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Properties

Label 18.0.24283924700...5603.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 37^{16}$
Root discriminant $42.91$
Ramified primes $3, 37$
Class number $171$ (GRH)
Class group $[171]$ (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![1, -7, 56, -97, 528, -129, 5069, 1505, 5540, -623, 3519, -218, 999, -76, 201, -6, 17, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 17*x^16 - 6*x^15 + 201*x^14 - 76*x^13 + 999*x^12 - 218*x^11 + 3519*x^10 - 623*x^9 + 5540*x^8 + 1505*x^7 + 5069*x^6 - 129*x^5 + 528*x^4 - 97*x^3 + 56*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 17 x^{16} - 6 x^{15} + 201 x^{14} - 76 x^{13} + 999 x^{12} - 218 x^{11} + 3519 x^{10} - 623 x^{9} + 5540 x^{8} + 1505 x^{7} + 5069 x^{6} - 129 x^{5} + 528 x^{4} - 97 x^{3} + 56 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-242839247007536485508643885603=-\,3^{9}\cdot 37^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.91$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(111=3\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(70,·)$, $\chi_{111}(7,·)$, $\chi_{111}(10,·)$, $\chi_{111}(16,·)$, $\chi_{111}(83,·)$, $\chi_{111}(86,·)$, $\chi_{111}(71,·)$, $\chi_{111}(26,·)$, $\chi_{111}(34,·)$, $\chi_{111}(100,·)$, $\chi_{111}(38,·)$, $\chi_{111}(107,·)$, $\chi_{111}(44,·)$, $\chi_{111}(46,·)$, $\chi_{111}(47,·)$, $\chi_{111}(49,·)$, $\chi_{111}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{12}{31} a^{13} + \frac{3}{31} a^{12} + \frac{10}{31} a^{11} - \frac{1}{31} a^{10} + \frac{3}{31} a^{9} - \frac{15}{31} a^{8} - \frac{5}{31} a^{7} - \frac{3}{31} a^{6} + \frac{10}{31} a^{4} - \frac{9}{31} a^{3} - \frac{5}{31} a + \frac{1}{31}$, $\frac{1}{1333} a^{15} - \frac{5}{1333} a^{14} + \frac{605}{1333} a^{13} - \frac{227}{1333} a^{12} + \frac{666}{1333} a^{11} - \frac{631}{1333} a^{10} - \frac{469}{1333} a^{9} - \frac{370}{1333} a^{8} - \frac{228}{1333} a^{7} - \frac{42}{1333} a^{6} - \frac{207}{1333} a^{5} + \frac{131}{1333} a^{4} + \frac{153}{1333} a^{3} - \frac{315}{1333} a^{2} + \frac{644}{1333} a + \frac{231}{1333}$, $\frac{1}{1333} a^{16} + \frac{21}{1333} a^{14} + \frac{89}{1333} a^{13} + \frac{520}{1333} a^{12} - \frac{225}{1333} a^{11} - \frac{399}{1333} a^{10} - \frac{393}{1333} a^{9} - \frac{358}{1333} a^{8} + \frac{280}{1333} a^{7} - \frac{73}{1333} a^{6} + \frac{429}{1333} a^{5} + \frac{550}{1333} a^{4} + \frac{149}{1333} a^{3} + \frac{402}{1333} a^{2} - \frac{419}{1333} a + \frac{596}{1333}$, $\frac{1}{145926445271948161921} a^{17} + \frac{26204538570308632}{145926445271948161921} a^{16} + \frac{20323563375242612}{145926445271948161921} a^{15} - \frac{1154432235088637782}{145926445271948161921} a^{14} + \frac{9260552738894825187}{145926445271948161921} a^{13} + \frac{54373274018046184763}{145926445271948161921} a^{12} + \frac{12610812834168077979}{145926445271948161921} a^{11} + \frac{47571110862336927666}{145926445271948161921} a^{10} - \frac{44556781532057638209}{145926445271948161921} a^{9} - \frac{35545052376878957565}{145926445271948161921} a^{8} - \frac{6640745932467620792}{145926445271948161921} a^{7} - \frac{50859099305401700166}{145926445271948161921} a^{6} + \frac{14281435877137699830}{145926445271948161921} a^{5} - \frac{7440189057094898258}{145926445271948161921} a^{4} - \frac{19713058543455673274}{145926445271948161921} a^{3} + \frac{21428572255115086920}{145926445271948161921} a^{2} - \frac{64687490838118477220}{145926445271948161921} a + \frac{68441920583577122445}{145926445271948161921}$

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

Class group and class number

$C_{171}$, which has order $171$ (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:  \( \frac{20025969607191039788}{145926445271948161921} a^{17} - \frac{20165950282932610000}{145926445271948161921} a^{16} + \frac{340315133056582183978}{145926445271948161921} a^{15} - \frac{121988076748962363963}{145926445271948161921} a^{14} + \frac{4021290298088786965742}{145926445271948161921} a^{13} - \frac{1543805908980430236049}{145926445271948161921} a^{12} + \frac{19962041145841092640272}{145926445271948161921} a^{11} - \frac{4429506202037532132371}{145926445271948161921} a^{10} + \frac{70222182484462262383690}{145926445271948161921} a^{9} - \frac{407706932425405162452}{4707304686191876191} a^{8} + \frac{110070180432231351495904}{145926445271948161921} a^{7} + \frac{30485040438604793540912}{145926445271948161921} a^{6} + \frac{99776704753695173021893}{145926445271948161921} a^{5} - \frac{2246476066609438087994}{145926445271948161921} a^{4} + \frac{9841717502401981062185}{145926445271948161921} a^{3} - \frac{550607737877098968877}{145926445271948161921} a^{2} + \frac{33525759933045100845}{4707304686191876191} a + \frac{16237411536556207962}{145926445271948161921} \) (order $6$)
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:  \( 409151.310213 \) (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

\(\Q(\sqrt{-3}) \), 3.3.1369.1, 6.0.50602347.1, 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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R $18$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ $18$ $18$

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
$37$37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$