Properties

Label 33.33.228...681.1
Degree $33$
Signature $[33, 0]$
Discriminant $2.283\times 10^{65}$
Root discriminant $95.62$
Ramified primes $13, 23$
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 - 48*x^31 + 488*x^30 + 889*x^29 - 13082*x^28 - 6681*x^27 + 203003*x^26 - 17780*x^25 - 2022483*x^24 + 832653*x^23 + 13572345*x^22 - 8326527*x^21 - 62657637*x^20 + 47704277*x^19 + 199663335*x^18 - 178099102*x^17 - 433344046*x^16 + 447058648*x^15 + 617907878*x^14 - 750473359*x^13 - 533823452*x^12 + 817138243*x^11 + 221279551*x^10 - 544192951*x^9 + 11597098*x^8 + 199867737*x^7 - 43176794*x^6 - 33036600*x^5 + 11713378*x^4 + 1505253*x^3 - 925381*x^2 + 72350*x + 1013)
 
gp: K = bnfinit(x^33 - 8*x^32 - 48*x^31 + 488*x^30 + 889*x^29 - 13082*x^28 - 6681*x^27 + 203003*x^26 - 17780*x^25 - 2022483*x^24 + 832653*x^23 + 13572345*x^22 - 8326527*x^21 - 62657637*x^20 + 47704277*x^19 + 199663335*x^18 - 178099102*x^17 - 433344046*x^16 + 447058648*x^15 + 617907878*x^14 - 750473359*x^13 - 533823452*x^12 + 817138243*x^11 + 221279551*x^10 - 544192951*x^9 + 11597098*x^8 + 199867737*x^7 - 43176794*x^6 - 33036600*x^5 + 11713378*x^4 + 1505253*x^3 - 925381*x^2 + 72350*x + 1013, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1013, 72350, -925381, 1505253, 11713378, -33036600, -43176794, 199867737, 11597098, -544192951, 221279551, 817138243, -533823452, -750473359, 617907878, 447058648, -433344046, -178099102, 199663335, 47704277, -62657637, -8326527, 13572345, 832653, -2022483, -17780, 203003, -6681, -13082, 889, 488, -48, -8, 1]);
 

\( x^{33} - 8 x^{32} - 48 x^{31} + 488 x^{30} + 889 x^{29} - 13082 x^{28} - 6681 x^{27} + 203003 x^{26} - 17780 x^{25} - 2022483 x^{24} + 832653 x^{23} + 13572345 x^{22} - 8326527 x^{21} - 62657637 x^{20} + 47704277 x^{19} + 199663335 x^{18} - 178099102 x^{17} - 433344046 x^{16} + 447058648 x^{15} + 617907878 x^{14} - 750473359 x^{13} - 533823452 x^{12} + 817138243 x^{11} + 221279551 x^{10} - 544192951 x^{9} + 11597098 x^{8} + 199867737 x^{7} - 43176794 x^{6} - 33036600 x^{5} + 11713378 x^{4} + 1505253 x^{3} - 925381 x^{2} + 72350 x + 1013 \)

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:  \(228\!\cdots\!681\)\(\medspace = 13^{22}\cdot 23^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $95.62$
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:  $13, 23$
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:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(256,·)$, $\chi_{299}(1,·)$, $\chi_{299}(3,·)$, $\chi_{299}(261,·)$, $\chi_{299}(9,·)$, $\chi_{299}(139,·)$, $\chi_{299}(269,·)$, $\chi_{299}(16,·)$, $\chi_{299}(146,·)$, $\chi_{299}(131,·)$, $\chi_{299}(282,·)$, $\chi_{299}(27,·)$, $\chi_{299}(29,·)$, $\chi_{299}(133,·)$, $\chi_{299}(289,·)$, $\chi_{299}(35,·)$, $\chi_{299}(165,·)$, $\chi_{299}(170,·)$, $\chi_{299}(48,·)$, $\chi_{299}(55,·)$, $\chi_{299}(185,·)$, $\chi_{299}(196,·)$, $\chi_{299}(81,·)$, $\chi_{299}(211,·)$, $\chi_{299}(87,·)$, $\chi_{299}(94,·)$, $\chi_{299}(144,·)$, $\chi_{299}(100,·)$, $\chi_{299}(209,·)$, $\chi_{299}(105,·)$, $\chi_{299}(243,·)$, $\chi_{299}(118,·)$, $\chi_{299}(248,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{47} a^{26} - \frac{6}{47} a^{24} + \frac{6}{47} a^{22} + \frac{14}{47} a^{21} - \frac{5}{47} a^{20} - \frac{20}{47} a^{19} - \frac{12}{47} a^{18} - \frac{12}{47} a^{17} + \frac{16}{47} a^{16} - \frac{16}{47} a^{15} - \frac{2}{47} a^{14} + \frac{21}{47} a^{13} + \frac{1}{47} a^{12} + \frac{5}{47} a^{11} - \frac{22}{47} a^{10} + \frac{18}{47} a^{9} + \frac{16}{47} a^{8} - \frac{12}{47} a^{7} + \frac{20}{47} a^{6} - \frac{5}{47} a^{5} + \frac{2}{47} a^{4} - \frac{15}{47} a^{3} - \frac{18}{47} a^{2} - \frac{21}{47} a - \frac{1}{47}$, $\frac{1}{47} a^{27} - \frac{6}{47} a^{25} + \frac{6}{47} a^{23} + \frac{14}{47} a^{22} - \frac{5}{47} a^{21} - \frac{20}{47} a^{20} - \frac{12}{47} a^{19} - \frac{12}{47} a^{18} + \frac{16}{47} a^{17} - \frac{16}{47} a^{16} - \frac{2}{47} a^{15} + \frac{21}{47} a^{14} + \frac{1}{47} a^{13} + \frac{5}{47} a^{12} - \frac{22}{47} a^{11} + \frac{18}{47} a^{10} + \frac{16}{47} a^{9} - \frac{12}{47} a^{8} + \frac{20}{47} a^{7} - \frac{5}{47} a^{6} + \frac{2}{47} a^{5} - \frac{15}{47} a^{4} - \frac{18}{47} a^{3} - \frac{21}{47} a^{2} - \frac{1}{47} a$, $\frac{1}{47} a^{28} + \frac{17}{47} a^{24} + \frac{14}{47} a^{23} - \frac{16}{47} a^{22} + \frac{17}{47} a^{21} + \frac{5}{47} a^{20} + \frac{9}{47} a^{19} - \frac{9}{47} a^{18} + \frac{6}{47} a^{17} + \frac{19}{47} a^{15} - \frac{11}{47} a^{14} - \frac{10}{47} a^{13} - \frac{16}{47} a^{12} + \frac{1}{47} a^{11} - \frac{22}{47} a^{10} + \frac{2}{47} a^{9} + \frac{22}{47} a^{8} + \frac{17}{47} a^{7} - \frac{19}{47} a^{6} + \frac{2}{47} a^{5} - \frac{6}{47} a^{4} - \frac{17}{47} a^{3} - \frac{15}{47} a^{2} + \frac{15}{47} a - \frac{6}{47}$, $\frac{1}{47} a^{29} + \frac{17}{47} a^{25} + \frac{14}{47} a^{24} - \frac{16}{47} a^{23} + \frac{17}{47} a^{22} + \frac{5}{47} a^{21} + \frac{9}{47} a^{20} - \frac{9}{47} a^{19} + \frac{6}{47} a^{18} + \frac{19}{47} a^{16} - \frac{11}{47} a^{15} - \frac{10}{47} a^{14} - \frac{16}{47} a^{13} + \frac{1}{47} a^{12} - \frac{22}{47} a^{11} + \frac{2}{47} a^{10} + \frac{22}{47} a^{9} + \frac{17}{47} a^{8} - \frac{19}{47} a^{7} + \frac{2}{47} a^{6} - \frac{6}{47} a^{5} - \frac{17}{47} a^{4} - \frac{15}{47} a^{3} + \frac{15}{47} a^{2} - \frac{6}{47} a$, $\frac{1}{47} a^{30} + \frac{14}{47} a^{25} - \frac{8}{47} a^{24} + \frac{17}{47} a^{23} - \frac{3}{47} a^{22} + \frac{6}{47} a^{21} - \frac{18}{47} a^{20} + \frac{17}{47} a^{19} + \frac{16}{47} a^{18} - \frac{12}{47} a^{17} - \frac{1}{47} a^{16} - \frac{20}{47} a^{15} + \frac{18}{47} a^{14} + \frac{20}{47} a^{13} + \frac{8}{47} a^{12} + \frac{11}{47} a^{11} + \frac{20}{47} a^{10} - \frac{7}{47} a^{9} - \frac{9}{47} a^{8} + \frac{18}{47} a^{7} - \frac{17}{47} a^{6} + \frac{21}{47} a^{5} - \frac{2}{47} a^{4} - \frac{12}{47} a^{3} + \frac{18}{47} a^{2} - \frac{19}{47} a + \frac{17}{47}$, $\frac{1}{10763} a^{31} - \frac{79}{10763} a^{30} - \frac{75}{10763} a^{29} - \frac{73}{10763} a^{28} + \frac{84}{10763} a^{27} + \frac{105}{10763} a^{26} + \frac{2042}{10763} a^{25} + \frac{2559}{10763} a^{24} + \frac{3566}{10763} a^{23} - \frac{3406}{10763} a^{22} - \frac{925}{10763} a^{21} + \frac{2964}{10763} a^{20} + \frac{1409}{10763} a^{19} - \frac{1383}{10763} a^{18} - \frac{226}{10763} a^{17} + \frac{1801}{10763} a^{16} + \frac{3078}{10763} a^{15} - \frac{1933}{10763} a^{14} + \frac{2917}{10763} a^{13} - \frac{3200}{10763} a^{12} - \frac{1699}{10763} a^{11} + \frac{1447}{10763} a^{10} + \frac{3281}{10763} a^{9} + \frac{3748}{10763} a^{8} - \frac{2594}{10763} a^{7} - \frac{3378}{10763} a^{6} + \frac{471}{10763} a^{5} - \frac{3167}{10763} a^{4} + \frac{5343}{10763} a^{3} + \frac{4057}{10763} a^{2} + \frac{194}{10763} a - \frac{1231}{10763}$, $\frac{1}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{32} + \frac{309668353915242236794360378330110192040752637963616955734563065477812718334631521418891}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{31} - \frac{115212464434768328088478069152133649616309605943119904966608146954543626850215699813023899}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{30} - \frac{34130240035098445456543526916263710903591002929257377137492426886769178425530148199305626}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{29} - \frac{35748761955822655764281144332301757073545620651842397690213255348390426943473149388642155}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{28} + \frac{748083867904177516027151756878761017790605410742401707257654710360592113716384925016235441}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{27} - \frac{744328330096284449191200889513379184944611522542028431679822413356263427585676806668059744}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{26} - \frac{12019997724326472206594607582744251493592226307280147043389137098739494865527050295345160279}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{25} - \frac{38539612898201091358924924830331221203795303778652329577288310991985178087451943858069050220}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{24} - \frac{31757004355357184745570327442834380982542205181848651471754807283354118893164560261397069348}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{23} - \frac{21314394884148334632608564299789542887049099214001546468906115970650878203966100071199925728}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{22} - \frac{3438255665699808732288925193230570789838370221025827324641000588157564245602252638305193149}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{21} + \frac{24011262102610940123322754289509624846993622068835624628489704918451585451737412748900390219}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{20} - \frac{20612938630815007181082443580646129160443179402607360320976558305310990257323018152374845823}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{19} - \frac{878888608813876974753295211257890103199653856823729832012296628725498920390857494670789895}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{18} - \frac{21341368337367318211445961089265020172790840140832098200261837096346977606107749777420305575}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{17} - \frac{2934130198933555874780837647571888206354827463553837671899372342404590587953237775708415010}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{16} - \frac{18819946108535110733883416212648498517472801132540415511161222683772559390117296514993095416}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{15} - \frac{28696596285005559501998558617476862584716906719486737651987910448926603211090481118176792854}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{14} - \frac{4095854954755836193587874799076573712192586347343750679768930551583220050231838692167562003}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{13} + \frac{18820684782636855028555485273447275341484080667122710373777857047444109535139643965026676302}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{12} + \frac{8049340407737548989701262862048967155606137293705079421414871431924452156213310683301295589}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{11} + \frac{31407404450496482210195094216666800476524528083031646282567273996154066713737607513722436504}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{10} - \frac{3892807527573884568504505808505744771362371909740739289479705664678373262196784979706138295}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{9} - \frac{14498867149716476808676883371673283904719717689992685509218049891655328734736300726361922693}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{8} + \frac{3773843741381163756372246016842249711740992990538318248272229554587084697714690200451731631}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{7} - \frac{7468496603911177223429283201774687547348523288049084296487220689871331808318079532379868891}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{6} - \frac{19175152643443531556257630412758816463949456963841936562426070300663896568901364220835770328}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{5} - \frac{31902327656792613599650741436857501562936412321713665995525979391487259616738075190824045381}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{4} - \frac{28117347616598141940941428661902991876822289869991995685631591465476253875835098829157319059}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{3} - \frac{9583964016509686798143139956744285154932776219865352122761619659397543141886340718066249024}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a^{2} + \frac{16252440302632526662573211804790080530318004634193411547836191806606851683755045859253408140}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389} a + \frac{2172093578364611562245970574379928515130227445649419247929046691999855440544641907353086103}{78260884649325858062943881052738015364258836423474450485179206591768772317025153368127664389}$

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:  \( 22765796489209540000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 22765796489209540000000 \cdot 1}{2\sqrt{228343593450302703244344174036290254973199912242460469577320120681}}\approx 0.204620101068219$ (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

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

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 $33$ $33$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{3}$ $33$ $33$ R $33$ $33$ R $33$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{3}$ $33$ $33$ $33$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{33}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
13Data not computed
23Data not computed