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Properties

Label 18.0.17548929959...6875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{9}\cdot 37^{17}$
Root discriminant $117.25$
Ramified primes $3, 5, 37$
Class number $261364$ (GRH)
Class group $[38, 6878]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![639840259, -40222117, 190294745, -165861112, 58201854, -81796806, 27823148, -14267296, 8052725, -1571735, 1562253, -140828, 143253, -6469, 6336, -132, 131, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 131*x^16 - 132*x^15 + 6336*x^14 - 6469*x^13 + 143253*x^12 - 140828*x^11 + 1562253*x^10 - 1571735*x^9 + 8052725*x^8 - 14267296*x^7 + 27823148*x^6 - 81796806*x^5 + 58201854*x^4 - 165861112*x^3 + 190294745*x^2 - 40222117*x + 639840259)
 
gp: K = bnfinit(x^18 - x^17 + 131*x^16 - 132*x^15 + 6336*x^14 - 6469*x^13 + 143253*x^12 - 140828*x^11 + 1562253*x^10 - 1571735*x^9 + 8052725*x^8 - 14267296*x^7 + 27823148*x^6 - 81796806*x^5 + 58201854*x^4 - 165861112*x^3 + 190294745*x^2 - 40222117*x + 639840259, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 131 x^{16} - 132 x^{15} + 6336 x^{14} - 6469 x^{13} + 143253 x^{12} - 140828 x^{11} + 1562253 x^{10} - 1571735 x^{9} + 8052725 x^{8} - 14267296 x^{7} + 27823148 x^{6} - 81796806 x^{5} + 58201854 x^{4} - 165861112 x^{3} + 190294745 x^{2} - 40222117 x + 639840259 \)

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:  \(-17548929959529003835585593295529296875=-\,3^{9}\cdot 5^{9}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.25$
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:  $3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(555=3\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{555}(256,·)$, $\chi_{555}(1,·)$, $\chi_{555}(451,·)$, $\chi_{555}(271,·)$, $\chi_{555}(16,·)$, $\chi_{555}(211,·)$, $\chi_{555}(344,·)$, $\chi_{555}(539,·)$, $\chi_{555}(284,·)$, $\chi_{555}(104,·)$, $\chi_{555}(554,·)$, $\chi_{555}(299,·)$, $\chi_{555}(46,·)$, $\chi_{555}(434,·)$, $\chi_{555}(181,·)$, $\chi_{555}(374,·)$, $\chi_{555}(121,·)$, $\chi_{555}(509,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{43} a^{14} + \frac{12}{43} a^{13} - \frac{13}{43} a^{12} + \frac{21}{43} a^{11} - \frac{12}{43} a^{10} + \frac{20}{43} a^{9} + \frac{2}{43} a^{8} - \frac{18}{43} a^{7} + \frac{21}{43} a^{6} - \frac{15}{43} a^{5} + \frac{21}{43} a^{4} + \frac{7}{43} a^{3} - \frac{21}{43} a^{2} - \frac{13}{43} a - \frac{16}{43}$, $\frac{1}{43} a^{15} + \frac{15}{43} a^{13} + \frac{5}{43} a^{12} - \frac{6}{43} a^{11} - \frac{8}{43} a^{10} + \frac{20}{43} a^{9} + \frac{1}{43} a^{8} - \frac{21}{43} a^{7} - \frac{9}{43} a^{6} - \frac{14}{43} a^{5} + \frac{13}{43} a^{4} - \frac{19}{43} a^{3} - \frac{19}{43} a^{2} + \frac{11}{43} a + \frac{20}{43}$, $\frac{1}{8213} a^{16} + \frac{60}{8213} a^{15} - \frac{29}{8213} a^{14} - \frac{182}{8213} a^{13} - \frac{3047}{8213} a^{12} - \frac{3055}{8213} a^{11} + \frac{1401}{8213} a^{10} - \frac{3033}{8213} a^{9} + \frac{3950}{8213} a^{8} + \frac{3350}{8213} a^{7} - \frac{2338}{8213} a^{6} - \frac{683}{8213} a^{5} + \frac{1299}{8213} a^{4} + \frac{3693}{8213} a^{3} + \frac{3665}{8213} a^{2} + \frac{3574}{8213} a + \frac{21}{43}$, $\frac{1}{861419340369076709815030640002664753627179772221100693572014898911473} a^{17} + \frac{6046483515028307772321076722441468606567192833974670838772016121}{861419340369076709815030640002664753627179772221100693572014898911473} a^{16} + \frac{5114584703840406881069692241826255879619676709396325030876944991405}{861419340369076709815030640002664753627179772221100693572014898911473} a^{15} - \frac{5923697200884849164923818270142383570892968565158997144365857650952}{861419340369076709815030640002664753627179772221100693572014898911473} a^{14} - \frac{3800641113875712405728352494133267124085588048219959430720994993678}{861419340369076709815030640002664753627179772221100693572014898911473} a^{13} + \frac{21373249733326741976448166776662032043361577446878797204694523913755}{861419340369076709815030640002664753627179772221100693572014898911473} a^{12} + \frac{400199322885662825190951573684689488040879520015119808948223844261883}{861419340369076709815030640002664753627179772221100693572014898911473} a^{11} + \frac{173378427302679102279364434841136033474644016067280265356140477241660}{861419340369076709815030640002664753627179772221100693572014898911473} a^{10} + \frac{19536381195458612599098292105673237549525570066907619686262827539441}{861419340369076709815030640002664753627179772221100693572014898911473} a^{9} - \frac{119703814469125594511621237125782409921851347196956198206509336450715}{861419340369076709815030640002664753627179772221100693572014898911473} a^{8} - \frac{374780004336453799515554082982537877069795295223267181518233508677600}{861419340369076709815030640002664753627179772221100693572014898911473} a^{7} + \frac{416632797088271843820126231716734897503875844335430771872813744955}{4510048902455899004267176125668401851451202995921993160062905229903} a^{6} - \frac{109945158769887944285566176870745652950154764925509539059204538399690}{861419340369076709815030640002664753627179772221100693572014898911473} a^{5} - \frac{351333576047217162609867803514439581779260204923278442903998455606243}{861419340369076709815030640002664753627179772221100693572014898911473} a^{4} - \frac{383205241086714382379919343080101308168150042877696925885270748164194}{861419340369076709815030640002664753627179772221100693572014898911473} a^{3} - \frac{341456297405974136404581661288918511900617253704985473141691589685245}{861419340369076709815030640002664753627179772221100693572014898911473} a^{2} + \frac{119642412754431342543685404850183367780507172924073702804853371635649}{861419340369076709815030640002664753627179772221100693572014898911473} a + \frac{1055137954564329304315278369803447890354031954230671261873932976854}{4510048902455899004267176125668401851451202995921993160062905229903}$

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

Class group and class number

$C_{38}\times C_{6878}$, which has order $261364$ (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:  \( 409151.3102125697 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-555}) \), 3.3.1369.1, 6.0.234035854875.1, 9.9.3512479453921.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R R $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R $18$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
5Data not computed
37Data not computed