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Properties

Label 18.0.23372182570...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 5^{9}\cdot 37^{17}$
Root discriminant $135.39$
Ramified primes $2, 5, 37$
Class number $1227856$ (GRH)
Class group $[2, 613928]$ (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![72265625, 0, 621484375, 0, 725546875, 0, 323171875, 0, 69375000, 0, 7677500, 0, 439375, 0, 12950, 0, 185, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 185*x^16 + 12950*x^14 + 439375*x^12 + 7677500*x^10 + 69375000*x^8 + 323171875*x^6 + 725546875*x^4 + 621484375*x^2 + 72265625)
 
gp: K = bnfinit(x^18 + 185*x^16 + 12950*x^14 + 439375*x^12 + 7677500*x^10 + 69375000*x^8 + 323171875*x^6 + 725546875*x^4 + 621484375*x^2 + 72265625, 1)
 

Normalized defining polynomial

\( x^{18} + 185 x^{16} + 12950 x^{14} + 439375 x^{12} + 7677500 x^{10} + 69375000 x^{8} + 323171875 x^{6} + 725546875 x^{4} + 621484375 x^{2} + 72265625 \)

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:  \(-233721825702929999566923221504000000000=-\,2^{18}\cdot 5^{9}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $135.39$
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:  $2, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(740=2^{2}\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{740}(1,·)$, $\chi_{740}(619,·)$, $\chi_{740}(581,·)$, $\chi_{740}(641,·)$, $\chi_{740}(201,·)$, $\chi_{740}(139,·)$, $\chi_{740}(81,·)$, $\chi_{740}(659,·)$, $\chi_{740}(121,·)$, $\chi_{740}(601,·)$, $\chi_{740}(539,·)$, $\chi_{740}(159,·)$, $\chi_{740}(99,·)$, $\chi_{740}(299,·)$, $\chi_{740}(559,·)$, $\chi_{740}(181,·)$, $\chi_{740}(739,·)$, $\chi_{740}(441,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{3359375} a^{14} + \frac{14}{134375} a^{10} - \frac{13}{26875} a^{8} - \frac{2}{1075} a^{6} + \frac{11}{1075} a^{4} - \frac{11}{215} a^{2} - \frac{21}{43}$, $\frac{1}{3359375} a^{15} + \frac{14}{134375} a^{11} - \frac{13}{26875} a^{9} - \frac{2}{1075} a^{7} + \frac{11}{1075} a^{5} - \frac{11}{215} a^{3} - \frac{21}{43} a$, $\frac{1}{1081764387109375} a^{16} + \frac{6296957}{43270575484375} a^{14} - \frac{237263789}{8654115096875} a^{12} + \frac{5680566}{1730823019375} a^{10} + \frac{454901551}{1730823019375} a^{8} + \frac{552171309}{346164603875} a^{6} - \frac{1045941801}{69232920775} a^{4} - \frac{852680188}{13846584155} a^{2} - \frac{7266499}{2769316831}$, $\frac{1}{1081764387109375} a^{17} + \frac{6296957}{43270575484375} a^{15} - \frac{237263789}{8654115096875} a^{13} + \frac{5680566}{1730823019375} a^{11} + \frac{454901551}{1730823019375} a^{9} + \frac{552171309}{346164603875} a^{7} - \frac{1045941801}{69232920775} a^{5} - \frac{852680188}{13846584155} a^{3} - \frac{7266499}{2769316831} a$

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

Class group and class number

$C_{2}\times C_{613928}$, which has order $1227856$ (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

\(\Q(\sqrt{-185}) \), 3.3.1369.1, 6.0.554751656000.1, 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
5Data not computed
37Data not computed