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Properties

Label 18.18.2211533774...1973.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{9}\cdot 19^{17}$
Root discriminant $42.68$
Ramified primes $7, 19$
Class number $1$ (GRH)
Class group Trivial (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![-797, -8931, 8931, 64029, -64029, -96483, 96483, 64029, -64029, -22915, 22915, 4749, -4749, -571, 571, 37, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 37*x^16 + 37*x^15 + 571*x^14 - 571*x^13 - 4749*x^12 + 4749*x^11 + 22915*x^10 - 22915*x^9 - 64029*x^8 + 64029*x^7 + 96483*x^6 - 96483*x^5 - 64029*x^4 + 64029*x^3 + 8931*x^2 - 8931*x - 797)
 
gp: K = bnfinit(x^18 - x^17 - 37*x^16 + 37*x^15 + 571*x^14 - 571*x^13 - 4749*x^12 + 4749*x^11 + 22915*x^10 - 22915*x^9 - 64029*x^8 + 64029*x^7 + 96483*x^6 - 96483*x^5 - 64029*x^4 + 64029*x^3 + 8931*x^2 - 8931*x - 797, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 37 x^{16} + 37 x^{15} + 571 x^{14} - 571 x^{13} - 4749 x^{12} + 4749 x^{11} + 22915 x^{10} - 22915 x^{9} - 64029 x^{8} + 64029 x^{7} + 96483 x^{6} - 96483 x^{5} - 64029 x^{4} + 64029 x^{3} + 8931 x^{2} - 8931 x - 797 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(221153377467012797984123331973=7^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.68$
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:  $7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(132,·)$, $\chi_{133}(69,·)$, $\chi_{133}(13,·)$, $\chi_{133}(85,·)$, $\chi_{133}(90,·)$, $\chi_{133}(27,·)$, $\chi_{133}(92,·)$, $\chi_{133}(97,·)$, $\chi_{133}(34,·)$, $\chi_{133}(99,·)$, $\chi_{133}(36,·)$, $\chi_{133}(41,·)$, $\chi_{133}(106,·)$, $\chi_{133}(43,·)$, $\chi_{133}(48,·)$, $\chi_{133}(120,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{457} a^{10} + \frac{80}{457} a^{9} - \frac{20}{457} a^{8} - \frac{69}{457} a^{7} + \frac{140}{457} a^{6} - \frac{43}{457} a^{5} + \frac{57}{457} a^{4} - \frac{6}{457} a^{3} - \frac{57}{457} a^{2} + \frac{95}{457} a - \frac{64}{457}$, $\frac{1}{457} a^{11} - \frac{22}{457} a^{9} + \frac{160}{457} a^{8} + \frac{176}{457} a^{7} + \frac{182}{457} a^{6} - \frac{159}{457} a^{5} + \frac{4}{457} a^{4} - \frac{34}{457} a^{3} + \frac{85}{457} a^{2} + \frac{105}{457} a + \frac{93}{457}$, $\frac{1}{457} a^{12} + \frac{92}{457} a^{9} + \frac{193}{457} a^{8} + \frac{35}{457} a^{7} + \frac{179}{457} a^{6} - \frac{28}{457} a^{5} - \frac{151}{457} a^{4} - \frac{47}{457} a^{3} + \frac{222}{457} a^{2} - \frac{102}{457} a - \frac{37}{457}$, $\frac{1}{457} a^{13} + \frac{145}{457} a^{9} + \frac{47}{457} a^{8} + \frac{129}{457} a^{7} - \frac{112}{457} a^{6} + \frac{149}{457} a^{5} + \frac{193}{457} a^{4} - \frac{140}{457} a^{3} + \frac{115}{457} a^{2} - \frac{94}{457} a - \frac{53}{457}$, $\frac{1}{457} a^{14} - \frac{128}{457} a^{9} - \frac{170}{457} a^{8} - \frac{161}{457} a^{7} - \frac{43}{457} a^{6} + \frac{30}{457} a^{5} - \frac{179}{457} a^{4} + \frac{71}{457} a^{3} - \frac{55}{457} a^{2} - \frac{118}{457} a + \frac{140}{457}$, $\frac{1}{457} a^{15} + \frac{16}{457} a^{9} + \frac{21}{457} a^{8} - \frac{192}{457} a^{7} + \frac{127}{457} a^{6} - \frac{199}{457} a^{5} + \frac{55}{457} a^{4} + \frac{91}{457} a^{3} - \frac{102}{457} a^{2} - \frac{39}{457} a + \frac{34}{457}$, $\frac{1}{457} a^{16} + \frac{112}{457} a^{9} + \frac{128}{457} a^{8} - \frac{140}{457} a^{7} - \frac{154}{457} a^{6} - \frac{171}{457} a^{5} + \frac{93}{457} a^{4} - \frac{6}{457} a^{3} - \frac{41}{457} a^{2} - \frac{115}{457} a + \frac{110}{457}$, $\frac{1}{457} a^{17} - \frac{149}{457} a^{9} - \frac{185}{457} a^{8} - \frac{195}{457} a^{7} + \frac{144}{457} a^{6} - \frac{118}{457} a^{5} + \frac{8}{457} a^{4} + \frac{174}{457} a^{3} - \frac{129}{457} a^{2} - \frac{19}{457} a - \frac{144}{457}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 428418281.929 \) (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

\(\Q(\sqrt{133}) \), 3.3.361.1, 6.6.849301957.1, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed