Normalized defining polynomial
\( x^{18} - 24 x^{16} - 3 x^{15} + 216 x^{14} + 42 x^{13} - 943 x^{12} - 180 x^{11} + 2190 x^{10} + 272 x^{9} - 2772 x^{8} - 54 x^{7} + 1819 x^{6} - 180 x^{5} - 522 x^{4} + 110 x^{3} + \cdots - 1 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(13314770360250302126953125\) \(\medspace = 3^{24}\cdot 5^{9}\cdot 17^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.88\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $3^{4/3}5^{1/2}17^{1/2}\approx 39.890652095882146$ | ||
Ramified primes: | \(3\), \(5\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}+\frac{5}{19}a^{14}+\frac{1}{19}a^{13}+\frac{7}{19}a^{12}-\frac{9}{19}a^{11}+\frac{2}{19}a^{10}-\frac{5}{19}a^{9}-\frac{3}{19}a^{8}+\frac{6}{19}a^{5}-\frac{5}{19}a^{4}+\frac{8}{19}a^{3}+\frac{4}{19}a^{2}+\frac{5}{19}a+\frac{4}{19}$, $\frac{1}{19}a^{16}-\frac{5}{19}a^{14}+\frac{2}{19}a^{13}-\frac{6}{19}a^{12}+\frac{9}{19}a^{11}+\frac{4}{19}a^{10}+\frac{3}{19}a^{9}-\frac{4}{19}a^{8}+\frac{6}{19}a^{6}+\frac{3}{19}a^{5}-\frac{5}{19}a^{4}+\frac{2}{19}a^{3}+\frac{4}{19}a^{2}-\frac{2}{19}a-\frac{1}{19}$, $\frac{1}{4993181}a^{17}+\frac{60376}{4993181}a^{16}-\frac{23577}{4993181}a^{15}-\frac{20807}{45809}a^{14}-\frac{428917}{4993181}a^{13}+\frac{971906}{4993181}a^{12}-\frac{1381394}{4993181}a^{11}+\frac{2262703}{4993181}a^{10}-\frac{11097}{262799}a^{9}+\frac{1178060}{4993181}a^{8}+\frac{261237}{4993181}a^{7}-\frac{1013721}{4993181}a^{6}+\frac{22033}{262799}a^{5}-\frac{2298243}{4993181}a^{4}+\frac{2305698}{4993181}a^{3}-\frac{1326521}{4993181}a^{2}+\frac{2168174}{4993181}a+\frac{761136}{4993181}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{598824}{262799}a^{17}+\frac{288198}{262799}a^{16}-\frac{14051118}{262799}a^{15}-\frac{78372}{2411}a^{14}+\frac{121170984}{262799}a^{13}+\frac{82447653}{262799}a^{12}-\frac{492479280}{262799}a^{11}-\frac{333001578}{262799}a^{10}+\frac{1033840199}{262799}a^{9}+\frac{614222676}{262799}a^{8}-\frac{1162421424}{262799}a^{7}-\frac{524286813}{262799}a^{6}+\frac{673745184}{262799}a^{5}+\frac{184566600}{262799}a^{4}-\frac{171360343}{262799}a^{3}-\frac{17405100}{262799}a^{2}+\frac{9833424}{262799}a+\frac{1321213}{262799}$, $\frac{11279978}{4993181}a^{17}+\frac{7906384}{4993181}a^{16}-\frac{266136276}{4993181}a^{15}-\frac{2021957}{45809}a^{14}+\frac{2305740430}{4993181}a^{13}+\frac{2093093894}{4993181}a^{12}-\frac{9392562412}{4993181}a^{11}-\frac{8660927344}{4993181}a^{10}+\frac{1034252688}{262799}a^{9}+\frac{17084355297}{4993181}a^{8}-\frac{21748423524}{4993181}a^{7}-\frac{863516500}{262799}a^{6}+\frac{12142672099}{4993181}a^{5}+\frac{7048125497}{4993181}a^{4}-\frac{2901184900}{4993181}a^{3}-\frac{990399906}{4993181}a^{2}+\frac{197591169}{4993181}a+\frac{43356668}{4993181}$, $\frac{2707318}{4993181}a^{17}-\frac{1581241}{4993181}a^{16}-\frac{69617508}{4993181}a^{15}+\frac{241237}{45809}a^{14}+\frac{692722516}{4993181}a^{13}-\frac{131485354}{4993181}a^{12}-\frac{3428195172}{4993181}a^{11}+\frac{130655702}{4993181}a^{10}+\frac{472586436}{262799}a^{9}+\frac{561581150}{4993181}a^{8}-\frac{12215817924}{4993181}a^{7}-\frac{64342444}{262799}a^{6}+\frac{7915227303}{4993181}a^{5}+\frac{567066195}{4993181}a^{4}-\frac{1905409012}{4993181}a^{3}+\frac{110356071}{4993181}a^{2}+\frac{55930940}{4993181}a+\frac{4742745}{4993181}$, $\frac{6677866}{4993181}a^{17}+\frac{5809781}{4993181}a^{16}-\frac{149647217}{4993181}a^{15}-\frac{1392262}{45809}a^{14}+\frac{1185478152}{4993181}a^{13}+\frac{1324988216}{4993181}a^{12}-\frac{4127881072}{4993181}a^{11}-\frac{4772957522}{4993181}a^{10}+\frac{6662825173}{4993181}a^{9}+\frac{7338259984}{4993181}a^{8}-\frac{5079976253}{4993181}a^{7}-\frac{4417598580}{4993181}a^{6}+\frac{1867585216}{4993181}a^{5}+\frac{508309419}{4993181}a^{4}-\frac{426416465}{4993181}a^{3}+\frac{139153262}{4993181}a^{2}+\frac{14852712}{4993181}a-\frac{9956505}{4993181}$, $\frac{26025251}{4993181}a^{17}+\frac{11930627}{4993181}a^{16}-\frac{619989322}{4993181}a^{15}-\frac{175462}{2411}a^{14}+\frac{5476021511}{4993181}a^{13}+\frac{3629000632}{4993181}a^{12}-\frac{1214300092}{262799}a^{11}-\frac{15468976716}{4993181}a^{10}+\frac{50739200739}{4993181}a^{9}+\frac{31054522566}{4993181}a^{8}-\frac{59755089769}{4993181}a^{7}-\frac{29863903950}{4993181}a^{6}+\frac{35771039317}{4993181}a^{5}+\frac{12318758601}{4993181}a^{4}-\frac{9073139270}{4993181}a^{3}-\frac{1366938487}{4993181}a^{2}+\frac{528764928}{4993181}a+\frac{79778901}{4993181}$, $\frac{6628371}{4993181}a^{17}+\frac{5512}{4993181}a^{16}-\frac{163196710}{4993181}a^{15}-\frac{222235}{45809}a^{14}+\frac{1524306959}{4993181}a^{13}+\frac{386876869}{4993181}a^{12}-\frac{6985351600}{4993181}a^{11}-\frac{2124897536}{4993181}a^{10}+\frac{17063809990}{4993181}a^{9}+\frac{5163521086}{4993181}a^{8}-\frac{22378934943}{4993181}a^{7}-\frac{5648801678}{4993181}a^{6}+\frac{14773825097}{4993181}a^{5}+\frac{2265424864}{4993181}a^{4}-\frac{4021563545}{4993181}a^{3}+\frac{3054909}{4993181}a^{2}+\frac{215567869}{4993181}a+\frac{16133725}{4993181}$, $\frac{8392602}{4993181}a^{17}+\frac{7935256}{4993181}a^{16}-\frac{194181157}{4993181}a^{15}-\frac{1926214}{45809}a^{14}+\frac{1623673734}{4993181}a^{13}+\frac{1913931176}{4993181}a^{12}-\frac{6224866928}{4993181}a^{11}-\frac{7611839811}{4993181}a^{10}+\frac{11902634658}{4993181}a^{9}+\frac{14375054486}{4993181}a^{8}-\frac{11866467853}{4993181}a^{7}-\frac{700115108}{262799}a^{6}+\frac{6131080762}{4993181}a^{5}+\frac{5740296067}{4993181}a^{4}-\frac{1576846721}{4993181}a^{3}-\frac{955888238}{4993181}a^{2}+\frac{179016060}{4993181}a+\frac{28628977}{4993181}$, $\frac{97678}{4993181}a^{17}-\frac{2430622}{4993181}a^{16}-\frac{834966}{4993181}a^{15}+\frac{532889}{45809}a^{14}-\frac{3491734}{4993181}a^{13}-\frac{526588487}{4993181}a^{12}+\frac{35456092}{4993181}a^{11}+\frac{2333897362}{4993181}a^{10}-\frac{412489}{262799}a^{9}-\frac{5414124453}{4993181}a^{8}-\frac{337583532}{4993181}a^{7}+\frac{339229687}{262799}a^{6}+\frac{658486397}{4993181}a^{5}-\frac{3541360097}{4993181}a^{4}-\frac{354661617}{4993181}a^{3}+\frac{659703006}{4993181}a^{2}-\frac{10756113}{4993181}a-\frac{13260440}{4993181}$, $\frac{1758953}{4993181}a^{17}-\frac{2471557}{4993181}a^{16}-\frac{43726119}{4993181}a^{15}+\frac{461806}{45809}a^{14}+\frac{420843292}{4993181}a^{13}-\frac{370763459}{4993181}a^{12}-\frac{2015296290}{4993181}a^{11}+\frac{66052347}{262799}a^{10}+\frac{5072093317}{4993181}a^{9}-\frac{2071980804}{4993181}a^{8}-\frac{6490005145}{4993181}a^{7}+\frac{1633763084}{4993181}a^{6}+\frac{3804568187}{4993181}a^{5}-\frac{444662771}{4993181}a^{4}-\frac{818499110}{4993181}a^{3}-\frac{2052472}{262799}a^{2}+\frac{69841869}{4993181}a+\frac{11576561}{4993181}$, $\frac{178156}{262799}a^{17}+\frac{4155717}{4993181}a^{16}-\frac{76970417}{4993181}a^{15}-\frac{994773}{45809}a^{14}+\frac{620943414}{4993181}a^{13}+\frac{990947380}{4993181}a^{12}-\frac{2204376376}{4993181}a^{11}-\frac{4030143986}{4993181}a^{10}+\frac{3536905741}{4993181}a^{9}+\frac{7965125450}{4993181}a^{8}-\frac{116132689}{262799}a^{7}-\frac{7872644249}{4993181}a^{6}-\frac{128620987}{4993181}a^{5}+\frac{3612696844}{4993181}a^{4}+\frac{510252111}{4993181}a^{3}-\frac{572444626}{4993181}a^{2}-\frac{59020627}{4993181}a+\frac{13826010}{4993181}$, $\frac{574391}{262799}a^{17}-\frac{2801411}{4993181}a^{16}-\frac{272725593}{4993181}a^{15}+\frac{185343}{45809}a^{14}+\frac{2609131757}{4993181}a^{13}+\frac{207714271}{4993181}a^{12}-\frac{12336346352}{4993181}a^{11}-\frac{2343745321}{4993181}a^{10}+\frac{31028519066}{4993181}a^{9}+\frac{7801128537}{4993181}a^{8}-\frac{2174401882}{262799}a^{7}-\frac{10642237480}{4993181}a^{6}+\frac{27187891530}{4993181}a^{5}+\frac{5595557994}{4993181}a^{4}-\frac{384176333}{262799}a^{3}-\frac{622975459}{4993181}a^{2}+\frac{471698056}{4993181}a+\frac{51326550}{4993181}$, $\frac{6392283}{4993181}a^{17}+\frac{5955762}{4993181}a^{16}-\frac{151469598}{4993181}a^{15}-\frac{1439080}{45809}a^{14}+\frac{1316878966}{4993181}a^{13}+\frac{75272150}{262799}a^{12}-\frac{5392385601}{4993181}a^{11}-\frac{5712872068}{4993181}a^{10}+\frac{11485856412}{4993181}a^{9}+\frac{10798591242}{4993181}a^{8}-\frac{13440290097}{4993181}a^{7}-\frac{9692233758}{4993181}a^{6}+\frac{8540343963}{4993181}a^{5}+\frac{3655269608}{4993181}a^{4}-\frac{2552489105}{4993181}a^{3}-\frac{383561279}{4993181}a^{2}+\frac{187214465}{4993181}a+\frac{22751158}{4993181}$, $\frac{320330}{4993181}a^{17}+\frac{2442464}{4993181}a^{16}-\frac{5095929}{4993181}a^{15}-\frac{499560}{45809}a^{14}+\frac{1590570}{4993181}a^{13}+\frac{418933658}{4993181}a^{12}+\frac{296767447}{4993181}a^{11}-\frac{1343244453}{4993181}a^{10}-\frac{1545253310}{4993181}a^{9}+\frac{1739888336}{4993181}a^{8}+\frac{2902365992}{4993181}a^{7}-\frac{628118381}{4993181}a^{6}-\frac{2261832612}{4993181}a^{5}-\frac{213389544}{4993181}a^{4}+\frac{32614865}{262799}a^{3}+\frac{73808071}{4993181}a^{2}-\frac{38695610}{4993181}a-\frac{12582511}{4993181}$, $\frac{13348555}{4993181}a^{17}+\frac{3246993}{4993181}a^{16}-\frac{319823183}{4993181}a^{15}-\frac{1119202}{45809}a^{14}+\frac{2858883870}{4993181}a^{13}+\frac{1348700279}{4993181}a^{12}-\frac{12282721557}{4993181}a^{11}-\frac{6125078486}{4993181}a^{10}+\frac{27647956616}{4993181}a^{9}+\frac{12940302582}{4993181}a^{8}-\frac{33121053818}{4993181}a^{7}-\frac{13020911531}{4993181}a^{6}+\frac{19856144477}{4993181}a^{5}+\frac{5614723717}{4993181}a^{4}-\frac{4926810142}{4993181}a^{3}-\frac{649877700}{4993181}a^{2}+\frac{286066838}{4993181}a+\frac{39346506}{4993181}$, $\frac{2792210}{4993181}a^{17}-\frac{4664334}{4993181}a^{16}-\frac{71478601}{4993181}a^{15}+\frac{879128}{45809}a^{14}+\frac{717656304}{4993181}a^{13}-\frac{701249479}{4993181}a^{12}-\frac{3618437728}{4993181}a^{11}+\frac{2288047890}{4993181}a^{10}+\frac{9606536632}{4993181}a^{9}-\frac{3513407275}{4993181}a^{8}-\frac{12947600948}{4993181}a^{7}+\frac{2684672102}{4993181}a^{6}+\frac{7949428486}{4993181}a^{5}-\frac{1213601387}{4993181}a^{4}-\frac{1580335223}{4993181}a^{3}+\frac{22499054}{262799}a^{2}-\frac{24073162}{4993181}a-\frac{7641626}{4993181}$, $\frac{869621}{4993181}a^{17}+\frac{413683}{4993181}a^{16}-\frac{17083870}{4993181}a^{15}-\frac{6913}{2411}a^{14}+\frac{5428307}{262799}a^{13}+\frac{149696764}{4993181}a^{12}-\frac{7361025}{262799}a^{11}-\frac{637088964}{4993181}a^{10}-\frac{569437631}{4993181}a^{9}+\frac{1268320720}{4993181}a^{8}+\frac{1924799905}{4993181}a^{7}-\frac{1328317563}{4993181}a^{6}-\frac{2031125437}{4993181}a^{5}+\frac{742091820}{4993181}a^{4}+\frac{786120389}{4993181}a^{3}-\frac{181973803}{4993181}a^{2}-\frac{59544674}{4993181}a+\frac{96910}{4993181}$, $\frac{11288610}{4993181}a^{17}+\frac{210506}{262799}a^{16}-\frac{271239695}{4993181}a^{15}-\frac{1210440}{45809}a^{14}+\frac{2430207924}{4993181}a^{13}+\frac{1385536302}{4993181}a^{12}-\frac{10456849544}{4993181}a^{11}-\frac{6176783379}{4993181}a^{10}+\frac{23562187687}{4993181}a^{9}+\frac{13049871664}{4993181}a^{8}-\frac{28251465214}{4993181}a^{7}-\frac{13328144584}{4993181}a^{6}+\frac{16924401918}{4993181}a^{5}+\frac{5934646269}{4993181}a^{4}-\frac{4171589596}{4993181}a^{3}-\frac{750315663}{4993181}a^{2}+\frac{253751740}{4993181}a+\frac{47975203}{4993181}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 3646318.70374 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 3646318.70374 \cdot 1}{2\cdot\sqrt{13314770360250302126953125}}\cr\approx \mathstrut & 0.130977805954 \end{aligned}\] (assuming GRH)
Galois group
$C_6\times S_3$ (as 18T6):
A solvable group of order 36 |
The 18 conjugacy class representatives for $S_3 \times C_6$ |
Character table for $S_3 \times C_6$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.6885.1, 6.6.820125.1, 6.6.237016125.1, 9.9.326371204125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 12 sibling: | 12.12.2474477972015625.1 |
Degree 18 sibling: | deg 18 |
Minimal sibling: | 12.12.2474477972015625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $18$ | $3$ | $6$ | $24$ | |||
\(5\) | 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |