3.3.214369.1, 11.11.452699390921229872008282849.1

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 $33$ $33$ $33$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{11}$ $33$ $33$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{3}$ $33$ $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{3}$ $33$ $33$

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
463Data not computed