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Properties

Label 18.0.44664131011...6579.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 19^{9}$
Root discriminant $15.95$
Ramified primes $7, 19$
Class number $1$
Class group Trivial
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, 1, 12, 8, 84, 59, 216, 169, 305, 118, 204, 33, 71, -22, 37, -11, 7, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 7*x^16 - 11*x^15 + 37*x^14 - 22*x^13 + 71*x^12 + 33*x^11 + 204*x^10 + 118*x^9 + 305*x^8 + 169*x^7 + 216*x^6 + 59*x^5 + 84*x^4 + 8*x^3 + 12*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 7 x^{16} - 11 x^{15} + 37 x^{14} - 22 x^{13} + 71 x^{12} + 33 x^{11} + 204 x^{10} + 118 x^{9} + 305 x^{8} + 169 x^{7} + 216 x^{6} + 59 x^{5} + 84 x^{4} + 8 x^{3} + 12 x^{2} + x + 1 \)

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:  \(-4466413101188628826579=-\,7^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.95$
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:  $7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{15} - \frac{6}{13} a^{14} - \frac{6}{13} a^{13} + \frac{1}{13} a^{12} + \frac{2}{13} a^{11} + \frac{2}{13} a^{10} + \frac{6}{13} a^{9} - \frac{6}{13} a^{8} + \frac{1}{13} a^{7} + \frac{4}{13} a^{6} + \frac{2}{13} a^{5} + \frac{6}{13} a^{2} + \frac{4}{13}$, $\frac{1}{390581495667853139} a^{17} - \frac{1445514591833235}{390581495667853139} a^{16} + \frac{127684316295014980}{390581495667853139} a^{15} - \frac{32747011183029448}{390581495667853139} a^{14} + \frac{69765186745153966}{390581495667853139} a^{13} - \frac{16228948298466863}{390581495667853139} a^{12} - \frac{189888712865148035}{390581495667853139} a^{11} + \frac{181965739436189637}{390581495667853139} a^{10} + \frac{13795726450731115}{390581495667853139} a^{9} - \frac{95194290284976635}{390581495667853139} a^{8} - \frac{41770204369500135}{390581495667853139} a^{7} + \frac{179384390028789259}{390581495667853139} a^{6} - \frac{8791174782984872}{30044730435988703} a^{5} + \frac{5100222016840991}{30044730435988703} a^{4} - \frac{177661710698938135}{390581495667853139} a^{3} + \frac{5525867689208321}{30044730435988703} a^{2} - \frac{164956053636172951}{390581495667853139} a - \frac{12163829605837008}{30044730435988703}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1060.85049787 \)
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

\(\Q(\sqrt{-19}) \), 3.1.931.1 x3, \(\Q(\zeta_{7})^+\), 6.0.16468459.2, 6.0.16468459.1, 6.0.336091.1 x2, 9.3.806954491.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.336091.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$