Normalized defining polynomial
\( x^{18} + 444 x^{16} + 78588 x^{14} + 7202864 x^{12} + 374901168 x^{10} + 11442434112 x^{8} + 203442083456 x^{6} + 2004094947840 x^{4} + 9552891988224 x^{2} + 14745141983744 \)
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-17304111519831580155910725615927954423788077056=-\,2^{27}\cdot 3^{24}\cdot 37^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $370.50$ | 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: | $2, 3, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2664=2^{3}\cdot 3^{2}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2664}(1,·)$, $\chi_{2664}(67,·)$, $\chi_{2664}(1921,·)$, $\chi_{2664}(1681,·)$, $\chi_{2664}(2371,·)$, $\chi_{2664}(691,·)$, $\chi_{2664}(625,·)$, $\chi_{2664}(601,·)$, $\chi_{2664}(2395,·)$, $\chi_{2664}(1825,·)$, $\chi_{2664}(739,·)$, $\chi_{2664}(1561,·)$, $\chi_{2664}(433,·)$, $\chi_{2664}(1003,·)$, $\chi_{2664}(1009,·)$, $\chi_{2664}(307,·)$, $\chi_{2664}(835,·)$, $\chi_{2664}(1915,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{352} a^{10} + \frac{1}{11}$, $\frac{1}{352} a^{11} + \frac{1}{11} a$, $\frac{1}{704} a^{12} + \frac{1}{22} a^{2}$, $\frac{1}{16192} a^{13} - \frac{3}{4048} a^{11} + \frac{1}{92} a^{9} - \frac{1}{184} a^{7} + \frac{5}{46} a^{5} - \frac{27}{253} a^{3} - \frac{61}{253} a$, $\frac{1}{48802688} a^{14} - \frac{6351}{12200672} a^{12} + \frac{333}{3050168} a^{10} - \frac{82}{34661} a^{8} + \frac{5}{138644} a^{6} + \frac{49055}{762542} a^{4} - \frac{143811}{762542} a^{2} - \frac{7292}{16577}$, $\frac{1}{59197660544} a^{15} - \frac{840045}{29598830272} a^{13} - \frac{647677}{3699853784} a^{11} - \frac{9315423}{336350344} a^{9} - \frac{3173737}{168175172} a^{7} - \frac{12814359}{462481723} a^{5} - \frac{110263315}{924963446} a^{3} + \frac{219272582}{462481723} a$, $\frac{1}{3623553127323996452095252016384} a^{16} - \frac{355379817134896890691}{78772894072260792436853304704} a^{14} + \frac{59742388597649752348908815}{452944140915499556511906502048} a^{12} + \frac{23702026700697384931538707}{19693223518065198109213326176} a^{10} - \frac{90531005261846131766171843}{5147092510403404051271664796} a^{8} - \frac{646113011843290987125461852}{14154504403609361140997078189} a^{6} + \frac{901285605525476751315407498}{14154504403609361140997078189} a^{4} + \frac{12833933713792428454310894}{488086358745150384172313041} a^{2} - \frac{80105542119214426216987}{507348091458810750958711}$, $\frac{1}{3623553127323996452095252016384} a^{17} - \frac{256966273577863189}{226472070457749778255953251024} a^{15} + \frac{12225691119785268484370679}{452944140915499556511906502048} a^{13} + \frac{22719405127169208665007949}{19693223518065198109213326176} a^{11} + \frac{35906999320229182377524456}{1286773127600851012817916199} a^{9} + \frac{908502881462879007842628741}{113236035228874889127976625512} a^{7} + \frac{3842187527378853726935764559}{56618017614437444563988312756} a^{5} - \frac{105116111103832202146386198}{488086358745150384172313041} a^{3} - \frac{5346097314978544323486237550}{14154504403609361140997078189} a$
Class group and class number
$C_{3}\times C_{72954210}$, which has order $218862630$ (assuming GRH)
Unit group
| Rank: | $8$ | 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: | \( 282437461.5224087 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-74}) \), 3.3.1369.1, 6.0.35504105984.1, 9.9.1866675593471230161.2 |
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 | R | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ | $18$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.9.12.2 | $x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ |
| 3.9.12.2 | $x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$ | $3$ | $3$ | $12$ | $C_9$ | $[2]^{3}$ | |
| 37 | Data not computed | ||||||