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Properties

Label 18.18.5633621358...3125.1
Degree $18$
Signature $[18, 0]$
Discriminant $5^{9}\cdot 19^{16}$
Root discriminant $30.63$
Ramified primes $5, 19$
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![-1, -5, 55, 30, -505, -131, 1711, 387, -2742, -566, 2291, 403, -1018, -137, 232, 20, -25, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*x - 1)
 
gp: K = bnfinit(x^18 - x^17 - 25*x^16 + 20*x^15 + 232*x^14 - 137*x^13 - 1018*x^12 + 403*x^11 + 2291*x^10 - 566*x^9 - 2742*x^8 + 387*x^7 + 1711*x^6 - 131*x^5 - 505*x^4 + 30*x^3 + 55*x^2 - 5*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 25 x^{16} + 20 x^{15} + 232 x^{14} - 137 x^{13} - 1018 x^{12} + 403 x^{11} + 2291 x^{10} - 566 x^{9} - 2742 x^{8} + 387 x^{7} + 1711 x^{6} - 131 x^{5} - 505 x^{4} + 30 x^{3} + 55 x^{2} - 5 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(563362135874260093126953125=5^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.63$
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:  $5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(95=5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{95}(64,·)$, $\chi_{95}(1,·)$, $\chi_{95}(66,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(9,·)$, $\chi_{95}(74,·)$, $\chi_{95}(11,·)$, $\chi_{95}(16,·)$, $\chi_{95}(81,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(36,·)$, $\chi_{95}(39,·)$, $\chi_{95}(44,·)$, $\chi_{95}(49,·)$, $\chi_{95}(54,·)$, $\chi_{95}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{4130024167189} a^{17} + \frac{920978222672}{4130024167189} a^{16} + \frac{606794707934}{4130024167189} a^{15} - \frac{492036479265}{4130024167189} a^{14} + \frac{629306648510}{4130024167189} a^{13} - \frac{1857958233227}{4130024167189} a^{12} - \frac{550922843041}{4130024167189} a^{11} - \frac{1554275344905}{4130024167189} a^{10} - \frac{1498768582140}{4130024167189} a^{9} + \frac{657444201028}{4130024167189} a^{8} + \frac{846780531398}{4130024167189} a^{7} - \frac{1207865625684}{4130024167189} a^{6} + \frac{446543373733}{4130024167189} a^{5} + \frac{491195884715}{4130024167189} a^{4} + \frac{670693194266}{4130024167189} a^{3} + \frac{1281532435586}{4130024167189} a^{2} + \frac{20048467778}{4130024167189} a + \frac{983742514991}{4130024167189}$

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:  \( 26179751.3516 \) (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

\(\Q(\sqrt{5}) \), 3.3.361.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\)

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 $18$ $18$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
5Data not computed
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$