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Properties

Label 18.0.24719087317...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 5^{9}\cdot 7^{12}\cdot 23^{6}$
Root discriminant $154.35$
Ramified primes $3, 5, 7, 23$
Class number $2277720$ (GRH)
Class group $[6, 6, 63270]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1073741824, 905969664, 767557632, 29097984, 17473536, 42470400, 10282368, -17911008, -394080, 1440664, 645948, -104844, -66495, 3969, 4146, 105, -90, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 90*x^16 + 105*x^15 + 4146*x^14 + 3969*x^13 - 66495*x^12 - 104844*x^11 + 645948*x^10 + 1440664*x^9 - 394080*x^8 - 17911008*x^7 + 10282368*x^6 + 42470400*x^5 + 17473536*x^4 + 29097984*x^3 + 767557632*x^2 + 905969664*x + 1073741824)
 
gp: K = bnfinit(x^18 - 3*x^17 - 90*x^16 + 105*x^15 + 4146*x^14 + 3969*x^13 - 66495*x^12 - 104844*x^11 + 645948*x^10 + 1440664*x^9 - 394080*x^8 - 17911008*x^7 + 10282368*x^6 + 42470400*x^5 + 17473536*x^4 + 29097984*x^3 + 767557632*x^2 + 905969664*x + 1073741824, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 90 x^{16} + 105 x^{15} + 4146 x^{14} + 3969 x^{13} - 66495 x^{12} - 104844 x^{11} + 645948 x^{10} + 1440664 x^{9} - 394080 x^{8} - 17911008 x^{7} + 10282368 x^{6} + 42470400 x^{5} + 17473536 x^{4} + 29097984 x^{3} + 767557632 x^{2} + 905969664 x + 1073741824 \)

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:  \(-2471908731721405920557482056194302734375=-\,3^{31}\cdot 5^{9}\cdot 7^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $154.35$
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:  $3, 5, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{24} a^{9} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{96} a^{10} + \frac{1}{96} a^{9} - \frac{1}{16} a^{8} - \frac{1}{32} a^{7} - \frac{1}{16} a^{6} + \frac{7}{32} a^{5} + \frac{3}{32} a^{4} - \frac{1}{8} a^{3} + \frac{1}{12} a + \frac{1}{3}$, $\frac{1}{192} a^{11} - \frac{1}{192} a^{10} - \frac{1}{48} a^{9} + \frac{3}{64} a^{8} - \frac{1}{16} a^{7} - \frac{5}{64} a^{6} - \frac{15}{64} a^{5} - \frac{5}{32} a^{4} + \frac{1}{8} a^{3} - \frac{11}{24} a^{2} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{1536} a^{12} - \frac{1}{1536} a^{11} + \frac{1}{384} a^{10} + \frac{17}{1536} a^{9} + \frac{7}{128} a^{8} + \frac{3}{512} a^{7} - \frac{15}{512} a^{6} - \frac{1}{256} a^{5} + \frac{7}{32} a^{4} - \frac{17}{192} a^{3} + \frac{1}{96} a^{2} + \frac{5}{24} a - \frac{1}{3}$, $\frac{1}{3072} a^{13} - \frac{1}{3072} a^{12} + \frac{1}{768} a^{11} - \frac{5}{1024} a^{10} + \frac{13}{768} a^{9} - \frac{61}{1024} a^{8} + \frac{17}{1024} a^{7} - \frac{33}{512} a^{6} + \frac{9}{64} a^{5} - \frac{5}{384} a^{4} - \frac{23}{192} a^{3} + \frac{5}{48} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{73728} a^{14} - \frac{11}{73728} a^{13} - \frac{1}{36864} a^{12} - \frac{7}{73728} a^{11} - \frac{11}{36864} a^{10} + \frac{305}{73728} a^{9} - \frac{335}{8192} a^{8} - \frac{45}{2048} a^{7} - \frac{33}{2048} a^{6} + \frac{967}{9216} a^{5} + \frac{7}{2304} a^{4} - \frac{191}{2304} a^{3} + \frac{257}{576} a^{2} + \frac{5}{36} a + \frac{2}{9}$, $\frac{1}{36569088} a^{15} + \frac{47}{12189696} a^{14} + \frac{291}{2031616} a^{13} - \frac{6839}{36569088} a^{12} + \frac{13963}{6094848} a^{11} - \frac{59893}{12189696} a^{10} + \frac{3343}{1179648} a^{9} - \frac{52583}{1015808} a^{8} + \frac{15175}{1015808} a^{7} - \frac{388085}{4571136} a^{6} + \frac{87611}{380928} a^{5} - \frac{19391}{126976} a^{4} + \frac{20671}{285696} a^{3} + \frac{1795}{11904} a^{2} + \frac{17}{1488} a - \frac{56}{279}$, $\frac{1}{585105408} a^{16} - \frac{1}{195035136} a^{15} - \frac{17}{3145728} a^{14} - \frac{11159}{585105408} a^{13} - \frac{4429}{97517568} a^{12} + \frac{303403}{195035136} a^{11} - \frac{830399}{585105408} a^{10} - \frac{217889}{48758784} a^{9} - \frac{522473}{16252928} a^{8} - \frac{945293}{73138176} a^{7} + \frac{48503}{6094848} a^{6} + \frac{1510451}{6094848} a^{5} - \frac{1039925}{4571136} a^{4} - \frac{7903}{190464} a^{3} - \frac{3401}{23808} a^{2} - \frac{583}{2232} a + \frac{13}{93}$, $\frac{1}{9161269299690852129368526181976725917567811584} a^{17} - \frac{5019400461211752456308297482932823171}{9161269299690852129368526181976725917567811584} a^{16} - \frac{566982902290179685955660126810909135}{1526878216615142021561421030329454319594635264} a^{15} + \frac{9924723643235758886256920792386764930851}{3053756433230284043122842060658908639189270528} a^{14} - \frac{341385990042712358432600410773319506334759}{4580634649845426064684263090988362958783905792} a^{13} - \frac{1465855022306967213795453617125916375226751}{9161269299690852129368526181976725917567811584} a^{12} + \frac{23802285581332501369872393460196001152448449}{9161269299690852129368526181976725917567811584} a^{11} + \frac{405350274191361649588017872624162161457023}{763439108307571010780710515164727159797317632} a^{10} - \frac{17074087346867764716864785297126525400992177}{2290317324922713032342131545494181479391952896} a^{9} + \frac{1468802203355547139351895287395514449094579}{1145158662461356516171065772747090739695976448} a^{8} - \frac{7313200468383605387743046914990107792961147}{286289665615339129042766443186772684923994112} a^{7} - \frac{10705574595793455525812149903562153396372557}{95429888538446376347588814395590894974664704} a^{6} - \frac{877670521899752135834815659107313374310343}{23857472134611594086897203598897723743666176} a^{5} + \frac{220944315624652540951749513120018089456147}{8946552050479347782586451349586646403874816} a^{4} + \frac{11149817301731186529375510454412536709813}{1118319006309918472823306418698330800484352} a^{3} - \frac{5725426785349112316054273740965708356731}{34947468947184952275728325584322837515136} a^{2} + \frac{32169879376276445123309673448242873253}{364036134866509919538836724836696224116} a - \frac{125643560252742571225934619140692225454}{273027101149882439654127543627522168087}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{63270}$, which has order $2277720$ (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:  \( 237885673.5279126 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.3.621.1, 3.3.3969.1, 6.0.144615375.1, 6.0.5907360375.1, 9.9.20539533187176381.1

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 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$