Properties

Label 30.30.150...497.1
Degree $30$
Signature $[30, 0]$
Discriminant $1.501\times 10^{52}$
Root discriminant $54.85$
Ramified primes $3, 7, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 - 52*x^28 + 51*x^27 + 1142*x^26 - 1092*x^25 - 13898*x^24 + 12856*x^23 + 103478*x^22 - 91664*x^21 - 491678*x^20 + 412467*x^19 + 1512708*x^18 - 1191563*x^17 - 3007832*x^16 + 2225799*x^15 + 3819257*x^14 - 2692049*x^13 - 3042474*x^12 + 2079355*x^11 + 1471733*x^10 - 992749*x^9 - 403229*x^8 + 274849*x^7 + 52970*x^6 - 38754*x^5 - 1790*x^4 + 2057*x^3 - 72*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^30 - x^29 - 52*x^28 + 51*x^27 + 1142*x^26 - 1092*x^25 - 13898*x^24 + 12856*x^23 + 103478*x^22 - 91664*x^21 - 491678*x^20 + 412467*x^19 + 1512708*x^18 - 1191563*x^17 - 3007832*x^16 + 2225799*x^15 + 3819257*x^14 - 2692049*x^13 - 3042474*x^12 + 2079355*x^11 + 1471733*x^10 - 992749*x^9 - 403229*x^8 + 274849*x^7 + 52970*x^6 - 38754*x^5 - 1790*x^4 + 2057*x^3 - 72*x^2 - 24*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -24, -72, 2057, -1790, -38754, 52970, 274849, -403229, -992749, 1471733, 2079355, -3042474, -2692049, 3819257, 2225799, -3007832, -1191563, 1512708, 412467, -491678, -91664, 103478, 12856, -13898, -1092, 1142, 51, -52, -1, 1]);
 

\( x^{30} - x^{29} - 52 x^{28} + 51 x^{27} + 1142 x^{26} - 1092 x^{25} - 13898 x^{24} + 12856 x^{23} + 103478 x^{22} - 91664 x^{21} - 491678 x^{20} + 412467 x^{19} + 1512708 x^{18} - 1191563 x^{17} - 3007832 x^{16} + 2225799 x^{15} + 3819257 x^{14} - 2692049 x^{13} - 3042474 x^{12} + 2079355 x^{11} + 1471733 x^{10} - 992749 x^{9} - 403229 x^{8} + 274849 x^{7} + 52970 x^{6} - 38754 x^{5} - 1790 x^{4} + 2057 x^{3} - 72 x^{2} - 24 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[30, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(15010049358097810645468215169836895080663686428828497\)\(\medspace = 3^{15}\cdot 7^{20}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $54.85$
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:  $3, 7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(128,·)$, $\chi_{231}(1,·)$, $\chi_{231}(2,·)$, $\chi_{231}(67,·)$, $\chi_{231}(4,·)$, $\chi_{231}(197,·)$, $\chi_{231}(134,·)$, $\chi_{231}(65,·)$, $\chi_{231}(8,·)$, $\chi_{231}(74,·)$, $\chi_{231}(130,·)$, $\chi_{231}(64,·)$, $\chi_{231}(16,·)$, $\chi_{231}(148,·)$, $\chi_{231}(149,·)$, $\chi_{231}(214,·)$, $\chi_{231}(25,·)$, $\chi_{231}(29,·)$, $\chi_{231}(95,·)$, $\chi_{231}(32,·)$, $\chi_{231}(163,·)$, $\chi_{231}(100,·)$, $\chi_{231}(37,·)$, $\chi_{231}(169,·)$, $\chi_{231}(107,·)$, $\chi_{231}(200,·)$, $\chi_{231}(50,·)$, $\chi_{231}(116,·)$, $\chi_{231}(58,·)$, $\chi_{231}(190,·)$$\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}{5354040114818371420711761027640775914454156062783718821} a^{29} - \frac{127462915631251464634916747641170270790658689143032723}{5354040114818371420711761027640775914454156062783718821} a^{28} + \frac{488830555180522058188974681748630255622071437435728984}{5354040114818371420711761027640775914454156062783718821} a^{27} - \frac{1379312936275859007490502783964633805135297281857386572}{5354040114818371420711761027640775914454156062783718821} a^{26} + \frac{948904476638615430077294314026027083853048845645754055}{5354040114818371420711761027640775914454156062783718821} a^{25} - \frac{340896130874224561184879036531662448136021020703198727}{5354040114818371420711761027640775914454156062783718821} a^{24} - \frac{356863840525894987336831623425549918596724515075288642}{5354040114818371420711761027640775914454156062783718821} a^{23} + \frac{1393452175575022780527525863177639581250460482051190500}{5354040114818371420711761027640775914454156062783718821} a^{22} - \frac{2451943452843471458225434457896535780010619628146425965}{5354040114818371420711761027640775914454156062783718821} a^{21} + \frac{465852364635296462054356134039793015221005992914900298}{5354040114818371420711761027640775914454156062783718821} a^{20} - \frac{387057696923409668490756986263499477265078696066086168}{5354040114818371420711761027640775914454156062783718821} a^{19} + \frac{2207394925433523933140308501793678516217736492205877577}{5354040114818371420711761027640775914454156062783718821} a^{18} - \frac{2065825092551414601752006559179237442840603169143075348}{5354040114818371420711761027640775914454156062783718821} a^{17} - \frac{559638018697565413326688574976522938146414343319108981}{5354040114818371420711761027640775914454156062783718821} a^{16} + \frac{1387674515288311091251020023620133782458619228897820333}{5354040114818371420711761027640775914454156062783718821} a^{15} + \frac{94311399947014795005762029657064962344106485321197665}{5354040114818371420711761027640775914454156062783718821} a^{14} + \frac{649573533887490363337765310036046471532720604144095091}{5354040114818371420711761027640775914454156062783718821} a^{13} + \frac{1959292588205974815665717236522328039282125808345728971}{5354040114818371420711761027640775914454156062783718821} a^{12} + \frac{2250335297428947034655835136449686833703397614947405495}{5354040114818371420711761027640775914454156062783718821} a^{11} + \frac{1322148792548555692621520670794901113502365815190968626}{5354040114818371420711761027640775914454156062783718821} a^{10} - \frac{957771657015028458383980058408959553313163900205371413}{5354040114818371420711761027640775914454156062783718821} a^{9} + \frac{2284982013840999800293503408743066739175243512823004019}{5354040114818371420711761027640775914454156062783718821} a^{8} - \frac{832021347016405398189585600087515913820000215897901451}{5354040114818371420711761027640775914454156062783718821} a^{7} + \frac{273541671637162485636043616011350333409743661254403352}{5354040114818371420711761027640775914454156062783718821} a^{6} + \frac{1347698694713779213874100437149732116268194103959271474}{5354040114818371420711761027640775914454156062783718821} a^{5} + \frac{345055934263418426967075814622495720404135587891923861}{5354040114818371420711761027640775914454156062783718821} a^{4} - \frac{1492573710330269403276905424579008793981486591841387728}{5354040114818371420711761027640775914454156062783718821} a^{3} - \frac{417877343111894384050690777395276213557919426402272766}{5354040114818371420711761027640775914454156062783718821} a^{2} + \frac{2447539843973921128349596409812802961722772939786539522}{5354040114818371420711761027640775914454156062783718821} a + \frac{1427468726486835844477959429006676820013028964227591749}{5354040114818371420711761027640775914454156062783718821}$

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:  $29$
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:  \( 25852981457107600 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 25852981457107600 \cdot 1}{2\sqrt{15010049358097810645468215169836895080663686428828497}}\approx 0.113289444884841$ (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.86284737.1, \(\Q(\zeta_{33})^+\), 15.15.886528337182930278529.1

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 $15^{2}$ R $30$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $15^{2}$ $30$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $30$ $30$ $30$

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
7Data not computed
11Data not computed