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Properties

Label 18.18.1292246553...6849.1
Degree $18$
Signature $[18, 0]$
Discriminant $11^{9}\cdot 19^{17}$
Root discriminant $53.51$
Ramified primes $11, 19$
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![44917, -418894, 418894, 1450991, -1450991, -1291507, 1291507, 536825, -536825, -123406, 123406, 16643, -16643, -1312, 1312, 56, -56, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917)
 
gp: K = bnfinit(x^18 - x^17 - 56*x^16 + 56*x^15 + 1312*x^14 - 1312*x^13 - 16643*x^12 + 16643*x^11 + 123406*x^10 - 123406*x^9 - 536825*x^8 + 536825*x^7 + 1291507*x^6 - 1291507*x^5 - 1450991*x^4 + 1450991*x^3 + 418894*x^2 - 418894*x + 44917, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 56 x^{16} + 56 x^{15} + 1312 x^{14} - 1312 x^{13} - 16643 x^{12} + 16643 x^{11} + 123406 x^{10} - 123406 x^{9} - 536825 x^{8} + 536825 x^{7} + 1291507 x^{6} - 1291507 x^{5} - 1450991 x^{4} + 1450991 x^{3} + 418894 x^{2} - 418894 x + 44917 \)

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:  \(12922465537100419689226617716849=11^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.51$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(1,·)$, $\chi_{209}(65,·)$, $\chi_{209}(10,·)$, $\chi_{209}(45,·)$, $\chi_{209}(208,·)$, $\chi_{209}(21,·)$, $\chi_{209}(23,·)$, $\chi_{209}(100,·)$, $\chi_{209}(32,·)$, $\chi_{209}(144,·)$, $\chi_{209}(98,·)$, $\chi_{209}(164,·)$, $\chi_{209}(199,·)$, $\chi_{209}(109,·)$, $\chi_{209}(111,·)$, $\chi_{209}(177,·)$, $\chi_{209}(186,·)$, $\chi_{209}(188,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{15467} a^{10} - \frac{6479}{15467} a^{9} - \frac{30}{15467} a^{8} + \frac{4796}{15467} a^{7} + \frac{315}{15467} a^{6} + \frac{3237}{15467} a^{5} - \frac{1350}{15467} a^{4} + \frac{4677}{15467} a^{3} + \frac{2025}{15467} a^{2} - \frac{5756}{15467} a - \frac{486}{15467}$, $\frac{1}{15467} a^{11} - \frac{33}{15467} a^{9} - \frac{3970}{15467} a^{8} + \frac{396}{15467} a^{7} + \frac{2478}{15467} a^{6} - \frac{2079}{15467} a^{5} - \frac{3118}{15467} a^{4} + \frac{4455}{15467} a^{3} - \frac{1797}{15467} a^{2} - \frac{2673}{15467} a + \frac{6474}{15467}$, $\frac{1}{15467} a^{12} - \frac{1239}{15467} a^{9} - \frac{594}{15467} a^{8} + \frac{6076}{15467} a^{7} - \frac{7151}{15467} a^{6} - \frac{4566}{15467} a^{5} + \frac{6306}{15467} a^{4} - \frac{2126}{15467} a^{3} + \frac{2284}{15467} a^{2} + \frac{2130}{15467} a - \frac{571}{15467}$, $\frac{1}{15467} a^{13} - \frac{702}{15467} a^{9} - \frac{160}{15467} a^{8} - \frac{4235}{15467} a^{7} - \frac{956}{15467} a^{6} - \frac{4471}{15467} a^{5} - \frac{4340}{15467} a^{4} - \frac{3038}{15467} a^{3} + \frac{5451}{15467} a^{2} - \frac{1968}{15467} a + \frac{1059}{15467}$, $\frac{1}{15467} a^{14} - \frac{1120}{15467} a^{9} + \frac{5639}{15467} a^{8} - \frac{5970}{15467} a^{7} + \frac{121}{15467} a^{6} - \frac{5615}{15467} a^{5} - \frac{7251}{15467} a^{4} - \frac{5766}{15467} a^{3} - \frac{3382}{15467} a^{2} - \frac{2766}{15467} a - \frac{898}{15467}$, $\frac{1}{15467} a^{15} + \frac{3182}{15467} a^{9} + \frac{6831}{15467} a^{8} + \frac{4592}{15467} a^{7} + \frac{6911}{15467} a^{6} - \frac{1089}{15467} a^{5} - \frac{2000}{15467} a^{4} + \frac{7012}{15467} a^{3} + \frac{7052}{15467} a^{2} + \frac{2121}{15467} a - \frac{2975}{15467}$, $\frac{1}{15467} a^{16} + \frac{5498}{15467} a^{9} + \frac{7250}{15467} a^{8} - \frac{3499}{15467} a^{7} + \frac{1936}{15467} a^{6} - \frac{1112}{15467} a^{5} + \frac{2886}{15467} a^{4} + \frac{4092}{15467} a^{3} - \frac{7157}{15467} a^{2} - \frac{311}{15467} a - \frac{248}{15467}$, $\frac{1}{15467} a^{17} - \frac{7176}{15467} a^{9} + \frac{6771}{15467} a^{8} + \frac{4763}{15467} a^{7} - \frac{678}{15467} a^{6} - \frac{7090}{15467} a^{5} + \frac{2232}{15467} a^{4} + \frac{318}{15467} a^{3} + \frac{2479}{15467} a^{2} + \frac{758}{15467} a - \frac{3763}{15467}$

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:  \( 3961676166.05 \) (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

\(\Q(\sqrt{209}) \), 3.3.361.1, 6.6.3295687769.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19Data not computed