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Properties

Label 18.0.37815202643...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{45}\cdot 5^{9}$
Root discriminant $64.55$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3\times D_9$ (as 18T50)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2656, -7848, 19764, -28872, 46152, -52002, 55053, -41922, 36450, -15620, 11979, -2916, 1764, -306, 243, -18, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 18*x^15 + 243*x^14 - 306*x^13 + 1764*x^12 - 2916*x^11 + 11979*x^10 - 15620*x^9 + 36450*x^8 - 41922*x^7 + 55053*x^6 - 52002*x^5 + 46152*x^4 - 28872*x^3 + 19764*x^2 - 7848*x + 2656)
 
gp: K = bnfinit(x^18 + 18*x^16 - 18*x^15 + 243*x^14 - 306*x^13 + 1764*x^12 - 2916*x^11 + 11979*x^10 - 15620*x^9 + 36450*x^8 - 41922*x^7 + 55053*x^6 - 52002*x^5 + 46152*x^4 - 28872*x^3 + 19764*x^2 - 7848*x + 2656, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} - 18 x^{15} + 243 x^{14} - 306 x^{13} + 1764 x^{12} - 2916 x^{11} + 11979 x^{10} - 15620 x^{9} + 36450 x^{8} - 41922 x^{7} + 55053 x^{6} - 52002 x^{5} + 46152 x^{4} - 28872 x^{3} + 19764 x^{2} - 7848 x + 2656 \)

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:  \(-378152026438506713426304000000000=-\,2^{16}\cdot 3^{45}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.55$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} + \frac{1}{64} a^{10} + \frac{1}{64} a^{9} + \frac{3}{64} a^{8} + \frac{5}{64} a^{7} + \frac{3}{64} a^{6} + \frac{3}{32} a^{5} - \frac{1}{16} a^{4} + \frac{7}{16} a^{3} + \frac{7}{16} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{128} a^{14} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{3}{64} a^{9} + \frac{3}{64} a^{8} + \frac{5}{64} a^{7} - \frac{9}{128} a^{6} + \frac{9}{64} a^{5} + \frac{3}{16} a^{4} + \frac{7}{16} a^{3} - \frac{13}{32} a^{2} + \frac{5}{16} a + \frac{1}{4}$, $\frac{1}{256} a^{15} - \frac{1}{256} a^{14} - \frac{1}{128} a^{13} - \frac{3}{128} a^{11} - \frac{1}{128} a^{10} + \frac{1}{64} a^{9} - \frac{15}{256} a^{7} - \frac{9}{256} a^{6} - \frac{13}{128} a^{5} - \frac{1}{8} a^{4} + \frac{21}{64} a^{3} - \frac{13}{64} a^{2} + \frac{11}{32} a - \frac{1}{8}$, $\frac{1}{512} a^{16} + \frac{1}{512} a^{14} - \frac{1}{256} a^{13} - \frac{7}{256} a^{12} + \frac{1}{256} a^{10} + \frac{3}{128} a^{9} - \frac{7}{512} a^{8} + \frac{1}{16} a^{7} - \frac{23}{512} a^{6} + \frac{15}{256} a^{5} - \frac{19}{128} a^{4} + \frac{7}{16} a^{3} - \frac{59}{128} a^{2} - \frac{5}{64} a + \frac{3}{16}$, $\frac{1}{26627064556052997143552} a^{17} + \frac{7508644524189408533}{26627064556052997143552} a^{16} - \frac{35826187311284599389}{26627064556052997143552} a^{15} + \frac{45528594494414389973}{26627064556052997143552} a^{14} - \frac{16593971734459548549}{6656766139013249285888} a^{13} - \frac{404460755919832253659}{13313532278026498571776} a^{12} + \frac{282168108152791958803}{13313532278026498571776} a^{11} - \frac{47312052257312760859}{13313532278026498571776} a^{10} - \frac{125190289915490865827}{26627064556052997143552} a^{9} - \frac{1624178158252853127843}{26627064556052997143552} a^{8} - \frac{3041415443179060211173}{26627064556052997143552} a^{7} - \frac{1785979683719060016067}{26627064556052997143552} a^{6} - \frac{559231786521834860217}{13313532278026498571776} a^{5} - \frac{1544860885518183376311}{6656766139013249285888} a^{4} + \frac{1915246250272994025159}{6656766139013249285888} a^{3} + \frac{690100075440970270985}{6656766139013249285888} a^{2} + \frac{716457486381459700581}{3328383069506624642944} a - \frac{117266735837170622231}{832095767376656160736}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $8$
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:  \( 369617552674 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_9$ (as 18T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $S_3\times D_9$
Character table for $S_3\times D_9$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.324.1, 6.0.39366000.3

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
5Data not computed