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Properties

Label 18.18.1416642414...8736.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 3^{27}\cdot 7^{12}$
Root discriminant $53.78$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, -4596, -11610, 38114, 64119, -115686, -96853, 145404, 53535, -84708, -10056, 24222, -360, -3486, 321, 238, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 33*x^16 + 238*x^15 + 321*x^14 - 3486*x^13 - 360*x^12 + 24222*x^11 - 10056*x^10 - 84708*x^9 + 53535*x^8 + 145404*x^7 - 96853*x^6 - 115686*x^5 + 64119*x^4 + 38114*x^3 - 11610*x^2 - 4596*x - 71)
 
gp: K = bnfinit(x^18 - 6*x^17 - 33*x^16 + 238*x^15 + 321*x^14 - 3486*x^13 - 360*x^12 + 24222*x^11 - 10056*x^10 - 84708*x^9 + 53535*x^8 + 145404*x^7 - 96853*x^6 - 115686*x^5 + 64119*x^4 + 38114*x^3 - 11610*x^2 - 4596*x - 71, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 33 x^{16} + 238 x^{15} + 321 x^{14} - 3486 x^{13} - 360 x^{12} + 24222 x^{11} - 10056 x^{10} - 84708 x^{9} + 53535 x^{8} + 145404 x^{7} - 96853 x^{6} - 115686 x^{5} + 64119 x^{4} + 38114 x^{3} - 11610 x^{2} - 4596 x - 71 \)

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:  \(14166424145858741301073711988736=2^{27}\cdot 3^{27}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.78$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(323,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(11,·)$, $\chi_{504}(337,·)$, $\chi_{504}(275,·)$, $\chi_{504}(25,·)$, $\chi_{504}(347,·)$, $\chi_{504}(491,·)$, $\chi_{504}(289,·)$, $\chi_{504}(155,·)$, $\chi_{504}(169,·)$, $\chi_{504}(107,·)$, $\chi_{504}(179,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{573531463012} a^{16} + \frac{19170608041}{286765731506} a^{15} - \frac{44507220027}{573531463012} a^{14} + \frac{3054893791}{286765731506} a^{13} + \frac{10586483009}{143382865753} a^{12} - \frac{2345649787}{286765731506} a^{11} + \frac{95979907729}{573531463012} a^{10} - \frac{14871133259}{286765731506} a^{9} + \frac{56324393061}{573531463012} a^{8} - \frac{15037611953}{286765731506} a^{7} + \frac{17643651921}{143382865753} a^{6} + \frac{4930139023}{286765731506} a^{5} - \frac{66849995417}{573531463012} a^{4} - \frac{137765424701}{286765731506} a^{3} - \frac{248086571915}{573531463012} a^{2} - \frac{47587659966}{143382865753} a - \frac{53910210429}{573531463012}$, $\frac{1}{44125565918952206308} a^{17} - \frac{5674529}{22062782959476103154} a^{16} - \frac{602401436088625975}{11031391479738051577} a^{15} - \frac{1311800010204671415}{11031391479738051577} a^{14} + \frac{2951721304933790113}{44125565918952206308} a^{13} + \frac{1163931158161106172}{11031391479738051577} a^{12} + \frac{1852769137520198802}{11031391479738051577} a^{11} - \frac{1823921197817683295}{22062782959476103154} a^{10} - \frac{1328107092870992425}{11031391479738051577} a^{9} + \frac{122584543249769927}{11031391479738051577} a^{8} - \frac{162542258428188617}{11031391479738051577} a^{7} + \frac{2367084600449536879}{22062782959476103154} a^{6} + \frac{4553563477497868375}{44125565918952206308} a^{5} + \frac{2142582079574140069}{22062782959476103154} a^{4} - \frac{2835405341032201962}{11031391479738051577} a^{3} - \frac{4999501644401867588}{11031391479738051577} a^{2} + \frac{1673435740929749511}{22062782959476103154} a - \frac{1380552258441031413}{11031391479738051577}$

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:  \( 6797133055.24 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.6.10077696.1, 6.6.33191424.1, 6.6.24196548096.1, 6.6.24196548096.2, 9.9.62523502209.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
7Data not computed