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Properties

Label 18.18.1773946662...1309.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{27}\cdot 7^{17}$
Root discriminant $32.65$
Ramified primes $3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_9:C_3$ (as 18T14)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, 126, 945, 3731, 7770, 5796, -8008, -18417, -5481, 12726, 8505, -3402, -3339, 378, 567, -14, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 42*x^16 - 14*x^15 + 567*x^14 + 378*x^13 - 3339*x^12 - 3402*x^11 + 8505*x^10 + 12726*x^9 - 5481*x^8 - 18417*x^7 - 8008*x^6 + 5796*x^5 + 7770*x^4 + 3731*x^3 + 945*x^2 + 126*x + 7)
 
gp: K = bnfinit(x^18 - 42*x^16 - 14*x^15 + 567*x^14 + 378*x^13 - 3339*x^12 - 3402*x^11 + 8505*x^10 + 12726*x^9 - 5481*x^8 - 18417*x^7 - 8008*x^6 + 5796*x^5 + 7770*x^4 + 3731*x^3 + 945*x^2 + 126*x + 7, 1)
 

Normalized defining polynomial

\( x^{18} - 42 x^{16} - 14 x^{15} + 567 x^{14} + 378 x^{13} - 3339 x^{12} - 3402 x^{11} + 8505 x^{10} + 12726 x^{9} - 5481 x^{8} - 18417 x^{7} - 8008 x^{6} + 5796 x^{5} + 7770 x^{4} + 3731 x^{3} + 945 x^{2} + 126 x + 7 \)

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:  \(1773946662392078824692561309=3^{27}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.65$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{43} a^{17} + \frac{21}{43} a^{16} + \frac{12}{43} a^{15} - \frac{20}{43} a^{14} + \frac{18}{43} a^{13} - \frac{18}{43} a^{12} - \frac{19}{43} a^{11} - \frac{17}{43} a^{10} + \frac{21}{43} a^{9} + \frac{9}{43} a^{8} - \frac{3}{43} a^{7} + \frac{10}{43} a^{6} - \frac{15}{43} a^{5} + \frac{20}{43} a^{4} + \frac{20}{43} a^{3} - \frac{20}{43} a^{2} + \frac{9}{43} a + \frac{14}{43}$

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:  \( 59140273.8142 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_9:C_3$ (as 18T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times C_9:C_3$
Character table for $C_2\times C_9:C_3$ is not computed

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 9.9.3063651608241.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
7Data not computed