Normalized defining polynomial
\( x^{30} - x^{29} - 52 x^{28} + 51 x^{27} + 1142 x^{26} - 1092 x^{25} - 13898 x^{24} + 12856 x^{23} + 103478 x^{22} - 91664 x^{21} - 491678 x^{20} + 412467 x^{19} + 1512708 x^{18} - 1191563 x^{17} - 3007832 x^{16} + 2225799 x^{15} + 3819257 x^{14} - 2692049 x^{13} - 3042474 x^{12} + 2079355 x^{11} + 1471733 x^{10} - 992749 x^{9} - 403229 x^{8} + 274849 x^{7} + 52970 x^{6} - 38754 x^{5} - 1790 x^{4} + 2057 x^{3} - 72 x^{2} - 24 x + 1 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[30, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(15010049358097810645468215169836895080663686428828497=3^{15}\cdot 7^{20}\cdot 11^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $54.85$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| 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}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $29$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 25852981457107600 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| 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.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 7 | Data not computed | ||||||
| 11 | Data not computed | ||||||