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Properties

Label 33.33.700...569.1
Degree $33$
Signature $[33, 0]$
Discriminant $7.001\times 10^{61}$
Root discriminant $74.83$
Ramified primes $3, 23$
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 - 63*x^31 - 4*x^30 + 1710*x^29 + 204*x^28 - 26254*x^27 - 4392*x^26 + 251754*x^25 + 52248*x^24 - 1572192*x^23 - 377544*x^22 + 6479940*x^21 + 1717920*x^20 - 17548167*x^19 - 4953616*x^18 + 30725595*x^17 + 8949468*x^16 - 34072421*x^15 - 9933084*x^14 + 23552382*x^13 + 6584132*x^12 - 10086444*x^11 - 2580438*x^10 + 2636960*x^9 + 587106*x^8 - 404862*x^7 - 74061*x^6 + 33768*x^5 + 4755*x^4 - 1326*x^3 - 135*x^2 + 18*x + 1)
 
gp: K = bnfinit(x^33 - 63*x^31 - 4*x^30 + 1710*x^29 + 204*x^28 - 26254*x^27 - 4392*x^26 + 251754*x^25 + 52248*x^24 - 1572192*x^23 - 377544*x^22 + 6479940*x^21 + 1717920*x^20 - 17548167*x^19 - 4953616*x^18 + 30725595*x^17 + 8949468*x^16 - 34072421*x^15 - 9933084*x^14 + 23552382*x^13 + 6584132*x^12 - 10086444*x^11 - 2580438*x^10 + 2636960*x^9 + 587106*x^8 - 404862*x^7 - 74061*x^6 + 33768*x^5 + 4755*x^4 - 1326*x^3 - 135*x^2 + 18*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 18, -135, -1326, 4755, 33768, -74061, -404862, 587106, 2636960, -2580438, -10086444, 6584132, 23552382, -9933084, -34072421, 8949468, 30725595, -4953616, -17548167, 1717920, 6479940, -377544, -1572192, 52248, 251754, -4392, -26254, 204, 1710, -4, -63, 0, 1]);
 

\(x^{33} - 63 x^{31} - 4 x^{30} + 1710 x^{29} + 204 x^{28} - 26254 x^{27} - 4392 x^{26} + 251754 x^{25} + 52248 x^{24} - 1572192 x^{23} - 377544 x^{22} + 6479940 x^{21} + 1717920 x^{20} - 17548167 x^{19} - 4953616 x^{18} + 30725595 x^{17} + 8949468 x^{16} - 34072421 x^{15} - 9933084 x^{14} + 23552382 x^{13} + 6584132 x^{12} - 10086444 x^{11} - 2580438 x^{10} + 2636960 x^{9} + 587106 x^{8} - 404862 x^{7} - 74061 x^{6} + 33768 x^{5} + 4755 x^{4} - 1326 x^{3} - 135 x^{2} + 18 x + 1\)  Toggle raw display

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:  \(700\!\cdots\!569\)\(\medspace = 3^{44}\cdot 23^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $74.83$
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:  $3, 23$
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:  \(207=3^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{207}(1,·)$, $\chi_{207}(4,·)$, $\chi_{207}(133,·)$, $\chi_{207}(139,·)$, $\chi_{207}(13,·)$, $\chi_{207}(142,·)$, $\chi_{207}(16,·)$, $\chi_{207}(151,·)$, $\chi_{207}(25,·)$, $\chi_{207}(154,·)$, $\chi_{207}(31,·)$, $\chi_{207}(163,·)$, $\chi_{207}(169,·)$, $\chi_{207}(49,·)$, $\chi_{207}(52,·)$, $\chi_{207}(55,·)$, $\chi_{207}(58,·)$, $\chi_{207}(187,·)$, $\chi_{207}(190,·)$, $\chi_{207}(64,·)$, $\chi_{207}(193,·)$, $\chi_{207}(196,·)$, $\chi_{207}(70,·)$, $\chi_{207}(73,·)$, $\chi_{207}(202,·)$, $\chi_{207}(82,·)$, $\chi_{207}(85,·)$, $\chi_{207}(94,·)$, $\chi_{207}(100,·)$, $\chi_{207}(118,·)$, $\chi_{207}(121,·)$, $\chi_{207}(124,·)$, $\chi_{207}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{32} - \frac{6780210491147747067436949760715781193095500383388176198273680318143503928}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{31} - \frac{52916933625934858050588721672295172552169596040125697127180072708062703318}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{30} + \frac{60050226259424403243592646026543076032545504596820005197809730313037085306}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{29} - \frac{95876552504366972146914723715515669931141301423806612187518667474359573495}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{28} + \frac{76962431332006724213700320487061805297993453462307626514100759513790410738}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{27} - \frac{35992845286188713128964740707382616499319889244728051344708900887863065634}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{26} + \frac{35581581837532597901199770933868354005806065964999388309201559505794893856}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{25} - \frac{156110845072161358424186125280549006571432181546998812273739716654195172114}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{24} + \frac{115408647410853784938555677365901965027169513080011785872989252036333597050}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{23} - \frac{90765092222979467393748246867438253957702053028729377500318104277929617721}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{22} - \frac{122023400663344809188316256930684069328451574931357103288087073002814389538}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{21} + \frac{91661959859164719457313852811970721785209339330579945349968427167764260769}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{20} - \frac{155349488590276948701554084165233171825968717377486634860267954047686199412}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{19} + \frac{129224134324656249694260904040506691467028629190254723242928101912413680809}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{18} - \frac{40254266637168771371080621541871787019512802644326211342831897761334296780}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{17} + \frac{111823285614662482842552520658490762085747410048504906256726869820791673630}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{16} + \frac{52573206417642030529586648247339992903063658072565621428004011790479091693}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{15} + \frac{10993301109098904561576379520330782153518698589525138989552500143560916679}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{14} - \frac{56298181650837752836037974538126769705907608981485399719021352408975377670}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{13} + \frac{22968504237290326712283219793731370587554667097411053268057599898943376687}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{12} - \frac{124104906277879312298002525425481693411270908436292141832404228512264250566}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{11} + \frac{8650050351697447591446241637940357910000517761115724805111973017984417847}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{10} + \frac{18669127128838002992702929242880699761746622276028259956385396328076547399}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{9} - \frac{150410320251912156668706186536469141285634926459368061999656872984161062972}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{8} - \frac{35202404756386676743853738550361705795608788113501042465633159863792963790}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{7} + \frac{40052502288447171571609893770087385606780620865518454130536749199762490683}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{6} - \frac{69962891754784360558019275668339944896548143842069483295900137000991477255}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{5} + \frac{71484330482486729245125354832758377175437385338908095037028208526836980612}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{4} + \frac{61369290070442566289733903444717198522333411746733431056368553101253441095}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{3} - \frac{135552926134166117545060617217780355964054016882098786574414111648118963779}{315060576932040188254188918553316867927726003844609577227312423072109353457} a^{2} + \frac{109861811613379077152658558152025661158479437539510888847696918327709137328}{315060576932040188254188918553316867927726003844609577227312423072109353457} a - \frac{102682777897231309888668977003867737961571190872975548750527681146600343145}{315060576932040188254188918553316867927726003844609577227312423072109353457}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 176588774716401650000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{33}\cdot(2\pi)^{0}\cdot 176588774716401650000 \cdot 1}{2\sqrt{70011645999218458416472683122408534303895571350166174758601569}}\approx 0.0906437387287649$ (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

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{23})^+\)

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 $33$ R $33$ $33$ $33$ $33$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{3}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{3}$ R $33$ $33$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{3}$ $33$ $33$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{11}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{3}$ $33$

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
3Data not computed
23Data not computed