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Properties

Label 18.0.74673302315...8751.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 31^{9}$
Root discriminant $24.09$
Ramified primes $3, 31$
Class number $3$ (GRH)
Class group $[3]$ (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, -9, 63, -252, 669, -1188, 1792, -1674, 1782, -918, 780, -162, 64, -36, 81, -9, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 15*x^16 - 9*x^15 + 81*x^14 - 36*x^13 + 64*x^12 - 162*x^11 + 780*x^10 - 918*x^9 + 1782*x^8 - 1674*x^7 + 1792*x^6 - 1188*x^5 + 669*x^4 - 252*x^3 + 63*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 15 x^{16} - 9 x^{15} + 81 x^{14} - 36 x^{13} + 64 x^{12} - 162 x^{11} + 780 x^{10} - 918 x^{9} + 1782 x^{8} - 1674 x^{7} + 1792 x^{6} - 1188 x^{5} + 669 x^{4} - 252 x^{3} + 63 x^{2} - 9 x + 1 \)

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:  \(-7467330231571086237938751=-\,3^{24}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.09$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{28} a^{12} + \frac{1}{14} a^{11} - \frac{5}{28} a^{10} - \frac{1}{28} a^{9} - \frac{3}{28} a^{8} + \frac{1}{14} a^{7} - \frac{2}{7} a^{6} + \frac{5}{28} a^{5} + \frac{1}{7} a^{4} - \frac{1}{28} a^{3} + \frac{5}{28} a^{2} - \frac{13}{28} a + \frac{2}{7}$, $\frac{1}{140} a^{13} - \frac{1}{140} a^{12} - \frac{1}{35} a^{11} + \frac{1}{5} a^{10} - \frac{1}{20} a^{9} + \frac{8}{35} a^{8} - \frac{1}{20} a^{7} - \frac{41}{140} a^{6} - \frac{11}{140} a^{5} - \frac{17}{70} a^{4} + \frac{16}{35} a^{3} + \frac{7}{20} a^{2} - \frac{29}{70} a - \frac{31}{140}$, $\frac{1}{140} a^{14} - \frac{1}{140} a^{11} - \frac{1}{35} a^{10} - \frac{3}{28} a^{9} - \frac{5}{28} a^{8} - \frac{3}{140} a^{7} - \frac{11}{70} a^{6} - \frac{1}{7} a^{5} + \frac{3}{28} a^{4} + \frac{19}{70} a^{3} - \frac{19}{140} a^{2} + \frac{3}{20} a - \frac{13}{70}$, $\frac{1}{280} a^{15} - \frac{1}{280} a^{14} + \frac{1}{70} a^{12} - \frac{1}{10} a^{11} + \frac{17}{140} a^{10} - \frac{5}{28} a^{9} - \frac{1}{10} a^{8} - \frac{11}{70} a^{7} - \frac{27}{70} a^{6} - \frac{2}{7} a^{5} - \frac{33}{70} a^{4} + \frac{39}{140} a^{3} + \frac{3}{28} a^{2} - \frac{11}{40} a - \frac{39}{280}$, $\frac{1}{120680} a^{16} - \frac{113}{120680} a^{15} + \frac{1}{12068} a^{14} - \frac{117}{60340} a^{13} + \frac{32}{2155} a^{12} + \frac{1368}{15085} a^{11} + \frac{617}{15085} a^{10} - \frac{10631}{60340} a^{9} - \frac{24}{2155} a^{8} + \frac{3474}{15085} a^{7} + \frac{5849}{60340} a^{6} + \frac{2199}{30170} a^{5} + \frac{13371}{30170} a^{4} + \frac{1989}{30170} a^{3} + \frac{33669}{120680} a^{2} - \frac{16503}{120680} a - \frac{11771}{60340}$, $\frac{1}{9318306200} a^{17} + \frac{24121}{9318306200} a^{16} - \frac{9258169}{9318306200} a^{15} + \frac{23382927}{9318306200} a^{14} + \frac{2197987}{2329576550} a^{13} + \frac{7588801}{4659153100} a^{12} - \frac{383874917}{4659153100} a^{11} + \frac{110525857}{4659153100} a^{10} - \frac{578881193}{4659153100} a^{9} - \frac{174015448}{1164788275} a^{8} - \frac{283947063}{2329576550} a^{7} + \frac{365394488}{1164788275} a^{6} - \frac{98363227}{4659153100} a^{5} - \frac{1076187701}{4659153100} a^{4} - \frac{2382409763}{9318306200} a^{3} - \frac{386272259}{1863661240} a^{2} - \frac{2615983827}{9318306200} a - \frac{565617651}{9318306200}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 42855.4917555 \) (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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, \(\Q(\zeta_{9})^+\), 6.0.29791.1, 6.0.195458751.2 x2, 6.0.195458751.1, 9.3.15832158831.4 x3

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

Sibling fields

Degree 6 sibling: 6.0.195458751.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$