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Properties

Label 18.0.20221511129...4067.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 41^{9}$
Root discriminant $42.47$
Ramified primes $3, 41$
Class number $152$ (GRH)
Class group $[2, 2, 38]$ (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![836587, -261210, -605661, 227274, 250608, -51519, -52743, 2436, 34635, -6847, -4575, -267, 3156, -1599, 267, 6, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 6*x^15 + 267*x^14 - 1599*x^13 + 3156*x^12 - 267*x^11 - 4575*x^10 - 6847*x^9 + 34635*x^8 + 2436*x^7 - 52743*x^6 - 51519*x^5 + 250608*x^4 + 227274*x^3 - 605661*x^2 - 261210*x + 836587)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 6*x^15 + 267*x^14 - 1599*x^13 + 3156*x^12 - 267*x^11 - 4575*x^10 - 6847*x^9 + 34635*x^8 + 2436*x^7 - 52743*x^6 - 51519*x^5 + 250608*x^4 + 227274*x^3 - 605661*x^2 - 261210*x + 836587, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 6 x^{15} + 267 x^{14} - 1599 x^{13} + 3156 x^{12} - 267 x^{11} - 4575 x^{10} - 6847 x^{9} + 34635 x^{8} + 2436 x^{7} - 52743 x^{6} - 51519 x^{5} + 250608 x^{4} + 227274 x^{3} - 605661 x^{2} - 261210 x + 836587 \)

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:  \(-202215111299126210886918044067=-\,3^{31}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.47$
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:  $3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{153} a^{14} - \frac{5}{153} a^{13} + \frac{2}{153} a^{12} - \frac{1}{153} a^{11} - \frac{7}{51} a^{10} + \frac{2}{17} a^{9} + \frac{2}{51} a^{8} - \frac{1}{51} a^{7} + \frac{1}{17} a^{6} + \frac{2}{9} a^{5} - \frac{59}{153} a^{4} - \frac{13}{153} a^{3} + \frac{62}{153} a^{2} + \frac{8}{17} a - \frac{1}{3}$, $\frac{1}{10863} a^{15} + \frac{1}{639} a^{14} + \frac{100}{3621} a^{13} + \frac{1}{1207} a^{12} + \frac{586}{10863} a^{11} + \frac{1375}{10863} a^{10} + \frac{997}{10863} a^{9} + \frac{502}{3621} a^{8} - \frac{461}{3621} a^{7} - \frac{1043}{10863} a^{6} + \frac{1964}{10863} a^{5} - \frac{743}{3621} a^{4} - \frac{409}{3621} a^{3} + \frac{4411}{10863} a^{2} - \frac{1238}{10863} a - \frac{238}{639}$, $\frac{1}{619191} a^{16} + \frac{13}{619191} a^{15} + \frac{10}{68799} a^{14} - \frac{18586}{619191} a^{13} - \frac{25081}{619191} a^{12} + \frac{3395}{68799} a^{11} - \frac{36524}{619191} a^{10} + \frac{11860}{619191} a^{9} + \frac{3508}{68799} a^{8} + \frac{102682}{619191} a^{7} - \frac{9626}{619191} a^{6} + \frac{10713}{22933} a^{5} - \frac{7921}{32589} a^{4} - \frac{253168}{619191} a^{3} - \frac{936}{22933} a^{2} + \frac{89656}{619191} a - \frac{3166}{36423}$, $\frac{1}{14666994566190475534663410824195496291} a^{17} + \frac{618503783140343825136108225434}{862764386246498560862553577893852723} a^{16} + \frac{8179477581257896219990432940162}{257315694143692553239708961827991163} a^{15} + \frac{22897727527433443280950292957042558}{14666994566190475534663410824195496291} a^{14} - \frac{212943735126335150467938940723314421}{14666994566190475534663410824195496291} a^{13} + \frac{171104486739581049069539013518705315}{4888998188730158511554470274731832097} a^{12} - \frac{737755302761903928413122279987987811}{14666994566190475534663410824195496291} a^{11} - \frac{1609742718414872801690704806520563167}{14666994566190475534663410824195496291} a^{10} - \frac{64593301323191473340718969238179556}{543222020970017612394941141636870233} a^{9} + \frac{98732753648360221262773405392018361}{14666994566190475534663410824195496291} a^{8} + \frac{2097673508914814620276519019852099491}{14666994566190475534663410824195496291} a^{7} - \frac{99050087783119413364014711599499397}{4888998188730158511554470274731832097} a^{6} - \frac{3512694559648457593294254067068106300}{14666994566190475534663410824195496291} a^{5} - \frac{6634007450424955679712136947120042400}{14666994566190475534663410824195496291} a^{4} + \frac{8880475998982683180616396072864526}{4888998188730158511554470274731832097} a^{3} + \frac{10986889892590532533910558461653274}{50750846249794032991914916346697219} a^{2} + \frac{1961963824034784298778340769827704131}{14666994566190475534663410824195496291} a + \frac{3383235015790805893480133338580135}{95862709582944284540283730877094747}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{38}$, which has order $152$ (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:  \( 269015.233286 \) (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

\(\Q(\sqrt{-123}) \), 3.1.1107.1 x3, \(\Q(\zeta_{9})^+\), 6.0.150730227.1, 6.0.1356572043.2 x2, 6.0.1356572043.1, 9.3.988941019347.3 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$41$41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
41.6.3.2$x^{6} - 1681 x^{2} + 895973$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$