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Properties

Label 18.18.2034121533...9184.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{21}\cdot 7^{15}$
Root discriminant $28.95$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -27, 243, -765, -84, 4158, -4241, -6174, 10665, 1316, -8874, 2646, 2437, -1323, -126, 178, -18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 18*x^16 + 178*x^15 - 126*x^14 - 1323*x^13 + 2437*x^12 + 2646*x^11 - 8874*x^10 + 1316*x^9 + 10665*x^8 - 6174*x^7 - 4241*x^6 + 4158*x^5 - 84*x^4 - 765*x^3 + 243*x^2 - 27*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 - 18*x^16 + 178*x^15 - 126*x^14 - 1323*x^13 + 2437*x^12 + 2646*x^11 - 8874*x^10 + 1316*x^9 + 10665*x^8 - 6174*x^7 - 4241*x^6 + 4158*x^5 - 84*x^4 - 765*x^3 + 243*x^2 - 27*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 18 x^{16} + 178 x^{15} - 126 x^{14} - 1323 x^{13} + 2437 x^{12} + 2646 x^{11} - 8874 x^{10} + 1316 x^{9} + 10665 x^{8} - 6174 x^{7} - 4241 x^{6} + 4158 x^{5} - 84 x^{4} - 765 x^{3} + 243 x^{2} - 27 x + 1 \)

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:  \(203412153331596396123869184=2^{12}\cdot 3^{21}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.95$
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 over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{16} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{157696981106818085} a^{17} + \frac{1745285859036472}{157696981106818085} a^{16} - \frac{10077962379774386}{157696981106818085} a^{15} - \frac{70682963858248184}{157696981106818085} a^{14} + \frac{10614083029818256}{157696981106818085} a^{13} - \frac{54447318681811563}{157696981106818085} a^{12} + \frac{20204041698890811}{157696981106818085} a^{11} + \frac{76219428119689271}{157696981106818085} a^{10} + \frac{16092472781310263}{157696981106818085} a^{9} - \frac{25678576267964801}{157696981106818085} a^{8} - \frac{77396383557634412}{157696981106818085} a^{7} - \frac{2448562140751291}{31539396221363617} a^{6} - \frac{10646525247676599}{157696981106818085} a^{5} - \frac{7186074568540490}{31539396221363617} a^{4} + \frac{68066917032892159}{157696981106818085} a^{3} + \frac{52355348767006797}{157696981106818085} a^{2} + \frac{700275840136333}{157696981106818085} a - \frac{7749310029014120}{31539396221363617}$

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:  \( 21322924.4726 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.3.756.1 x3, \(\Q(\zeta_{7})^+\), 6.6.12002256.1, 6.6.588110544.1 x2, \(\Q(\zeta_{21})^+\), 9.9.1037426999616.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: 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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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

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
2Data not computed
3Data not computed
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$