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Properties

Label 18.18.1062971709...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 19^{6}$
Root discriminant $40.98$
Ramified primes $2, 3, 5, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-125, 750, 450, -6175, -570, 19710, 3176, -29310, -9144, 20685, 9132, -6570, -3614, 870, 612, -35, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 35*x^15 + 612*x^14 + 870*x^13 - 3614*x^12 - 6570*x^11 + 9132*x^10 + 20685*x^9 - 9144*x^8 - 29310*x^7 + 3176*x^6 + 19710*x^5 - 570*x^4 - 6175*x^3 + 450*x^2 + 750*x - 125)
 
gp: K = bnfinit(x^18 - 42*x^16 - 35*x^15 + 612*x^14 + 870*x^13 - 3614*x^12 - 6570*x^11 + 9132*x^10 + 20685*x^9 - 9144*x^8 - 29310*x^7 + 3176*x^6 + 19710*x^5 - 570*x^4 - 6175*x^3 + 450*x^2 + 750*x - 125, 1)
 

Normalized defining polynomial

\( x^{18} - 42 x^{16} - 35 x^{15} + 612 x^{14} + 870 x^{13} - 3614 x^{12} - 6570 x^{11} + 9132 x^{10} + 20685 x^{9} - 9144 x^{8} - 29310 x^{7} + 3176 x^{6} + 19710 x^{5} - 570 x^{4} - 6175 x^{3} + 450 x^{2} + 750 x - 125 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(106297170913362278088000000000=2^{12}\cdot 3^{24}\cdot 5^{9}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.98$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{35} a^{15} - \frac{12}{35} a^{13} + \frac{3}{7} a^{12} + \frac{1}{5} a^{11} - \frac{3}{7} a^{10} + \frac{6}{35} a^{9} + \frac{3}{7} a^{8} - \frac{13}{35} a^{7} + \frac{3}{7} a^{6} + \frac{1}{35} a^{5} - \frac{1}{7} a^{4} + \frac{1}{35} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{175} a^{16} - \frac{12}{175} a^{14} + \frac{3}{35} a^{13} + \frac{11}{25} a^{12} + \frac{4}{35} a^{11} - \frac{29}{175} a^{10} - \frac{4}{35} a^{9} - \frac{13}{175} a^{8} + \frac{17}{35} a^{7} - \frac{34}{175} a^{6} + \frac{13}{35} a^{5} - \frac{69}{175} a^{4} + \frac{2}{35} a^{3} + \frac{17}{35} a^{2} - \frac{2}{7} a$, $\frac{1}{25010359768015825} a^{17} - \frac{19142063071707}{25010359768015825} a^{16} + \frac{280753850309028}{25010359768015825} a^{15} - \frac{1184648843218606}{25010359768015825} a^{14} + \frac{12354763192299217}{25010359768015825} a^{13} + \frac{11322188015840641}{25010359768015825} a^{12} + \frac{8315436914690536}{25010359768015825} a^{11} - \frac{644096987994411}{3572908538287975} a^{10} - \frac{6514173571990158}{25010359768015825} a^{9} - \frac{4712217360274079}{25010359768015825} a^{8} - \frac{10217933113356099}{25010359768015825} a^{7} + \frac{3653665847005443}{25010359768015825} a^{6} - \frac{93549039727287}{3572908538287975} a^{5} - \frac{457759607358416}{3572908538287975} a^{4} + \frac{770375039485101}{5002071953603165} a^{3} - \frac{302495622929460}{1000414390720633} a^{2} + \frac{213298051483674}{1000414390720633} a - \frac{94763558146312}{1000414390720633}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 441529173.991 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, 6.6.4737042000.1, 6.6.296065125.3, 6.6.722000.1, 6.6.13122000.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$