Normalized defining polynomial
\( x^{30} - 3 x^{29} - 66 x^{28} + 175 x^{27} + 1845 x^{26} - 4269 x^{25} - 28987 x^{24} + 57798 x^{23} + 285615 x^{22} - 483539 x^{21} - 1861710 x^{20} + 2628246 x^{19} + 8242129 x^{18} - 9503007 x^{17} - 25018704 x^{16} + 23065539 x^{15} + 51874065 x^{14} - 37627170 x^{13} - 72317630 x^{12} + 41223645 x^{11} + 65947266 x^{10} - 30352364 x^{9} - 37774599 x^{8} + 14865612 x^{7} + 12736408 x^{6} - 4591656 x^{5} - 2216865 x^{4} + 783780 x^{3} + 135282 x^{2} - 54867 x + 3761 \)
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: | \(3654472495532549188442614010323982591205830108642578125=3^{40}\cdot 5^{15}\cdot 11^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.88$ | 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: | $3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(495=3^{2}\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{495}(256,·)$, $\chi_{495}(1,·)$, $\chi_{495}(4,·)$, $\chi_{495}(454,·)$, $\chi_{495}(199,·)$, $\chi_{495}(136,·)$, $\chi_{495}(394,·)$, $\chi_{495}(331,·)$, $\chi_{495}(64,·)$, $\chi_{495}(16,·)$, $\chi_{495}(466,·)$, $\chi_{495}(334,·)$, $\chi_{495}(214,·)$, $\chi_{495}(196,·)$, $\chi_{495}(346,·)$, $\chi_{495}(91,·)$, $\chi_{495}(31,·)$, $\chi_{495}(289,·)$, $\chi_{495}(34,·)$, $\chi_{495}(229,·)$, $\chi_{495}(421,·)$, $\chi_{495}(166,·)$, $\chi_{495}(169,·)$, $\chi_{495}(364,·)$, $\chi_{495}(301,·)$, $\chi_{495}(49,·)$, $\chi_{495}(181,·)$, $\chi_{495}(361,·)$, $\chi_{495}(379,·)$, $\chi_{495}(124,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{36079} a^{27} + \frac{15857}{36079} a^{26} - \frac{1663}{36079} a^{25} + \frac{10210}{36079} a^{24} - \frac{17112}{36079} a^{23} - \frac{3240}{36079} a^{22} - \frac{14882}{36079} a^{21} - \frac{6566}{36079} a^{20} + \frac{14917}{36079} a^{19} - \frac{568}{36079} a^{18} - \frac{16616}{36079} a^{17} - \frac{8599}{36079} a^{16} - \frac{15928}{36079} a^{15} + \frac{16159}{36079} a^{14} - \frac{2910}{36079} a^{13} - \frac{1251}{36079} a^{12} - \frac{3828}{36079} a^{11} - \frac{12343}{36079} a^{10} + \frac{13693}{36079} a^{9} + \frac{1847}{36079} a^{8} + \frac{5241}{36079} a^{7} + \frac{17183}{36079} a^{6} - \frac{5281}{36079} a^{5} - \frac{10578}{36079} a^{4} - \frac{5601}{36079} a^{3} - \frac{3919}{36079} a^{2} + \frac{12355}{36079} a + \frac{6073}{36079}$, $\frac{1}{633244212479} a^{28} + \frac{6172976}{633244212479} a^{27} + \frac{153501116398}{633244212479} a^{26} + \frac{211101417623}{633244212479} a^{25} + \frac{108263239474}{633244212479} a^{24} + \frac{207309407051}{633244212479} a^{23} + \frac{192921962381}{633244212479} a^{22} + \frac{245386682033}{633244212479} a^{21} - \frac{196706883652}{633244212479} a^{20} - \frac{52324749838}{633244212479} a^{19} - \frac{181374451114}{633244212479} a^{18} + \frac{253826008303}{633244212479} a^{17} + \frac{232073966731}{633244212479} a^{16} + \frac{51556578238}{633244212479} a^{15} + \frac{151697067350}{633244212479} a^{14} + \frac{27657017600}{633244212479} a^{13} - \frac{219829328829}{633244212479} a^{12} - \frac{295483140776}{633244212479} a^{11} + \frac{64085111858}{633244212479} a^{10} - \frac{101447592587}{633244212479} a^{9} + \frac{175043360848}{633244212479} a^{8} - \frac{64778341868}{633244212479} a^{7} - \frac{162222765831}{633244212479} a^{6} - \frac{169817204679}{633244212479} a^{5} - \frac{754096313}{7115103511} a^{4} + \frac{288192029149}{633244212479} a^{3} + \frac{1740610230}{5809579931} a^{2} - \frac{157960056768}{633244212479} a - \frac{160358222394}{633244212479}$, $\frac{1}{13787163140472731130779706170819052526458477360748852916295617051} a^{29} + \frac{9209513570089388415004668393681024364078583860502988}{13787163140472731130779706170819052526458477360748852916295617051} a^{28} - \frac{184507603492570964719865116753405086129199691657739856252481}{13787163140472731130779706170819052526458477360748852916295617051} a^{27} + \frac{1020554921963238818567019771880210548484547046694843590161083448}{13787163140472731130779706170819052526458477360748852916295617051} a^{26} - \frac{2298625557041958303903028555398141201252540132422990198477034592}{13787163140472731130779706170819052526458477360748852916295617051} a^{25} - \frac{2052433653651329288929392516725483693411670331480771449814959252}{13787163140472731130779706170819052526458477360748852916295617051} a^{24} + \frac{2927466466283274028188443509872133997655017804136356343485742212}{13787163140472731130779706170819052526458477360748852916295617051} a^{23} + \frac{1905136876678963876695990044638985638710313743201167608003539615}{13787163140472731130779706170819052526458477360748852916295617051} a^{22} - \frac{6662172756771772454575236798648176576354003921441873881008809798}{13787163140472731130779706170819052526458477360748852916295617051} a^{21} + \frac{4467486494566751764488070956027940008454880676727647123901364302}{13787163140472731130779706170819052526458477360748852916295617051} a^{20} - \frac{6808747315996153615934975215355716307930452219628787278123296585}{13787163140472731130779706170819052526458477360748852916295617051} a^{19} + \frac{1342827551845073245846565056693171298435311515117129163572199195}{13787163140472731130779706170819052526458477360748852916295617051} a^{18} - \frac{2910784429045507562879481410163034047316898429909530197546044914}{13787163140472731130779706170819052526458477360748852916295617051} a^{17} - \frac{2645406782356401867113751899996798786851328090694379332076823332}{13787163140472731130779706170819052526458477360748852916295617051} a^{16} + \frac{1864537517173005568133773936552803413242241976510934818944022553}{13787163140472731130779706170819052526458477360748852916295617051} a^{15} - \frac{1601047133697678843550665556532635620277284131813022851930725991}{13787163140472731130779706170819052526458477360748852916295617051} a^{14} - \frac{417979537461606264666969329700767154773358252523254027206732917}{13787163140472731130779706170819052526458477360748852916295617051} a^{13} - \frac{4419899007166105372865259915634970208929910354166944445646799618}{13787163140472731130779706170819052526458477360748852916295617051} a^{12} + \frac{4007003400917832670840795515745602301496778631561518722928228281}{13787163140472731130779706170819052526458477360748852916295617051} a^{11} - \frac{5562259328158421397503066955686839117619474543722719294949323501}{13787163140472731130779706170819052526458477360748852916295617051} a^{10} + \frac{1315374479571832511252838573268096054387319477189120795961696630}{13787163140472731130779706170819052526458477360748852916295617051} a^{9} - \frac{679996974909486590787756750275454742320644191359898931217548224}{13787163140472731130779706170819052526458477360748852916295617051} a^{8} + \frac{6039860274217365074721449523995711440912968011904040647623525927}{13787163140472731130779706170819052526458477360748852916295617051} a^{7} - \frac{4180863908152999470566869008169703634497407698271363788555559533}{13787163140472731130779706170819052526458477360748852916295617051} a^{6} + \frac{174506056204996016584285955084248617246005553446364814990392989}{13787163140472731130779706170819052526458477360748852916295617051} a^{5} - \frac{745687711036791166309630593829255336766781563843122033228242400}{13787163140472731130779706170819052526458477360748852916295617051} a^{4} + \frac{121532846561531045151408222845482199462572030650247178616669818}{13787163140472731130779706170819052526458477360748852916295617051} a^{3} + \frac{1748646658869528912166770449817181064911693759087769784710879439}{13787163140472731130779706170819052526458477360748852916295617051} a^{2} + \frac{5412541689677617708664986623221827142216356696379843959713438786}{13787163140472731130779706170819052526458477360748852916295617051} a - \frac{2435999811549608643867466235936154319354981639667260611864159526}{13787163140472731130779706170819052526458477360748852916295617051}$
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: | \( 636236955303921700 \) (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.6.820125.1, 10.10.669871503125.1, 15.15.10943023107606534329121.1 |
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 | R | $30$ | R | $30$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$ | $15^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ | $30$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ | $15^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $11$ | 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ |
| 11.15.12.1 | $x^{15} + 165 x^{10} + 5324 x^{5} + 323433$ | $5$ | $3$ | $12$ | $C_{15}$ | $[\ ]_{5}^{3}$ | |