Properties

Label 33.33.964...489.1
Degree $33$
Signature $[33, 0]$
Discriminant $9.647\times 10^{68}$
Root discriminant $123.15$
Ramified primes $19, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{33}$ (as 33T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 8*x^32 - 70*x^31 + 734*x^30 + 1439*x^29 - 27994*x^28 + 11289*x^27 + 571079*x^26 - 1005928*x^25 - 6614661*x^24 + 19440415*x^23 + 40980613*x^22 - 194665653*x^21 - 82258867*x^20 + 1125872365*x^19 - 536982871*x^18 - 3781576926*x^17 + 4155619240*x^16 + 6998281772*x^15 - 12201861484*x^14 - 5994133153*x^13 + 18812121252*x^12 + 75160599*x^11 - 16337602363*x^10 + 3902636199*x^9 + 8090155706*x^8 - 3031301627*x^7 - 2187750082*x^6 + 1012187926*x^5 + 277584302*x^4 - 153651191*x^3 - 8485173*x^2 + 8057502*x - 470213)
 
gp: K = bnfinit(x^33 - 8*x^32 - 70*x^31 + 734*x^30 + 1439*x^29 - 27994*x^28 + 11289*x^27 + 571079*x^26 - 1005928*x^25 - 6614661*x^24 + 19440415*x^23 + 40980613*x^22 - 194665653*x^21 - 82258867*x^20 + 1125872365*x^19 - 536982871*x^18 - 3781576926*x^17 + 4155619240*x^16 + 6998281772*x^15 - 12201861484*x^14 - 5994133153*x^13 + 18812121252*x^12 + 75160599*x^11 - 16337602363*x^10 + 3902636199*x^9 + 8090155706*x^8 - 3031301627*x^7 - 2187750082*x^6 + 1012187926*x^5 + 277584302*x^4 - 153651191*x^3 - 8485173*x^2 + 8057502*x - 470213, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-470213, 8057502, -8485173, -153651191, 277584302, 1012187926, -2187750082, -3031301627, 8090155706, 3902636199, -16337602363, 75160599, 18812121252, -5994133153, -12201861484, 6998281772, 4155619240, -3781576926, -536982871, 1125872365, -82258867, -194665653, 40980613, 19440415, -6614661, -1005928, 571079, 11289, -27994, 1439, 734, -70, -8, 1]);
 

\( x^{33} - 8 x^{32} - 70 x^{31} + 734 x^{30} + 1439 x^{29} - 27994 x^{28} + 11289 x^{27} + 571079 x^{26} - 1005928 x^{25} - 6614661 x^{24} + 19440415 x^{23} + 40980613 x^{22} - 194665653 x^{21} - 82258867 x^{20} + 1125872365 x^{19} - 536982871 x^{18} - 3781576926 x^{17} + 4155619240 x^{16} + 6998281772 x^{15} - 12201861484 x^{14} - 5994133153 x^{13} + 18812121252 x^{12} + 75160599 x^{11} - 16337602363 x^{10} + 3902636199 x^{9} + 8090155706 x^{8} - 3031301627 x^{7} - 2187750082 x^{6} + 1012187926 x^{5} + 277584302 x^{4} - 153651191 x^{3} - 8485173 x^{2} + 8057502 x - 470213 \)

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

Invariants

Degree:  $33$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[33, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(964\!\cdots\!489\)\(\medspace = 19^{22}\cdot 23^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $123.15$
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:  $19, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $33$
This field is Galois and abelian over $\Q$.
Conductor:  \(437=19\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(140,·)$, $\chi_{437}(144,·)$, $\chi_{437}(277,·)$, $\chi_{437}(26,·)$, $\chi_{437}(163,·)$, $\chi_{437}(292,·)$, $\chi_{437}(39,·)$, $\chi_{437}(305,·)$, $\chi_{437}(311,·)$, $\chi_{437}(58,·)$, $\chi_{437}(315,·)$, $\chi_{437}(64,·)$, $\chi_{437}(324,·)$, $\chi_{437}(197,·)$, $\chi_{437}(330,·)$, $\chi_{437}(77,·)$, $\chi_{437}(334,·)$, $\chi_{437}(210,·)$, $\chi_{437}(87,·)$, $\chi_{437}(216,·)$, $\chi_{437}(49,·)$, $\chi_{437}(220,·)$, $\chi_{437}(349,·)$, $\chi_{437}(96,·)$, $\chi_{437}(353,·)$, $\chi_{437}(239,·)$, $\chi_{437}(400,·)$, $\chi_{437}(372,·)$, $\chi_{437}(248,·)$, $\chi_{437}(121,·)$, $\chi_{437}(381,·)$, $\chi_{437}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{277} a^{29} + \frac{83}{277} a^{28} + \frac{17}{277} a^{27} - \frac{47}{277} a^{26} + \frac{50}{277} a^{25} + \frac{115}{277} a^{24} + \frac{120}{277} a^{23} + \frac{22}{277} a^{22} - \frac{55}{277} a^{21} - \frac{95}{277} a^{20} + \frac{112}{277} a^{19} + \frac{63}{277} a^{18} - \frac{20}{277} a^{17} + \frac{22}{277} a^{16} + \frac{106}{277} a^{15} - \frac{24}{277} a^{14} + \frac{76}{277} a^{13} - \frac{121}{277} a^{12} - \frac{14}{277} a^{11} - \frac{129}{277} a^{10} + \frac{60}{277} a^{9} - \frac{82}{277} a^{8} + \frac{127}{277} a^{7} - \frac{88}{277} a^{6} - \frac{39}{277} a^{5} + \frac{122}{277} a^{4} - \frac{79}{277} a^{3} - \frac{33}{277} a^{2} - \frac{138}{277} a - \frac{11}{277}$, $\frac{1}{277} a^{30} + \frac{53}{277} a^{28} - \frac{73}{277} a^{27} + \frac{73}{277} a^{26} + \frac{120}{277} a^{25} - \frac{7}{277} a^{24} + \frac{34}{277} a^{23} + \frac{58}{277} a^{22} + \frac{38}{277} a^{21} - \frac{36}{277} a^{20} - \frac{92}{277} a^{19} + \frac{14}{277} a^{18} + \frac{20}{277} a^{17} - \frac{58}{277} a^{16} + \frac{42}{277} a^{15} + \frac{129}{277} a^{14} - \frac{58}{277} a^{13} + \frac{57}{277} a^{12} - \frac{75}{277} a^{11} - \frac{36}{277} a^{10} - \frac{76}{277} a^{9} + \frac{8}{277} a^{8} - \frac{103}{277} a^{7} + \frac{63}{277} a^{6} + \frac{35}{277} a^{5} + \frac{44}{277} a^{4} - \frac{124}{277} a^{3} + \frac{108}{277} a^{2} + \frac{86}{277} a + \frac{82}{277}$, $\frac{1}{277} a^{31} - \frac{40}{277} a^{28} + \frac{3}{277} a^{27} + \frac{118}{277} a^{26} + \frac{113}{277} a^{25} + \frac{33}{277} a^{24} + \frac{69}{277} a^{23} - \frac{20}{277} a^{22} + \frac{109}{277} a^{21} - \frac{43}{277} a^{20} - \frac{105}{277} a^{19} + \frac{5}{277} a^{18} - \frac{106}{277} a^{17} - \frac{16}{277} a^{16} + \frac{51}{277} a^{15} + \frac{106}{277} a^{14} - \frac{93}{277} a^{13} - \frac{33}{277} a^{12} - \frac{125}{277} a^{11} + \frac{113}{277} a^{10} - \frac{125}{277} a^{9} + \frac{88}{277} a^{8} - \frac{20}{277} a^{7} - \frac{10}{277} a^{6} - \frac{105}{277} a^{5} + \frac{58}{277} a^{4} - \frac{137}{277} a^{3} - \frac{104}{277} a^{2} - \frac{83}{277} a + \frac{29}{277}$, $\frac{1}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{32} - \frac{153222654234554209105602667300497964440569267885814315940387571070693122986625524065740482654982275545447540324642792788}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{31} + \frac{375395039513225772279154382339668618913524418197210851585624227855891415364262176780553154241337808889003517557012315357}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{30} + \frac{211969398119538889642998574557432628542820988025007265694911650042923086562325292409358438784952896434777623270476686864}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{29} - \frac{72075934459092328720339186628363694681666466773665432547627559427361499240689369945766258329087884983433780982212960481581}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{28} + \frac{77179077848678198723893842911730083758882586707120796222133268048498667067822767187108059211019704807927186181012902442941}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{27} - \frac{56743053461517266703898621136157121080683140231073259341151404972182690380436683452255087410095935679668239089016585618499}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{26} + \frac{103484805391608627633871318666985980911104425339332754836331383384312400861217514860324597154985103172863848038915645733841}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{25} - \frac{119667777375746060730127513954361249241611244564666004135761065276407972854402273746745075205071475830889092124790465760748}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{24} - \frac{2688779606485824349593153048446020247471351389998542397479972962799453934272320094797154591355573515888309455163517352085}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{23} - \frac{15493162757095906184672150367592740681264721495421424153153693005686348332648758772047199866427793216458605205013026780992}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{22} + \frac{24555401519235302141567129566633657687364591887432360305617957955143031618293076283049342924173115081801045931270477657155}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{21} + \frac{96801572883419214778829752750540164504918568182067358232208227538149491195864898452714515326906839311837944203730001378191}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{20} - \frac{105892500302150492977366194169692117295976460644956557719057350525280704572034336891823500780308005034995376800530967142622}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{19} + \frac{20778455200380947529385850048434201654548295849602023972946432054805714330221419498634668018752521510110692335481344629398}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{18} - \frac{100906976730493768983656364383788174390908355852902064291505600180601745224772664789735036435395780781488945835986191373950}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{17} + \frac{129109697373270966339236778180688434544839343559819325815348935486384074957352434703429409554454309723438204291359522903674}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{16} + \frac{39167804087224855168388434219835901681237640800289903569635774370486187613557072984792863198761206544005957513231809645673}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{15} - \frac{130114953514404325194128130579042649431175025340124867988340014778751537619586395251373888637952729143044917307432585728354}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{14} + \frac{42498143065508241400643149009775384390554021060133944296757351247457309768914117812258498883028250299724429969021498618459}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{13} + \frac{74155451654996645536377639353955807796770953634106107117420030715855916900200771283111856847835502434266636546219750949334}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{12} + \frac{126073037101832852527824118550895382331633796178668442167129972752463194201391051955468335151256051231600525113014609433220}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{11} - \frac{78056770297877445248980926212277034006634835615275061509526280946545484580533893688335294231831401300467057093682355886885}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{10} - \frac{7134786161061926339661780658691355288811259768376034051048502289046632969419219698892709220520777195361235277889525580597}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{9} - \frac{95243888858886694974111035054532756513982997212918883953867912726444828832634477324019003494681012653277826135605019657142}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{8} - \frac{131403819603361881216464706281458055685483904037028400924544494804228545818284965787247298743225734680224477965216007879418}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{7} + \frac{57471456680816117286943200060503543180215712347658642260482259099420812686150998634869655325416443899712361752526184473765}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{6} + \frac{93616918350344065895140864412952747103345843247284122293659749840878926220450356809350004730222858737994895743956460461539}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{5} - \frac{124658261266607110427170623450447469536012464100488837801217361786702333826658845184569608375369273540982833966608830898683}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{4} - \frac{24060385800071189102895346015981211693026673337999226814831667740764846979888942116237518112790101973168046115956350039651}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{3} - \frac{111004881096609644839369442310911969537250154216404894084032842143176070519023412413004064477877252902605858890148210764258}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a^{2} + \frac{137256327289717028819592741912111789983093166200329558769618186039842343282357176190943742992487918591043756672509895909293}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993} a + \frac{138357970789074308708698072859864353914115186847785537485416648074901256299224391663423041989954283469433421006432729284208}{282078349870259062300720349979738234839445961011254957816118061949638701997148262699431408093526674851867962491079545689993}$

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:  $32$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 859641744829651100000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 859641744829651100000000 \cdot 1}{2\sqrt{964748920938762847635574420140466077720720339834593232479190796888489}}\approx 0.118869437471761$ (assuming GRH)

Galois group

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

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

Intermediate fields

3.3.361.1, \(\Q(\zeta_{23})^+\)

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 $33$ $33$ $33$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{3}$ $33$ $33$ R R $33$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{3}$ $33$ $33$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{11}$ $33$ $33$

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
19Data not computed
23Data not computed