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Properties

Label 18.0.39531097362...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{6}$
Root discriminant $18.00$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -66, 192, -306, 339, -150, 54, 16, -138, 90, -33, -12, 36, -6, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 6*x^15 + 36*x^14 - 12*x^13 - 33*x^12 + 90*x^11 - 138*x^10 + 16*x^9 + 54*x^8 - 150*x^7 + 339*x^6 - 306*x^5 + 192*x^4 - 66*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 + 6*x^16 - 6*x^15 + 36*x^14 - 12*x^13 - 33*x^12 + 90*x^11 - 138*x^10 + 16*x^9 + 54*x^8 - 150*x^7 + 339*x^6 - 306*x^5 + 192*x^4 - 66*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 6 x^{15} + 36 x^{14} - 12 x^{13} - 33 x^{12} + 90 x^{11} - 138 x^{10} + 16 x^{9} + 54 x^{8} - 150 x^{7} + 339 x^{6} - 306 x^{5} + 192 x^{4} - 66 x^{3} + 18 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-39531097362172608000000=-\,2^{12}\cdot 3^{31}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.00$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{18} a^{9} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a - \frac{1}{18}$, $\frac{1}{18} a^{10} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{18} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{18} a^{2} - \frac{1}{3}$, $\frac{1}{18} a^{12} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} - \frac{1}{36} a^{10} - \frac{1}{36} a^{9} - \frac{5}{12} a^{8} + \frac{5}{12} a^{7} + \frac{1}{4} a^{6} - \frac{5}{12} a^{5} - \frac{1}{36} a^{4} + \frac{1}{4} a^{3} + \frac{1}{36} a^{2} + \frac{1}{9} a - \frac{11}{36}$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{36} a^{10} - \frac{1}{36} a^{9} + \frac{5}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} - \frac{1}{36} a^{5} - \frac{1}{12} a^{4} - \frac{5}{36} a^{3} + \frac{1}{9} a^{2} + \frac{13}{36} a - \frac{7}{18}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{12} + \frac{1}{18} a^{6} + \frac{7}{36} a^{3} - \frac{1}{4}$, $\frac{1}{8244} a^{16} - \frac{26}{2061} a^{15} + \frac{1}{1374} a^{14} + \frac{73}{8244} a^{13} - \frac{22}{2061} a^{12} + \frac{1}{2061} a^{11} - \frac{34}{2061} a^{10} - \frac{44}{2061} a^{9} + \frac{56}{229} a^{8} + \frac{829}{4122} a^{7} - \frac{763}{2061} a^{6} - \frac{197}{687} a^{5} - \frac{1501}{8244} a^{4} + \frac{1000}{2061} a^{3} - \frac{1301}{4122} a^{2} + \frac{2437}{8244} a - \frac{943}{2061}$, $\frac{1}{22283532} a^{17} - \frac{1199}{22283532} a^{16} + \frac{6563}{11141766} a^{15} - \frac{27637}{11141766} a^{14} + \frac{160885}{22283532} a^{13} - \frac{393467}{22283532} a^{12} - \frac{201227}{22283532} a^{11} + \frac{358279}{22283532} a^{10} + \frac{605791}{22283532} a^{9} - \frac{5204617}{22283532} a^{8} + \frac{7482731}{22283532} a^{7} - \frac{6122017}{22283532} a^{6} + \frac{644537}{5570883} a^{5} - \frac{567706}{5570883} a^{4} - \frac{40723}{97308} a^{3} - \frac{46709}{1310796} a^{2} - \frac{1510985}{11141766} a + \frac{676490}{5570883}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{6542}{8109} a^{17} + \frac{811}{3604} a^{16} + \frac{159623}{32436} a^{15} - \frac{18703}{5406} a^{14} + \frac{152411}{5406} a^{13} - \frac{60649}{32436} a^{12} - \frac{859697}{32436} a^{11} + \frac{705841}{10812} a^{10} - \frac{3034741}{32436} a^{9} - \frac{381203}{32436} a^{8} + \frac{138133}{3604} a^{7} - \frac{3588653}{32436} a^{6} + \frac{2628743}{10812} a^{5} - \frac{490231}{2703} a^{4} + \frac{1774793}{16218} a^{3} - \frac{50285}{1908} a^{2} + \frac{110615}{10812} a - \frac{18893}{8109} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 49776.35407190846 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1620.1, 3.1.108.1 x3, 6.0.7873200.1, 6.0.34992.1, 9.3.114791256000.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$