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Properties

Label 18.6.13418074738...3008.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 13^{15}$
Root discriminant $16.96$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, -3, 16, 125, 225, 457, 283, 261, -49, -48, -74, -67, -16, -24, 3, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 3*x^15 - 24*x^14 - 16*x^13 - 67*x^12 - 74*x^11 - 48*x^10 - 49*x^9 + 261*x^8 + 283*x^7 + 457*x^6 + 225*x^5 + 125*x^4 + 16*x^3 - 3*x^2 + 4*x - 1)
 
gp: K = bnfinit(x^18 - x^17 + x^16 + 3*x^15 - 24*x^14 - 16*x^13 - 67*x^12 - 74*x^11 - 48*x^10 - 49*x^9 + 261*x^8 + 283*x^7 + 457*x^6 + 225*x^5 + 125*x^4 + 16*x^3 - 3*x^2 + 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + x^{16} + 3 x^{15} - 24 x^{14} - 16 x^{13} - 67 x^{12} - 74 x^{11} - 48 x^{10} - 49 x^{9} + 261 x^{8} + 283 x^{7} + 457 x^{6} + 225 x^{5} + 125 x^{4} + 16 x^{3} - 3 x^{2} + 4 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13418074738285807403008=2^{18}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.96$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24177} a^{16} - \frac{2101}{24177} a^{15} + \frac{4718}{24177} a^{14} - \frac{3849}{8059} a^{13} - \frac{8633}{24177} a^{12} + \frac{11047}{24177} a^{11} + \frac{2670}{8059} a^{10} - \frac{697}{8059} a^{9} - \frac{4418}{24177} a^{8} - \frac{5578}{24177} a^{7} - \frac{11012}{24177} a^{6} - \frac{3923}{24177} a^{5} - \frac{6536}{24177} a^{4} + \frac{7484}{24177} a^{3} + \frac{8392}{24177} a^{2} + \frac{5875}{24177} a - \frac{329}{8059}$, $\frac{1}{2092426183616199} a^{17} + \frac{38255094076}{2092426183616199} a^{16} - \frac{278883923896906}{2092426183616199} a^{15} - \frac{986050182790204}{2092426183616199} a^{14} + \frac{197159564945522}{2092426183616199} a^{13} + \frac{9031464874519}{22499206275443} a^{12} + \frac{458085727289818}{2092426183616199} a^{11} + \frac{675007529601299}{2092426183616199} a^{10} - \frac{887937781568744}{2092426183616199} a^{9} - \frac{153737918444696}{2092426183616199} a^{8} + \frac{757890355656016}{2092426183616199} a^{7} + \frac{466999671397657}{2092426183616199} a^{6} - \frac{683161031918161}{2092426183616199} a^{5} - \frac{238830874558572}{697475394538733} a^{4} - \frac{513606285784262}{2092426183616199} a^{3} + \frac{42387346106851}{2092426183616199} a^{2} + \frac{173735890120561}{2092426183616199} a + \frac{1022062588153988}{2092426183616199}$

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:  $11$
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:  \( 11167.782400474298 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.1.104.1, \(\Q(\zeta_{13})^+\), 6.2.140608.1, 9.3.32127240704.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$