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Properties

Label 18.0.61599256770...4751.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,13^{12}\cdot 31^{9}$
Root discriminant $30.78$
Ramified primes $13, 31$
Class number $12$ (GRH)
Class group $[2, 6]$ (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![39944, -56832, 44674, -59237, 61742, -25667, 36528, -12017, 13950, -3596, 4781, -921, 1085, -138, 193, -8, 22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 22*x^16 - 8*x^15 + 193*x^14 - 138*x^13 + 1085*x^12 - 921*x^11 + 4781*x^10 - 3596*x^9 + 13950*x^8 - 12017*x^7 + 36528*x^6 - 25667*x^5 + 61742*x^4 - 59237*x^3 + 44674*x^2 - 56832*x + 39944)
 
gp: K = bnfinit(x^18 + 22*x^16 - 8*x^15 + 193*x^14 - 138*x^13 + 1085*x^12 - 921*x^11 + 4781*x^10 - 3596*x^9 + 13950*x^8 - 12017*x^7 + 36528*x^6 - 25667*x^5 + 61742*x^4 - 59237*x^3 + 44674*x^2 - 56832*x + 39944, 1)
 

Normalized defining polynomial

\( x^{18} + 22 x^{16} - 8 x^{15} + 193 x^{14} - 138 x^{13} + 1085 x^{12} - 921 x^{11} + 4781 x^{10} - 3596 x^{9} + 13950 x^{8} - 12017 x^{7} + 36528 x^{6} - 25667 x^{5} + 61742 x^{4} - 59237 x^{3} + 44674 x^{2} - 56832 x + 39944 \)

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:  \(-615992567705547976896144751=-\,13^{12}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.78$
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:  $13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{15} - \frac{1}{8} a^{14} + \frac{1}{40} a^{13} - \frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{1}{8} a^{10} + \frac{1}{20} a^{9} - \frac{3}{40} a^{8} + \frac{19}{40} a^{7} + \frac{3}{8} a^{6} - \frac{2}{5} a^{5} - \frac{3}{20} a^{4} + \frac{9}{40} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{7634680} a^{16} + \frac{1959}{763468} a^{15} + \frac{92859}{1908670} a^{14} - \frac{556599}{7634680} a^{13} - \frac{608359}{3817340} a^{12} - \frac{86141}{1526936} a^{11} - \frac{1150553}{7634680} a^{10} - \frac{1079723}{7634680} a^{9} + \frac{512467}{3817340} a^{8} + \frac{8331}{24628} a^{7} + \frac{3090719}{7634680} a^{6} + \frac{460041}{1908670} a^{5} + \frac{1015419}{7634680} a^{4} - \frac{33}{1048} a^{3} - \frac{1218653}{3817340} a^{2} + \frac{428163}{954335} a + \frac{8203}{190867}$, $\frac{1}{28747934229673362918475527202520} a^{17} - \frac{1116333544592261542430191}{28747934229673362918475527202520} a^{16} + \frac{164152138180380498869763722077}{28747934229673362918475527202520} a^{15} - \frac{85901026173766517767429735051}{2874793422967336291847552720252} a^{14} + \frac{1531938595831547991164312468871}{14373967114836681459237763601260} a^{13} - \frac{4404133523964198093467965058411}{28747934229673362918475527202520} a^{12} - \frac{3037152477822038572021757438623}{14373967114836681459237763601260} a^{11} - \frac{849328724647926127437203802367}{5749586845934672583695105440504} a^{10} - \frac{5989534888970169649233576697571}{28747934229673362918475527202520} a^{9} + \frac{3175495208576986243166418530243}{28747934229673362918475527202520} a^{8} - \frac{4447539118967822590767980095241}{14373967114836681459237763601260} a^{7} + \frac{148684375682409473082575849010}{718698355741834072961888180063} a^{6} + \frac{66205221527979219564698035969}{927352717086237513499210554920} a^{5} - \frac{350634878873281381456675757597}{718698355741834072961888180063} a^{4} - \frac{793395682838380852230298481263}{7186983557418340729618881800630} a^{3} + \frac{345313030978813471802246196959}{718698355741834072961888180063} a^{2} + \frac{537427688416883522078765363573}{1437396711483668145923776360126} a - \frac{345263906356451795132927714}{719705944063523005169124955}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $8$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 138932.419034 \) (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

\(\Q(\sqrt{-31}) \), 3.1.31.1 x3, 3.3.169.1, 6.0.29791.1, 6.0.850860751.1, 6.0.850860751.2 x2, 9.3.143795466919.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$