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Properties

Label 18.0.13445405384...0192.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{43}$
Root discriminant $21.90$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![175, -810, 2223, -4125, 6111, -7497, 8241, -8145, 7425, -6118, 4572, -3051, 1815, -945, 423, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 423*x^14 - 945*x^13 + 1815*x^12 - 3051*x^11 + 4572*x^10 - 6118*x^9 + 7425*x^8 - 8145*x^7 + 8241*x^6 - 7497*x^5 + 6111*x^4 - 4125*x^3 + 2223*x^2 - 810*x + 175)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 423*x^14 - 945*x^13 + 1815*x^12 - 3051*x^11 + 4572*x^10 - 6118*x^9 + 7425*x^8 - 8145*x^7 + 8241*x^6 - 7497*x^5 + 6111*x^4 - 4125*x^3 + 2223*x^2 - 810*x + 175, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 423 x^{14} - 945 x^{13} + 1815 x^{12} - 3051 x^{11} + 4572 x^{10} - 6118 x^{9} + 7425 x^{8} - 8145 x^{7} + 8241 x^{6} - 7497 x^{5} + 6111 x^{4} - 4125 x^{3} + 2223 x^{2} - 810 x + 175 \)

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:  \(-1344540538448023869960192=-\,2^{12}\cdot 3^{43}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.90$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{3}{10} a^{6} + \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{14} + \frac{3}{10} a^{10} + \frac{1}{5} a^{9} + \frac{3}{10} a^{8} + \frac{2}{5} a^{7} - \frac{1}{10} a^{4} - \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{15} + \frac{3}{10} a^{11} + \frac{1}{5} a^{10} + \frac{3}{10} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{5} - \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a$, $\frac{1}{382550} a^{16} - \frac{4}{191275} a^{15} - \frac{4903}{382550} a^{14} - \frac{271}{27325} a^{13} + \frac{1017}{191275} a^{12} + \frac{18377}{191275} a^{11} + \frac{23777}{54650} a^{10} - \frac{74591}{191275} a^{9} - \frac{12409}{76510} a^{8} - \frac{6547}{27325} a^{7} + \frac{55141}{191275} a^{6} - \frac{11522}{27325} a^{5} + \frac{111093}{382550} a^{4} - \frac{58032}{191275} a^{3} - \frac{70699}{191275} a^{2} + \frac{56241}{191275} a + \frac{1754}{5465}$, $\frac{1}{148811950} a^{17} + \frac{93}{74405975} a^{16} - \frac{47197}{29762390} a^{15} + \frac{402377}{74405975} a^{14} + \frac{3894853}{148811950} a^{13} + \frac{3662058}{14881195} a^{12} - \frac{521267}{14881195} a^{11} + \frac{1188797}{74405975} a^{10} - \frac{20663763}{148811950} a^{9} - \frac{29782294}{74405975} a^{8} - \frac{3587831}{29762390} a^{7} + \frac{164684}{14881195} a^{6} + \frac{34312318}{74405975} a^{5} + \frac{29003879}{74405975} a^{4} - \frac{27434262}{74405975} a^{3} + \frac{1126134}{2976239} a^{2} - \frac{42132837}{148811950} a + \frac{143536}{2125885}$

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:  \( -\frac{910398}{10629425} a^{17} + \frac{7738383}{10629425} a^{16} - \frac{36420108}{10629425} a^{15} + \frac{4735326}{425177} a^{14} - \frac{301676358}{10629425} a^{13} + \frac{634880454}{10629425} a^{12} - \frac{1153637526}{10629425} a^{11} + \frac{1835920812}{10629425} a^{10} - \frac{2606567848}{10629425} a^{9} + \frac{3291047799}{10629425} a^{8} - \frac{3780361518}{10629425} a^{7} + \frac{3912652242}{10629425} a^{6} - \frac{3757401006}{10629425} a^{5} + \frac{3174333099}{10629425} a^{4} - \frac{2325846462}{10629425} a^{3} + \frac{1299594732}{10629425} a^{2} - \frac{509004312}{10629425} a + \frac{20790429}{2125885} \) (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:  \( 337293.4983275989 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.972.2 x3, 6.0.2834352.2, 9.3.669462604992.4

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed