Properties

Label 33.33.365...601.1
Degree $33$
Signature $[33, 0]$
Discriminant $3.658\times 10^{73}$
Root discriminant $169.51$
Ramified prime $199$
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 - x^32 - 96*x^31 + 217*x^30 + 3795*x^29 - 13403*x^28 - 74197*x^27 + 394231*x^26 + 599821*x^25 - 6350170*x^24 + 2832807*x^23 + 56803105*x^22 - 104532088*x^21 - 244229488*x^20 + 932758015*x^19 + 5618002*x^18 - 3890173018*x^17 + 4529747891*x^16 + 6495127532*x^15 - 18110944809*x^14 + 4574986912*x^13 + 26694143816*x^12 - 29645825157*x^11 - 6037403432*x^10 + 30417132332*x^9 - 15468969217*x^8 - 6737165737*x^7 + 8515927088*x^6 - 1017730658*x^5 - 1409177433*x^4 + 395068072*x^3 + 75602260*x^2 - 21210003*x - 2947097)
 
gp: K = bnfinit(x^33 - x^32 - 96*x^31 + 217*x^30 + 3795*x^29 - 13403*x^28 - 74197*x^27 + 394231*x^26 + 599821*x^25 - 6350170*x^24 + 2832807*x^23 + 56803105*x^22 - 104532088*x^21 - 244229488*x^20 + 932758015*x^19 + 5618002*x^18 - 3890173018*x^17 + 4529747891*x^16 + 6495127532*x^15 - 18110944809*x^14 + 4574986912*x^13 + 26694143816*x^12 - 29645825157*x^11 - 6037403432*x^10 + 30417132332*x^9 - 15468969217*x^8 - 6737165737*x^7 + 8515927088*x^6 - 1017730658*x^5 - 1409177433*x^4 + 395068072*x^3 + 75602260*x^2 - 21210003*x - 2947097, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2947097, -21210003, 75602260, 395068072, -1409177433, -1017730658, 8515927088, -6737165737, -15468969217, 30417132332, -6037403432, -29645825157, 26694143816, 4574986912, -18110944809, 6495127532, 4529747891, -3890173018, 5618002, 932758015, -244229488, -104532088, 56803105, 2832807, -6350170, 599821, 394231, -74197, -13403, 3795, 217, -96, -1, 1]);
 

\( x^{33} - x^{32} - 96 x^{31} + 217 x^{30} + 3795 x^{29} - 13403 x^{28} - 74197 x^{27} + 394231 x^{26} + 599821 x^{25} - 6350170 x^{24} + 2832807 x^{23} + 56803105 x^{22} - 104532088 x^{21} - 244229488 x^{20} + 932758015 x^{19} + 5618002 x^{18} - 3890173018 x^{17} + 4529747891 x^{16} + 6495127532 x^{15} - 18110944809 x^{14} + 4574986912 x^{13} + 26694143816 x^{12} - 29645825157 x^{11} - 6037403432 x^{10} + 30417132332 x^{9} - 15468969217 x^{8} - 6737165737 x^{7} + 8515927088 x^{6} - 1017730658 x^{5} - 1409177433 x^{4} + 395068072 x^{3} + 75602260 x^{2} - 21210003 x - 2947097 \)

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:  \(365\!\cdots\!601\)\(\medspace = 199^{32}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $169.51$
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:  $199$
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:  \(199\)
Dirichlet character group:    $\lbrace$$\chi_{199}(1,·)$, $\chi_{199}(132,·)$, $\chi_{199}(5,·)$, $\chi_{199}(8,·)$, $\chi_{199}(139,·)$, $\chi_{199}(140,·)$, $\chi_{199}(144,·)$, $\chi_{199}(18,·)$, $\chi_{199}(25,·)$, $\chi_{199}(28,·)$, $\chi_{199}(157,·)$, $\chi_{199}(40,·)$, $\chi_{199}(172,·)$, $\chi_{199}(52,·)$, $\chi_{199}(182,·)$, $\chi_{199}(187,·)$, $\chi_{199}(188,·)$, $\chi_{199}(61,·)$, $\chi_{199}(62,·)$, $\chi_{199}(63,·)$, $\chi_{199}(64,·)$, $\chi_{199}(90,·)$, $\chi_{199}(92,·)$, $\chi_{199}(98,·)$, $\chi_{199}(103,·)$, $\chi_{199}(106,·)$, $\chi_{199}(111,·)$, $\chi_{199}(114,·)$, $\chi_{199}(116,·)$, $\chi_{199}(117,·)$, $\chi_{199}(121,·)$, $\chi_{199}(123,·)$, $\chi_{199}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{107} a^{28} - \frac{11}{107} a^{27} - \frac{43}{107} a^{26} - \frac{11}{107} a^{25} - \frac{25}{107} a^{24} + \frac{30}{107} a^{23} - \frac{12}{107} a^{22} - \frac{50}{107} a^{21} - \frac{37}{107} a^{19} + \frac{31}{107} a^{18} + \frac{33}{107} a^{16} + \frac{39}{107} a^{15} + \frac{9}{107} a^{14} + \frac{28}{107} a^{13} - \frac{34}{107} a^{12} - \frac{4}{107} a^{11} + \frac{13}{107} a^{10} - \frac{51}{107} a^{9} + \frac{4}{107} a^{8} + \frac{27}{107} a^{7} + \frac{45}{107} a^{6} - \frac{45}{107} a^{5} - \frac{14}{107} a^{4} + \frac{48}{107} a^{3} - \frac{42}{107} a^{2} + \frac{37}{107} a + \frac{34}{107}$, $\frac{1}{107} a^{29} + \frac{50}{107} a^{27} + \frac{51}{107} a^{26} - \frac{39}{107} a^{25} - \frac{31}{107} a^{24} - \frac{3}{107} a^{23} + \frac{32}{107} a^{22} - \frac{15}{107} a^{21} - \frac{37}{107} a^{20} + \frac{52}{107} a^{19} + \frac{20}{107} a^{18} + \frac{33}{107} a^{17} - \frac{26}{107} a^{16} + \frac{10}{107} a^{15} + \frac{20}{107} a^{14} - \frac{47}{107} a^{13} + \frac{50}{107} a^{12} - \frac{31}{107} a^{11} - \frac{15}{107} a^{10} - \frac{22}{107} a^{9} - \frac{36}{107} a^{8} + \frac{21}{107} a^{7} + \frac{22}{107} a^{6} + \frac{26}{107} a^{5} + \frac{1}{107} a^{4} - \frac{49}{107} a^{3} + \frac{3}{107} a^{2} + \frac{13}{107} a + \frac{53}{107}$, $\frac{1}{107} a^{30} - \frac{41}{107} a^{27} - \frac{29}{107} a^{26} - \frac{16}{107} a^{25} - \frac{37}{107} a^{24} + \frac{30}{107} a^{23} + \frac{50}{107} a^{22} + \frac{2}{107} a^{21} + \frac{52}{107} a^{20} + \frac{51}{107} a^{19} - \frac{19}{107} a^{18} - \frac{26}{107} a^{17} - \frac{35}{107} a^{16} - \frac{4}{107} a^{15} + \frac{38}{107} a^{14} + \frac{41}{107} a^{13} - \frac{43}{107} a^{12} - \frac{29}{107} a^{11} - \frac{30}{107} a^{10} + \frac{53}{107} a^{9} + \frac{35}{107} a^{8} - \frac{44}{107} a^{7} + \frac{23}{107} a^{6} + \frac{4}{107} a^{5} + \frac{9}{107} a^{4} - \frac{43}{107} a^{3} - \frac{27}{107} a^{2} + \frac{22}{107} a + \frac{12}{107}$, $\frac{1}{85279} a^{31} + \frac{200}{85279} a^{30} - \frac{310}{85279} a^{29} - \frac{350}{85279} a^{28} - \frac{1}{797} a^{27} + \frac{38955}{85279} a^{26} + \frac{35792}{85279} a^{25} - \frac{19460}{85279} a^{24} + \frac{21143}{85279} a^{23} - \frac{7552}{85279} a^{22} + \frac{18198}{85279} a^{21} - \frac{4936}{85279} a^{20} + \frac{40911}{85279} a^{19} - \frac{16288}{85279} a^{18} + \frac{34825}{85279} a^{17} - \frac{26368}{85279} a^{16} - \frac{23938}{85279} a^{15} + \frac{265}{85279} a^{14} - \frac{11498}{85279} a^{13} + \frac{1678}{85279} a^{12} + \frac{1913}{85279} a^{11} + \frac{30745}{85279} a^{10} - \frac{4236}{85279} a^{9} - \frac{18537}{85279} a^{8} + \frac{22380}{85279} a^{7} + \frac{33206}{85279} a^{6} - \frac{9075}{85279} a^{5} - \frac{41842}{85279} a^{4} - \frac{39299}{85279} a^{3} - \frac{40303}{85279} a^{2} + \frac{25329}{85279} a - \frac{5597}{85279}$, $\frac{1}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{32} - \frac{438897633968529796139850794391781373725155282827429592545742453754653758}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{31} - \frac{110402546553769652630517426987523221265406509511904028477833912681295184433}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{30} + \frac{320936223084897199657985786918830123142659206716931515171038855868669434494}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{29} - \frac{374908578497306880128163102914250800633485143768155075842994856790433045172}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{28} - \frac{30743493205113985994167010070949657887864805163144174475109645892975829004212}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{27} + \frac{25356037735169413811459530807110345442706848648638267190150784388291670032583}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{26} + \frac{23470517440383994415605927999524298738720707322662561013555149009748330766351}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{25} + \frac{12766944573729428846774254960177509984844734718343700669965983248748774722568}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{24} + \frac{14975292975365784641082475344702493089707650002408493449000817364947429348061}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{23} + \frac{14924745911544692110799056676101361776099767828562346927514756440921114882622}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{22} + \frac{8216190614053508038159652735556314707653996180651639837177153203563934013960}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{21} - \frac{22970992314794669892396731847871379849059884104718625186556050762178277931014}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{20} + \frac{49955306356803724194778993886028674938590021815018447937533143805121191780330}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{19} - \frac{5180635917730927010684352507506494639542192931970873600630632495259793960619}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{18} + \frac{48517887467994969535184665359906021871428962537739738059228752747612860407782}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{17} + \frac{32812338187970572656159713612624074377771435314801166903073838250896951444758}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{16} - \frac{2616901621921753030397829196403387332080446536734032200863868763926503885258}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{15} - \frac{21532901663148743214811689407860864906719378229739621308361510383169945793789}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{14} + \frac{12054516188201319321522699841002176634071112345030785941738168900616811360095}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{13} + \frac{31264541748829993117378410044826614269958018386165198311503746225434683453463}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{12} - \frac{47169536983914598731633922943580927997368856169749918614618794685152540242283}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{11} - \frac{43168804543004074938266684638479210697079315118805737814490130352217998791075}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{10} + \frac{18614299057042667178075574727084464563060072240468685207631630064232530572928}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{9} + \frac{33903009564876630326881250934235490396854251770235014190930531384625129631744}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{8} + \frac{37454674054966490087073218660817366298874310763514168341148253768557006921844}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{7} - \frac{2141071672468326349367177597543086434747190323340729965850954481465294789876}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{6} + \frac{17367023243537628972790939335411424812139394598424997369813227291411361484558}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{5} - \frac{47586298302744340489101125823909598017395052331123812728399255864118785771119}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{4} - \frac{49768997760892950697834957884665557209186192196605180706922785696559925948084}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{3} - \frac{25129704869123703134655166045989838573994498991929010347482234215437846523494}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a^{2} + \frac{38404255889277382429064943474916715592440230230959646726738884900693020155634}{102671498988363166441560189308232262476210556522544195280794021105929195422227} a + \frac{36884515875702917506075331585791530761741418441044738793672822864730304131594}{102671498988363166441560189308232262476210556522544195280794021105929195422227}$

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:  \( 194708110268326050000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 194708110268326050000000000 \cdot 1}{2\sqrt{36584611296554742180833097810429342639777502523008874222975105176339833601}}\approx 0.138259404012701$ (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

3.3.39601.1, 11.11.97393677359695041798001.1

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 $33$ $33$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{3}$ $33$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{11}$ $33$ $33$ $33$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{11}$ $33$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{11}$ $33$ $33$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{3}$

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
199Data not computed