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Properties

Label 18.0.43398229994...5887.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 17^{9}\cdot 31^{9}$
Root discriminant $84.00$
Ramified primes $7, 17, 31$
Class number $13122$ (GRH)
Class group $[3, 3, 9, 9, 18]$ (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![2299968, 0, 6075300, -293353, 6472620, -360675, 3604930, -164045, 1167950, -36104, 230055, -4025, 27573, -210, 1925, -4, 70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 70*x^16 - 4*x^15 + 1925*x^14 - 210*x^13 + 27573*x^12 - 4025*x^11 + 230055*x^10 - 36104*x^9 + 1167950*x^8 - 164045*x^7 + 3604930*x^6 - 360675*x^5 + 6472620*x^4 - 293353*x^3 + 6075300*x^2 + 2299968)
 
gp: K = bnfinit(x^18 + 70*x^16 - 4*x^15 + 1925*x^14 - 210*x^13 + 27573*x^12 - 4025*x^11 + 230055*x^10 - 36104*x^9 + 1167950*x^8 - 164045*x^7 + 3604930*x^6 - 360675*x^5 + 6472620*x^4 - 293353*x^3 + 6075300*x^2 + 2299968, 1)
 

Normalized defining polynomial

\( x^{18} + 70 x^{16} - 4 x^{15} + 1925 x^{14} - 210 x^{13} + 27573 x^{12} - 4025 x^{11} + 230055 x^{10} - 36104 x^{9} + 1167950 x^{8} - 164045 x^{7} + 3604930 x^{6} - 360675 x^{5} + 6472620 x^{4} - 293353 x^{3} + 6075300 x^{2} + 2299968 \)

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:  \(-43398229994269610052656606538315887=-\,7^{12}\cdot 17^{9}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.00$
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:  $7, 17, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{348} a^{14} - \frac{7}{174} a^{13} - \frac{2}{87} a^{12} + \frac{71}{348} a^{11} + \frac{2}{29} a^{10} - \frac{59}{348} a^{9} + \frac{1}{348} a^{8} - \frac{107}{348} a^{7} - \frac{13}{29} a^{6} + \frac{21}{58} a^{5} - \frac{67}{348} a^{4} - \frac{5}{29} a^{3} - \frac{91}{348} a^{2} + \frac{83}{348} a + \frac{7}{29}$, $\frac{1}{54636} a^{15} + \frac{67}{54636} a^{14} + \frac{163}{54636} a^{13} - \frac{1289}{27318} a^{12} - \frac{483}{4553} a^{11} - \frac{463}{1884} a^{10} + \frac{2068}{13659} a^{9} - \frac{5333}{54636} a^{8} + \frac{4019}{18212} a^{7} - \frac{3967}{18212} a^{6} + \frac{3557}{13659} a^{5} + \frac{3363}{9106} a^{4} - \frac{27049}{54636} a^{3} + \frac{3920}{13659} a^{2} - \frac{13}{58} a + \frac{857}{4553}$, $\frac{1}{20952523548} a^{16} + \frac{9379}{1904774868} a^{15} - \frac{2095048}{5238130887} a^{14} - \frac{150913639}{3492087258} a^{13} + \frac{7968547}{1904774868} a^{12} + \frac{1236640039}{5238130887} a^{11} + \frac{2443406423}{20952523548} a^{10} + \frac{4781350487}{20952523548} a^{9} + \frac{2406628555}{20952523548} a^{8} + \frac{264322813}{3492087258} a^{7} + \frac{1145955809}{5238130887} a^{6} - \frac{8593222585}{20952523548} a^{5} + \frac{1597518260}{5238130887} a^{4} - \frac{1680286637}{20952523548} a^{3} + \frac{213720910}{476193717} a^{2} - \frac{7928609}{1164029086} a - \frac{20600829}{52910413}$, $\frac{1}{877527777258270929107152} a^{17} - \frac{4493599979}{906536959977552612714} a^{16} - \frac{1933396579295456537}{438763888629135464553576} a^{15} + \frac{625107062395669450}{630407885961401529531} a^{14} + \frac{6786262654989917996203}{79775252478024629918832} a^{13} + \frac{4016652163170582281483}{438763888629135464553576} a^{12} - \frac{112463107384086732851395}{877527777258270929107152} a^{11} + \frac{33434136567174329535539}{877527777258270929107152} a^{10} + \frac{205737108475561400413567}{877527777258270929107152} a^{9} + \frac{2419168688133226598363}{24375771590507525808532} a^{8} + \frac{44810706450503180459917}{438763888629135464553576} a^{7} - \frac{434780044234183665002713}{877527777258270929107152} a^{6} + \frac{175611140895864309084817}{438763888629135464553576} a^{5} - \frac{3062204979912076963235}{877527777258270929107152} a^{4} + \frac{246476749433455375753}{4985953279876539369927} a^{3} + \frac{68836546987163799578957}{292509259086090309702384} a^{2} - \frac{384931665576190875722}{1661984426625513123309} a - \frac{300337927816768981}{50363164443197367373}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{18}$, which has order $13122$ (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:  \( 1201631.239 \) (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

\(\Q(\sqrt{-527}) \), 3.1.527.1 x3, \(\Q(\zeta_{7})^+\), 6.0.146363183.2, 6.0.351418002383.1 x2, 6.0.351418002383.2, 9.3.17219482116767.4 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$