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Properties

Label 18.18.2637116370...1277.1
Degree $18$
Signature $[18, 0]$
Discriminant $13^{15}\cdot 61^{6}$
Root discriminant $33.37$
Ramified primes $13, 61$
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, 128, 195, -1621, -3305, 4714, 14804, 2957, -15065, -7031, 7196, 3762, -2020, -876, 346, 89, -31, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 31*x^16 + 89*x^15 + 346*x^14 - 876*x^13 - 2020*x^12 + 3762*x^11 + 7196*x^10 - 7031*x^9 - 15065*x^8 + 2957*x^7 + 14804*x^6 + 4714*x^5 - 3305*x^4 - 1621*x^3 + 195*x^2 + 128*x - 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 31*x^16 + 89*x^15 + 346*x^14 - 876*x^13 - 2020*x^12 + 3762*x^11 + 7196*x^10 - 7031*x^9 - 15065*x^8 + 2957*x^7 + 14804*x^6 + 4714*x^5 - 3305*x^4 - 1621*x^3 + 195*x^2 + 128*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 31 x^{16} + 89 x^{15} + 346 x^{14} - 876 x^{13} - 2020 x^{12} + 3762 x^{11} + 7196 x^{10} - 7031 x^{9} - 15065 x^{8} + 2957 x^{7} + 14804 x^{6} + 4714 x^{5} - 3305 x^{4} - 1621 x^{3} + 195 x^{2} + 128 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2637116370088050448669881277=13^{15}\cdot 61^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.37$
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:  $13, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{372} a^{16} - \frac{19}{93} a^{15} + \frac{20}{93} a^{14} + \frac{121}{372} a^{13} - \frac{23}{372} a^{12} - \frac{61}{186} a^{11} - \frac{41}{124} a^{10} + \frac{131}{372} a^{9} + \frac{11}{31} a^{8} - \frac{83}{186} a^{7} - \frac{67}{372} a^{6} + \frac{53}{186} a^{5} + \frac{89}{372} a^{4} - \frac{17}{372} a^{3} - \frac{125}{372} a^{2} - \frac{45}{124} a - \frac{13}{372}$, $\frac{1}{873555687700308} a^{17} - \frac{702787312703}{873555687700308} a^{16} - \frac{10117477591795}{145592614616718} a^{15} + \frac{43287548519243}{873555687700308} a^{14} - \frac{37306695925577}{145592614616718} a^{13} + \frac{970845307405}{291185229233436} a^{12} - \frac{103433377361167}{873555687700308} a^{11} + \frac{140526805337023}{436777843850154} a^{10} + \frac{201993911833657}{873555687700308} a^{9} - \frac{36720525146332}{218388921925077} a^{8} + \frac{11280723663145}{291185229233436} a^{7} + \frac{178245896159321}{873555687700308} a^{6} + \frac{125444837994811}{873555687700308} a^{5} + \frac{114193068019567}{436777843850154} a^{4} + \frac{21974979155381}{145592614616718} a^{3} - \frac{1313508740611}{218388921925077} a^{2} - \frac{109710709524887}{436777843850154} a - \frac{63432840032513}{873555687700308}$

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:  \( 66833881.58911677 \) (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

\(\Q(\sqrt{13}) \), 3.3.10309.1, 3.3.169.1, 6.6.1381581253.1, \(\Q(\zeta_{13})^+\), 9.9.1095593933629.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
13Data not computed
$61$61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
61.6.3.1$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$