Normalized defining polynomial
\( x^{30} - x^{29} - 58 x^{28} + 57 x^{27} + 1427 x^{26} - 1371 x^{25} - 19532 x^{24} + 18280 x^{23} + 163955 x^{22} - 148985 x^{21} - 878198 x^{20} + 771582 x^{19} + 3033009 x^{18} - 2562602 x^{17} - 6690617 x^{16} + 5402217 x^{15} + 9167438 x^{14} - 7021358 x^{13} - 7422000 x^{12} + 5341306 x^{11} + 3295409 x^{10} - 2212210 x^{9} - 740486 x^{8} + 479104 x^{7} + 71252 x^{6} - 50085 x^{5} - 1394 x^{4} + 2108 x^{3} - 72 x^{2} - 24 x + 1 \)
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: | \(189537114621750490998034780134822461022325002110684109=3^{15}\cdot 7^{25}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $59.69$ | 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$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(130,·)$, $\chi_{231}(67,·)$, $\chi_{231}(4,·)$, $\chi_{231}(5,·)$, $\chi_{231}(16,·)$, $\chi_{231}(146,·)$, $\chi_{231}(20,·)$, $\chi_{231}(214,·)$, $\chi_{231}(89,·)$, $\chi_{231}(152,·)$, $\chi_{231}(25,·)$, $\chi_{231}(26,·)$, $\chi_{231}(122,·)$, $\chi_{231}(80,·)$, $\chi_{231}(148,·)$, $\chi_{231}(163,·)$, $\chi_{231}(100,·)$, $\chi_{231}(37,·)$, $\chi_{231}(38,·)$, $\chi_{231}(104,·)$, $\chi_{231}(169,·)$, $\chi_{231}(47,·)$, $\chi_{231}(185,·)$, $\chi_{231}(58,·)$, $\chi_{231}(59,·)$, $\chi_{231}(188,·)$, $\chi_{231}(125,·)$, $\chi_{231}(190,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{43} a^{24} + \frac{17}{43} a^{23} - \frac{19}{43} a^{22} + \frac{10}{43} a^{21} + \frac{5}{43} a^{20} - \frac{13}{43} a^{19} + \frac{5}{43} a^{18} - \frac{14}{43} a^{17} + \frac{11}{43} a^{16} - \frac{5}{43} a^{15} + \frac{14}{43} a^{14} - \frac{1}{43} a^{13} - \frac{5}{43} a^{12} - \frac{8}{43} a^{11} - \frac{17}{43} a^{10} - \frac{13}{43} a^{9} - \frac{4}{43} a^{8} - \frac{5}{43} a^{7} + \frac{12}{43} a^{6} - \frac{21}{43} a^{5} + \frac{17}{43} a^{4} + \frac{7}{43} a^{3} - \frac{3}{43} a + \frac{17}{43}$, $\frac{1}{43} a^{25} - \frac{7}{43} a^{23} - \frac{11}{43} a^{22} + \frac{7}{43} a^{21} - \frac{12}{43} a^{20} + \frac{11}{43} a^{19} - \frac{13}{43} a^{18} - \frac{9}{43} a^{17} - \frac{20}{43} a^{16} + \frac{13}{43} a^{15} + \frac{19}{43} a^{14} + \frac{12}{43} a^{13} - \frac{9}{43} a^{12} - \frac{10}{43} a^{11} + \frac{18}{43} a^{10} + \frac{2}{43} a^{9} + \frac{20}{43} a^{8} + \frac{11}{43} a^{7} - \frac{10}{43} a^{6} - \frac{13}{43} a^{5} + \frac{19}{43} a^{4} + \frac{10}{43} a^{3} - \frac{3}{43} a^{2} - \frac{18}{43} a + \frac{12}{43}$, $\frac{1}{43} a^{26} - \frac{21}{43} a^{23} + \frac{3}{43} a^{22} + \frac{15}{43} a^{21} + \frac{3}{43} a^{20} - \frac{18}{43} a^{19} - \frac{17}{43} a^{18} + \frac{11}{43} a^{17} + \frac{4}{43} a^{16} - \frac{16}{43} a^{15} - \frac{19}{43} a^{14} - \frac{16}{43} a^{13} - \frac{2}{43} a^{12} + \frac{5}{43} a^{11} + \frac{12}{43} a^{10} + \frac{15}{43} a^{9} - \frac{17}{43} a^{8} - \frac{2}{43} a^{7} - \frac{15}{43} a^{6} + \frac{1}{43} a^{5} + \frac{3}{43} a^{3} - \frac{18}{43} a^{2} - \frac{9}{43} a - \frac{10}{43}$, $\frac{1}{43} a^{27} + \frac{16}{43} a^{23} + \frac{3}{43} a^{22} - \frac{2}{43} a^{21} + \frac{1}{43} a^{20} + \frac{11}{43} a^{19} - \frac{13}{43} a^{18} + \frac{11}{43} a^{17} + \frac{5}{43} a^{15} + \frac{20}{43} a^{14} + \frac{20}{43} a^{13} - \frac{14}{43} a^{12} + \frac{16}{43} a^{11} + \frac{2}{43} a^{10} + \frac{11}{43} a^{9} + \frac{9}{43} a^{7} - \frac{5}{43} a^{6} - \frac{11}{43} a^{5} + \frac{16}{43} a^{4} - \frac{9}{43} a^{2} + \frac{13}{43} a + \frac{13}{43}$, $\frac{1}{43} a^{28} - \frac{11}{43} a^{23} + \frac{1}{43} a^{22} + \frac{13}{43} a^{21} + \frac{17}{43} a^{20} - \frac{20}{43} a^{19} + \frac{17}{43} a^{18} + \frac{9}{43} a^{17} + \frac{1}{43} a^{16} + \frac{14}{43} a^{15} + \frac{11}{43} a^{14} + \frac{2}{43} a^{13} + \frac{10}{43} a^{12} + \frac{1}{43} a^{11} - \frac{18}{43} a^{10} - \frac{7}{43} a^{9} - \frac{13}{43} a^{8} - \frac{11}{43} a^{7} + \frac{12}{43} a^{6} + \frac{8}{43} a^{5} - \frac{14}{43} a^{4} + \frac{8}{43} a^{3} + \frac{13}{43} a^{2} + \frac{18}{43} a - \frac{14}{43}$, $\frac{1}{142483128259185440034779578687195722757995403307190087060719} a^{29} + \frac{1457130403966900445135717330288808947600580238349911093427}{142483128259185440034779578687195722757995403307190087060719} a^{28} - \frac{1036142612981511805310554997616339694403241231520775046690}{142483128259185440034779578687195722757995403307190087060719} a^{27} + \frac{675540698171458793216607848599696566032684208817600775366}{142483128259185440034779578687195722757995403307190087060719} a^{26} - \frac{12454333472252619508663380139483656651041587097106843880}{142483128259185440034779578687195722757995403307190087060719} a^{25} + \frac{681086964552741032291090859526721390529311324855595612238}{142483128259185440034779578687195722757995403307190087060719} a^{24} - \frac{32744596202058599825829801648159367479809420146705124372452}{142483128259185440034779578687195722757995403307190087060719} a^{23} + \frac{24766129111276535687404428751556822618346162255020910907766}{142483128259185440034779578687195722757995403307190087060719} a^{22} + \frac{44335768394326033312783993695904663102502137294795462198629}{142483128259185440034779578687195722757995403307190087060719} a^{21} - \frac{70672792843314587048572119322024388428430055433866755154137}{142483128259185440034779578687195722757995403307190087060719} a^{20} - \frac{51012521301812564668676814258380194010684918256877836976845}{142483128259185440034779578687195722757995403307190087060719} a^{19} - \frac{28498415409210075573866529689792429985658771402655893879031}{142483128259185440034779578687195722757995403307190087060719} a^{18} + \frac{35979483307708909807840950331064341474733979035635305565847}{142483128259185440034779578687195722757995403307190087060719} a^{17} + \frac{12121506158308198185465680558295204956740638859384907346914}{142483128259185440034779578687195722757995403307190087060719} a^{16} + \frac{41567964207545875093375800275744800074726463564386384981395}{142483128259185440034779578687195722757995403307190087060719} a^{15} + \frac{14896949214977884423903895745756743949398464424465769178069}{142483128259185440034779578687195722757995403307190087060719} a^{14} - \frac{521840706286988477933662383335321544365487971950377914}{1461650252451097547571112100688295388414105346756702199} a^{13} - \frac{70372240851746789162881864437757827648317736461533405860660}{142483128259185440034779578687195722757995403307190087060719} a^{12} + \frac{667598049971030569418221154403146802343624959173152503098}{142483128259185440034779578687195722757995403307190087060719} a^{11} + \frac{33426863467438735126875735395951236823909901803814797326088}{142483128259185440034779578687195722757995403307190087060719} a^{10} - \frac{26143559611330217181702655430458030085114329989249618383181}{142483128259185440034779578687195722757995403307190087060719} a^{9} - \frac{45501915406875634068504543332056221096294189758429830677195}{142483128259185440034779578687195722757995403307190087060719} a^{8} - \frac{56032234376064270349079841902730919702401153549381854224282}{142483128259185440034779578687195722757995403307190087060719} a^{7} + \frac{52619393550868653187881094128058688620611131321040700374375}{142483128259185440034779578687195722757995403307190087060719} a^{6} + \frac{62761863285063550726889312856659762153162284734505595768537}{142483128259185440034779578687195722757995403307190087060719} a^{5} + \frac{34352228403997917533346784390445398837335806158542627453657}{142483128259185440034779578687195722757995403307190087060719} a^{4} + \frac{54248304953772207819030135629317058667011011859009259850606}{142483128259185440034779578687195722757995403307190087060719} a^{3} - \frac{14474423132980454065977739602272560761544849301561048495166}{142483128259185440034779578687195722757995403307190087060719} a^{2} - \frac{66424312037719503897395122272059594830356008502001532605837}{142483128259185440034779578687195722757995403307190087060719} a + \frac{53404676729991894691340975608096194874622931400376614663482}{142483128259185440034779578687195722757995403307190087060719}$
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: | \( 162927449513789120 \) (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
| \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{21})^+\), 10.10.875463320250981.1, 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 | $30$ | R | $15^{2}$ | R | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{30}$ | $15^{2}$ | $30$ | $15^{2}$ |
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 | ||||||