Normalized defining polynomial
\( x^{18} - 6 x^{17} - 93 x^{16} + 854 x^{15} - 3061 x^{14} - 41886 x^{13} + 612274 x^{12} + 982052 x^{11} - 25727268 x^{10} - 18424230 x^{9} + 481948659 x^{8} + 393242142 x^{7} - 4171202649 x^{6} - 5609525286 x^{5} + 12169951277 x^{4} + 35299641836 x^{3} + 19566863748 x^{2} - 62055591982 x - 61237993441 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(43985025462771439670319408614090211328=2^{18}\cdot 37^{6}\cdot 97^{3}\cdot 16361^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $123.39$ | 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: | $2, 37, 97, 16361$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{17} + \frac{52731107040192433096441931769926493656830411913151127105830907401771717713356216999047680702}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{16} - \frac{20548329092191807570455757078597309932663020038554425439164316787717240867845158208188225381}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{15} - \frac{105671257217612485766969767642458782396502608226820672070679398861495356297627792334585520020}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{14} + \frac{101958227087279285324631565741977526643448455165281215936912177453836052771165114193155889168}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{13} + \frac{21015908874596374442741290549949795158925647073100788490992569008306501811481362932648410621}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{12} - \frac{115667825830483814256323312414230802161000846133695643313830037000432890203089486616755445180}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{11} + \frac{17391489523979811625742455201996501505767366738489864833260014209526700787483367552687446064}{38230338682146427385405220046784704049888211322101338064657046186394782561141995408388840423} a^{10} + \frac{31815977852857231082450595109361508494338387212096493078566578306958433766938436803213650995}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a^{9} + \frac{18963628424397868126759030542660511995882813121264535861634767651308184937545761610972074376}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a^{8} + \frac{25320684494469964887330513130501302017059292359364167798114049714347839964679362521518291965}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a^{7} + \frac{11427023732719070193171320922715486308561452948746588686625050558570684112161817217187037077}{38230338682146427385405220046784704049888211322101338064657046186394782561141995408388840423} a^{6} - \frac{18022013683484008572421997554978512623079687322319423468647299234913822955827539306038632224}{38230338682146427385405220046784704049888211322101338064657046186394782561141995408388840423} a^{5} + \frac{11644116129188705184771393784037691230821128698009308128086661006318775798354572408241424282}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a^{4} - \frac{410892243401652226245306480862879538168300688056770881740714955226385146905450831282021793}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807} a^{3} + \frac{12918532331009843035285664669163091114757411100582726768396949888106507563513420684948588162}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a^{2} + \frac{56205354190531845570238634835857139269289925780588923832657128255231796525545343131145489991}{114691016046439282156215660140354112149664633966304014193971138559184347683425986225166521269} a - \frac{121048363890704812343792147255289455388225944305064302327981056064804069300534880387846408086}{344073048139317846468646980421062336448993901898912042581913415677553043050277958675499563807}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 4085256787540 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 165888 |
| The 130 conjugacy class representatives for t18n837 are not computed |
| Character table for t18n837 is not computed |
Intermediate fields
| 3.3.148.1, 9.9.53038958912.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ | $18$ | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}$ | R | $18$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.