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Properties

Label 18.18.1192171739...5301.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{45}\cdot 7^{9}$
Root discriminant $41.24$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1511, -720, 20736, 1200, -69120, -540, 88704, 90, -57024, -5, 20592, 0, -4368, 0, 540, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511)
 
gp: K = bnfinit(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 5*x^9 - 57024*x^8 + 90*x^7 + 88704*x^6 - 540*x^5 - 69120*x^4 + 1200*x^3 + 20736*x^2 - 720*x - 1511, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 5 x^{9} - 57024 x^{8} + 90 x^{7} + 88704 x^{6} - 540 x^{5} - 69120 x^{4} + 1200 x^{3} + 20736 x^{2} - 720 x - 1511 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(119217173915258668597396055301=3^{45}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.24$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(146,·)$, $\chi_{189}(83,·)$, $\chi_{189}(20,·)$, $\chi_{189}(85,·)$, $\chi_{189}(22,·)$, $\chi_{189}(167,·)$, $\chi_{189}(104,·)$, $\chi_{189}(41,·)$, $\chi_{189}(106,·)$, $\chi_{189}(43,·)$, $\chi_{189}(169,·)$, $\chi_{189}(148,·)$, $\chi_{189}(188,·)$, $\chi_{189}(125,·)$, $\chi_{189}(62,·)$, $\chi_{189}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} - \frac{1}{17} a^{7} + \frac{6}{17} a^{5} - \frac{2}{17} a^{3} + \frac{8}{17} a + \frac{6}{17}$, $\frac{1}{17} a^{10} - \frac{1}{17} a^{8} + \frac{6}{17} a^{6} - \frac{2}{17} a^{4} + \frac{8}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{17} a^{11} + \frac{5}{17} a^{7} + \frac{4}{17} a^{5} + \frac{6}{17} a^{3} + \frac{6}{17} a^{2} + \frac{8}{17} a + \frac{6}{17}$, $\frac{1}{17} a^{12} + \frac{5}{17} a^{8} + \frac{4}{17} a^{6} + \frac{6}{17} a^{4} + \frac{6}{17} a^{3} + \frac{8}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{17} a^{13} - \frac{8}{17} a^{7} - \frac{7}{17} a^{5} + \frac{6}{17} a^{4} + \frac{1}{17} a^{3} + \frac{6}{17} a^{2} - \frac{6}{17} a + \frac{4}{17}$, $\frac{1}{8279} a^{14} + \frac{91}{8279} a^{13} - \frac{28}{8279} a^{12} + \frac{69}{8279} a^{11} - \frac{179}{8279} a^{10} - \frac{203}{8279} a^{9} - \frac{1193}{8279} a^{8} - \frac{3019}{8279} a^{7} + \frac{1782}{8279} a^{6} - \frac{910}{8279} a^{5} + \frac{2981}{8279} a^{4} - \frac{3960}{8279} a^{3} - \frac{2708}{8279} a^{2} + \frac{42}{487} a + \frac{3640}{8279}$, $\frac{1}{8279} a^{15} - \frac{30}{8279} a^{13} + \frac{182}{8279} a^{12} - \frac{127}{8279} a^{11} + \frac{15}{8279} a^{10} + \frac{235}{8279} a^{9} + \frac{1813}{8279} a^{8} + \frac{2330}{8279} a^{7} + \frac{3969}{8279} a^{6} + \frac{566}{8279} a^{5} - \frac{1050}{8279} a^{4} + \frac{1168}{8279} a^{3} - \frac{3663}{8279} a^{2} - \frac{1433}{8279} a + \frac{1868}{8279}$, $\frac{1}{8279} a^{16} - \frac{10}{8279} a^{13} + \frac{7}{8279} a^{12} + \frac{137}{8279} a^{11} + \frac{222}{8279} a^{10} + \frac{106}{8279} a^{9} - \frac{831}{8279} a^{8} - \frac{2837}{8279} a^{7} - \frac{1005}{8279} a^{6} + \frac{2331}{8279} a^{5} + \frac{1964}{8279} a^{4} + \frac{44}{487} a^{3} - \frac{3292}{8279} a^{2} - \frac{1062}{8279} a - \frac{3784}{8279}$, $\frac{1}{8279} a^{17} - \frac{57}{8279} a^{13} - \frac{143}{8279} a^{12} - \frac{62}{8279} a^{11} - \frac{223}{8279} a^{10} + \frac{61}{8279} a^{9} + \frac{330}{8279} a^{8} + \frac{113}{487} a^{7} + \frac{240}{487} a^{6} - \frac{3240}{8279} a^{5} - \frac{3045}{8279} a^{4} + \frac{2399}{8279} a^{3} - \frac{3305}{8279} a^{2} + \frac{434}{8279} a + \frac{2797}{8279}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
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:  \( 550518949.413 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), 6.6.6751269.1, \(\Q(\zeta_{27})^+\)

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
7Data not computed