Normalized defining polynomial
\( x^{30} - x^{29} + 44 x^{28} - 37 x^{27} + 940 x^{26} - 674 x^{25} + 12608 x^{24} - 7673 x^{23} + 117056 x^{22} - 59978 x^{21} + 786528 x^{20} - 334398 x^{19} + 3902536 x^{18} - 1351484 x^{17} + 14360768 x^{16} - 3945405 x^{15} + 38900494 x^{14} - 8209064 x^{13} + 76055528 x^{12} - 11738464 x^{11} + 103832512 x^{10} - 11108608 x^{9} + 93815424 x^{8} - 6087168 x^{7} + 51674112 x^{6} - 2150400 x^{5} + 14938112 x^{4} + 114688 x^{3} + 1736704 x^{2} - 131072 x + 32768 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 15]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2720182407758598253643604359419657198151006682611956663=-\,7^{15}\cdot 31^{28}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.24$ | 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: | $7, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(217=7\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{217}(64,·)$, $\chi_{217}(1,·)$, $\chi_{217}(195,·)$, $\chi_{217}(132,·)$, $\chi_{217}(69,·)$, $\chi_{217}(134,·)$, $\chi_{217}(71,·)$, $\chi_{217}(8,·)$, $\chi_{217}(76,·)$, $\chi_{217}(202,·)$, $\chi_{217}(204,·)$, $\chi_{217}(78,·)$, $\chi_{217}(211,·)$, $\chi_{217}(20,·)$, $\chi_{217}(169,·)$, $\chi_{217}(90,·)$, $\chi_{217}(160,·)$, $\chi_{217}(97,·)$, $\chi_{217}(162,·)$, $\chi_{217}(36,·)$, $\chi_{217}(41,·)$, $\chi_{217}(174,·)$, $\chi_{217}(111,·)$, $\chi_{217}(113,·)$, $\chi_{217}(50,·)$, $\chi_{217}(118,·)$, $\chi_{217}(183,·)$, $\chi_{217}(188,·)$, $\chi_{217}(125,·)$, $\chi_{217}(190,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{2} a^{14} + \frac{1}{4} a^{13} - \frac{1}{8} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{18} - \frac{1}{16} a^{16} + \frac{1}{4} a^{15} - \frac{3}{8} a^{14} - \frac{1}{2} a^{13} - \frac{1}{16} a^{12} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{3}{8} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{20} - \frac{1}{32} a^{19} - \frac{1}{32} a^{17} + \frac{1}{8} a^{16} - \frac{3}{16} a^{15} - \frac{1}{4} a^{14} - \frac{1}{32} a^{13} - \frac{1}{2} a^{12} + \frac{1}{16} a^{11} + \frac{1}{4} a^{10} - \frac{3}{16} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{13}{32} a^{5} + \frac{7}{16} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{21} - \frac{1}{64} a^{20} - \frac{1}{64} a^{18} + \frac{1}{16} a^{17} - \frac{3}{32} a^{16} + \frac{3}{8} a^{15} + \frac{31}{64} a^{14} + \frac{1}{4} a^{13} + \frac{1}{32} a^{12} - \frac{3}{8} a^{11} + \frac{13}{32} a^{10} + \frac{3}{8} a^{9} - \frac{5}{16} a^{8} + \frac{1}{4} a^{7} + \frac{19}{64} a^{6} - \frac{9}{32} a^{5} + \frac{5}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{22} - \frac{1}{128} a^{21} - \frac{1}{128} a^{19} + \frac{1}{32} a^{18} - \frac{3}{64} a^{17} + \frac{3}{16} a^{16} - \frac{33}{128} a^{15} - \frac{3}{8} a^{14} + \frac{1}{64} a^{13} + \frac{5}{16} a^{12} + \frac{13}{64} a^{11} + \frac{3}{16} a^{10} - \frac{5}{32} a^{9} - \frac{3}{8} a^{8} + \frac{19}{128} a^{7} - \frac{9}{64} a^{6} - \frac{11}{32} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{256} a^{23} - \frac{1}{256} a^{22} - \frac{1}{256} a^{20} + \frac{1}{64} a^{19} - \frac{3}{128} a^{18} + \frac{3}{32} a^{17} - \frac{33}{256} a^{16} + \frac{5}{16} a^{15} - \frac{63}{128} a^{14} + \frac{5}{32} a^{13} + \frac{13}{128} a^{12} - \frac{13}{32} a^{11} + \frac{27}{64} a^{10} + \frac{5}{16} a^{9} - \frac{109}{256} a^{8} - \frac{9}{128} a^{7} - \frac{11}{64} a^{6} + \frac{13}{32} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{512} a^{24} - \frac{1}{512} a^{23} - \frac{1}{512} a^{21} + \frac{1}{128} a^{20} - \frac{3}{256} a^{19} + \frac{3}{64} a^{18} - \frac{33}{512} a^{17} + \frac{5}{32} a^{16} - \frac{63}{256} a^{15} + \frac{5}{64} a^{14} - \frac{115}{256} a^{13} + \frac{19}{64} a^{12} - \frac{37}{128} a^{11} + \frac{5}{32} a^{10} - \frac{109}{512} a^{9} - \frac{9}{256} a^{8} - \frac{11}{128} a^{7} - \frac{19}{64} a^{6} - \frac{5}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1024} a^{25} - \frac{1}{1024} a^{24} - \frac{1}{1024} a^{22} + \frac{1}{256} a^{21} - \frac{3}{512} a^{20} + \frac{3}{128} a^{19} - \frac{33}{1024} a^{18} + \frac{5}{64} a^{17} - \frac{63}{512} a^{16} - \frac{59}{128} a^{15} + \frac{141}{512} a^{14} - \frac{45}{128} a^{13} + \frac{91}{256} a^{12} + \frac{5}{64} a^{11} - \frac{109}{1024} a^{10} - \frac{9}{512} a^{9} - \frac{11}{256} a^{8} + \frac{45}{128} a^{7} - \frac{5}{64} a^{6} - \frac{1}{32} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2048} a^{26} - \frac{1}{2048} a^{25} - \frac{1}{2048} a^{23} + \frac{1}{512} a^{22} - \frac{3}{1024} a^{21} + \frac{3}{256} a^{20} - \frac{33}{2048} a^{19} + \frac{5}{128} a^{18} - \frac{63}{1024} a^{17} - \frac{59}{256} a^{16} + \frac{141}{1024} a^{15} + \frac{83}{256} a^{14} - \frac{165}{512} a^{13} + \frac{5}{128} a^{12} + \frac{915}{2048} a^{11} - \frac{9}{1024} a^{10} + \frac{245}{512} a^{9} + \frac{45}{256} a^{8} - \frac{5}{128} a^{7} + \frac{31}{64} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4096} a^{27} - \frac{1}{4096} a^{26} - \frac{1}{4096} a^{24} + \frac{1}{1024} a^{23} - \frac{3}{2048} a^{22} + \frac{3}{512} a^{21} - \frac{33}{4096} a^{20} + \frac{5}{256} a^{19} - \frac{63}{2048} a^{18} - \frac{59}{512} a^{17} + \frac{141}{2048} a^{16} - \frac{173}{512} a^{15} + \frac{347}{1024} a^{14} + \frac{5}{256} a^{13} + \frac{915}{4096} a^{12} - \frac{9}{2048} a^{11} + \frac{245}{1024} a^{10} - \frac{211}{512} a^{9} + \frac{123}{256} a^{8} - \frac{33}{128} a^{7} - \frac{7}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8192} a^{28} - \frac{1}{8192} a^{27} - \frac{1}{8192} a^{25} + \frac{1}{2048} a^{24} - \frac{3}{4096} a^{23} + \frac{3}{1024} a^{22} - \frac{33}{8192} a^{21} + \frac{5}{512} a^{20} - \frac{63}{4096} a^{19} - \frac{59}{1024} a^{18} + \frac{141}{4096} a^{17} - \frac{173}{1024} a^{16} + \frac{347}{2048} a^{15} + \frac{5}{512} a^{14} + \frac{915}{8192} a^{13} - \frac{9}{4096} a^{12} + \frac{245}{2048} a^{11} + \frac{301}{1024} a^{10} - \frac{133}{512} a^{9} - \frac{33}{256} a^{8} + \frac{9}{32} a^{7} + \frac{1}{4} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1301413606121861983433868572440137363347988877653225500736355693365645426688} a^{29} + \frac{41710703264102873858429635814922838164639611301146159663794391345078393}{1301413606121861983433868572440137363347988877653225500736355693365645426688} a^{28} + \frac{73985612661013081857180206168744559292072266218086462781468521015778795}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{27} + \frac{192998741549783602023927179466658322186195852996252719926062040918481495}{1301413606121861983433868572440137363347988877653225500736355693365645426688} a^{26} - \frac{104556429983768571264402406293600323233747215463987313023983986249827783}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{25} + \frac{379585519611124048565437659226600035211893532231309750792614632167163793}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{24} - \frac{482515579937847136165385334739507136661182524002481564242915569483373955}{325353401530465495858467143110034340836997219413306375184088923341411356672} a^{23} + \frac{528209111784868096900086649340337736529232982705366432956399242598103999}{1301413606121861983433868572440137363347988877653225500736355693365645426688} a^{22} - \frac{3028805328456836580369190082468429503171146658384343951480401098700219805}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{21} + \frac{2589568205746653745392698615311697459307977362604922253408596829414298745}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{20} + \frac{8941126033543303995776865070294169658467434680710858038608985346281010477}{325353401530465495858467143110034340836997219413306375184088923341411356672} a^{19} - \frac{4994521472568714633969721729526604621053045174545456034173352679241029731}{650706803060930991716934286220068681673994438826612750368177846682822713344} a^{18} - \frac{40343621500332440269938230044271210493140926620021911778095595695164943253}{325353401530465495858467143110034340836997219413306375184088923341411356672} a^{17} + \frac{16026001349793880309035503435516607330143782761991147539515700378438780551}{325353401530465495858467143110034340836997219413306375184088923341411356672} a^{16} - \frac{7833256449531911983137994939970982508771356418098934954735853608181288301}{162676700765232747929233571555017170418498609706653187592044461670705678336} a^{15} + \frac{634160874260471297942265386295827538833114716409928907033130746213042755923}{1301413606121861983433868572440137363347988877653225500736355693365645426688} a^{14} + \frac{17072855319967728378304966265374595684916480125125349107823454880416607079}{325353401530465495858467143110034340836997219413306375184088923341411356672} a^{13} - \frac{24612421699458167815817318583173757917554275908803132433445901441894299297}{81338350382616373964616785777508585209249304853326593796022230835352839168} a^{12} - \frac{7012622405090649023305655485713277249684746320484660922691120435784408615}{162676700765232747929233571555017170418498609706653187592044461670705678336} a^{11} - \frac{9927157004986643569565286489880106914089877851904819504054312286729293741}{40669175191308186982308392888754292604624652426663296898011115417676419584} a^{10} - \frac{10164549120147940036488570558346636699786794573242382521586396523384981723}{40669175191308186982308392888754292604624652426663296898011115417676419584} a^{9} + \frac{7337270312523632189541843309734369216833027184922395048413659010892569719}{20334587595654093491154196444377146302312326213331648449005557708838209792} a^{8} - \frac{1923815745635503724741994233677305834977407971752658065456274236610150877}{10167293797827046745577098222188573151156163106665824224502778854419104896} a^{7} + \frac{352340594825052931573968338159504045048692264613754384259186897861791775}{5083646898913523372788549111094286575578081553332912112251389427209552448} a^{6} + \frac{428873697037751707952280187780811454757070826150827629324365582848772533}{1270911724728380843197137277773571643894520388333228028062847356802388112} a^{5} + \frac{19427144791351616614034596257050913335593439959047835165459718173969593}{79431982795523802699821079860848227743407524270826751753927959800149257} a^{4} - \frac{253225504649895334573765745422294649962269592496004544060170447296971041}{635455862364190421598568638886785821947260194166614014031423678401194056} a^{3} - \frac{34626945229987982730903496428827950558376092353075952152767854970512017}{158863965591047605399642159721696455486815048541653503507855919600298514} a^{2} - \frac{34078481510317076286986004711839020654938825498131879379884638045490137}{79431982795523802699821079860848227743407524270826751753927959800149257} a + \frac{3458323823864030381813043317261021103689823697998308031562901786252788}{79431982795523802699821079860848227743407524270826751753927959800149257}$
Class group and class number
$C_{98087}$, which has order $98087$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 4316173757.895952 \) (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
| \(\Q(\sqrt{-7}) \), 3.3.961.1, 5.5.923521.1, 6.0.316767703.1, 10.0.14334539666270887.1, \(\Q(\zeta_{31})^+\) |
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 | ${\href{/LocalNumberField/2.5.0.1}{5} }^{6}$ | $30$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ | R | $15^{2}$ | $30$ | $30$ | $30$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ | R | ${\href{/LocalNumberField/37.3.0.1}{3} }^{10}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
| 7 | Data not computed | ||||||
| 31 | Data not computed | ||||||