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Properties

Label 18.0.31616164379...7879.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 13^{15}$
Root discriminant $56.23$
Ramified primes $3, 13$
Class number $1404$ (GRH)
Class group $[3, 468]$ (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![123904, 563136, 946224, 730068, 260853, -140562, -251841, -85296, 34182, 28042, 2442, -3222, -1032, 192, 162, -6, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 - 6*x^15 + 162*x^14 + 192*x^13 - 1032*x^12 - 3222*x^11 + 2442*x^10 + 28042*x^9 + 34182*x^8 - 85296*x^7 - 251841*x^6 - 140562*x^5 + 260853*x^4 + 730068*x^3 + 946224*x^2 + 563136*x + 123904)
 
gp: K = bnfinit(x^18 - 15*x^16 - 6*x^15 + 162*x^14 + 192*x^13 - 1032*x^12 - 3222*x^11 + 2442*x^10 + 28042*x^9 + 34182*x^8 - 85296*x^7 - 251841*x^6 - 140562*x^5 + 260853*x^4 + 730068*x^3 + 946224*x^2 + 563136*x + 123904, 1)
 

Normalized defining polynomial

\( x^{18} - 15 x^{16} - 6 x^{15} + 162 x^{14} + 192 x^{13} - 1032 x^{12} - 3222 x^{11} + 2442 x^{10} + 28042 x^{9} + 34182 x^{8} - 85296 x^{7} - 251841 x^{6} - 140562 x^{5} + 260853 x^{4} + 730068 x^{3} + 946224 x^{2} + 563136 x + 123904 \)

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:  \(-31616164379840194492072800177879=-\,3^{31}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.23$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{36} a^{9} - \frac{1}{12} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{12} a^{3} - \frac{1}{12} a + \frac{2}{9}$, $\frac{1}{36} a^{10} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{6} a^{5} - \frac{1}{12} a^{4} - \frac{1}{12} a^{2} + \frac{17}{36} a$, $\frac{1}{72} a^{11} - \frac{1}{24} a^{8} - \frac{1}{6} a^{6} + \frac{5}{24} a^{5} - \frac{1}{6} a^{3} + \frac{35}{72} a^{2} + \frac{1}{3}$, $\frac{1}{144} a^{12} - \frac{1}{144} a^{11} + \frac{1}{144} a^{9} + \frac{1}{48} a^{8} + \frac{1}{12} a^{7} - \frac{7}{48} a^{6} + \frac{7}{48} a^{5} - \frac{1}{6} a^{4} - \frac{37}{144} a^{3} + \frac{1}{144} a^{2} + \frac{1}{12} a - \frac{4}{9}$, $\frac{1}{144} a^{13} - \frac{1}{144} a^{11} + \frac{1}{144} a^{10} + \frac{5}{48} a^{8} + \frac{1}{48} a^{7} - \frac{1}{6} a^{6} - \frac{1}{48} a^{5} - \frac{13}{144} a^{4} - \frac{1}{6} a^{3} + \frac{13}{144} a^{2} + \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{288} a^{14} - \frac{1}{72} a^{10} + \frac{1}{16} a^{8} + \frac{1}{24} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{5}{24} a^{4} - \frac{1}{6} a^{3} - \frac{11}{32} a^{2} - \frac{35}{72} a + \frac{1}{3}$, $\frac{1}{12384} a^{15} - \frac{19}{12384} a^{14} + \frac{5}{2064} a^{13} + \frac{1}{3096} a^{12} - \frac{13}{2064} a^{11} + \frac{29}{6192} a^{10} - \frac{7}{2064} a^{9} + \frac{29}{344} a^{8} + \frac{67}{688} a^{7} - \frac{205}{3096} a^{6} - \frac{325}{6192} a^{5} + \frac{325}{2064} a^{4} + \frac{4505}{12384} a^{3} - \frac{25}{96} a^{2} + \frac{137}{3096} a - \frac{19}{129}$, $\frac{1}{148608} a^{16} - \frac{1}{148608} a^{15} - \frac{1}{2752} a^{14} - \frac{61}{18576} a^{13} + \frac{169}{74304} a^{12} + \frac{5}{24768} a^{11} + \frac{329}{74304} a^{10} - \frac{25}{9288} a^{9} + \frac{853}{8256} a^{8} + \frac{547}{18576} a^{7} + \frac{9323}{74304} a^{6} - \frac{257}{2752} a^{5} + \frac{11741}{148608} a^{4} + \frac{4249}{148608} a^{3} - \frac{3853}{12384} a^{2} + \frac{1165}{4644} a - \frac{364}{1161}$, $\frac{1}{3436276923890658121049856} a^{17} - \frac{228829472149653947}{572712820648443020174976} a^{16} - \frac{21515862088301815777}{3436276923890658121049856} a^{15} - \frac{808122273073773758227}{1718138461945329060524928} a^{14} + \frac{1491270124429793709847}{572712820648443020174976} a^{13} - \frac{2191783360128930071953}{859069230972664530262464} a^{12} - \frac{1992674128074679808183}{859069230972664530262464} a^{11} - \frac{505812966661453815737}{190904273549481006724992} a^{10} - \frac{4470514817182547234879}{1718138461945329060524928} a^{9} - \frac{1022587680735476557207}{29120990880429306110592} a^{8} + \frac{11252017110046472305541}{572712820648443020174976} a^{7} - \frac{168339063661734858689855}{859069230972664530262464} a^{6} - \frac{807087623848311951909661}{3436276923890658121049856} a^{5} + \frac{32852326026622913583061}{286356410324221510087488} a^{4} - \frac{1435132662084473800738817}{3436276923890658121049856} a^{3} + \frac{160866963215212599778813}{429534615486332265131232} a^{2} - \frac{8290350866163508129829}{23863034193685125840624} a + \frac{6161613785149604655292}{13422956733947883285351}$

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

Class group and class number

$C_{3}\times C_{468}$, which has order $1404$ (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:  \( 63087923.79573219 \) (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

\(\Q(\sqrt{-39}) \), 3.3.13689.1, 3.1.4563.1, 6.0.812017791.2, 6.0.7308160119.1, 9.3.69259433447763.8

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
3Data not computed
13Data not computed