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Properties

Label 18.18.1020942315...1584.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{31}\cdot 7^{9}$
Root discriminant $27.86$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![487, 654, -6102, 1416, 18096, -12441, -20619, 20628, 9870, -15058, -1083, 5580, -675, -1056, 240, 93, -27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487)
 
gp: K = bnfinit(x^18 - 3*x^17 - 27*x^16 + 93*x^15 + 240*x^14 - 1056*x^13 - 675*x^12 + 5580*x^11 - 1083*x^10 - 15058*x^9 + 9870*x^8 + 20628*x^7 - 20619*x^6 - 12441*x^5 + 18096*x^4 + 1416*x^3 - 6102*x^2 + 654*x + 487, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 27 x^{16} + 93 x^{15} + 240 x^{14} - 1056 x^{13} - 675 x^{12} + 5580 x^{11} - 1083 x^{10} - 15058 x^{9} + 9870 x^{8} + 20628 x^{7} - 20619 x^{6} - 12441 x^{5} + 18096 x^{4} + 1416 x^{3} - 6102 x^{2} + 654 x + 487 \)

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:  \(102094231502838405721411584=2^{12}\cdot 3^{31}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.86$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{445} a^{16} - \frac{37}{89} a^{15} + \frac{2}{445} a^{14} - \frac{11}{445} a^{13} - \frac{14}{89} a^{12} - \frac{15}{89} a^{11} + \frac{33}{89} a^{9} + \frac{37}{445} a^{8} + \frac{2}{5} a^{7} + \frac{117}{445} a^{6} + \frac{46}{445} a^{5} - \frac{38}{445} a^{4} + \frac{39}{445} a^{3} + \frac{191}{445} a^{2} - \frac{22}{89} a + \frac{47}{445}$, $\frac{1}{713403238525} a^{17} - \frac{593223989}{713403238525} a^{16} - \frac{8233897298}{713403238525} a^{15} - \frac{158042164929}{713403238525} a^{14} + \frac{223171034209}{713403238525} a^{13} + \frac{6626973894}{142680647705} a^{12} + \frac{775282589}{1603153345} a^{11} + \frac{10861242508}{28536129541} a^{10} + \frac{32226621467}{713403238525} a^{9} - \frac{91136511}{1603153345} a^{8} + \frac{41814917763}{142680647705} a^{7} - \frac{98973818062}{713403238525} a^{6} - \frac{84676301362}{713403238525} a^{5} + \frac{49926545241}{713403238525} a^{4} - \frac{18638122556}{142680647705} a^{3} - \frac{239664829004}{713403238525} a^{2} + \frac{143097891767}{713403238525} a - \frac{29880597}{8015766725}$

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:  \( 15149384.4329 \) (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

\(\Q(\sqrt{21}) \), 3.3.756.1 x3, \(\Q(\zeta_{9})^+\), 6.6.12002256.1, 6.6.108020304.1 x2, 6.6.6751269.1, 9.9.314987206464.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.108020304.1
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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$