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Properties

Label 18.18.2110600968...2841.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $25.52$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -45, -100, 474, 573, -2235, -453, 4395, -1647, -3117, 2301, 529, -861, 111, 96, -24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 24*x^16 + 96*x^15 + 111*x^14 - 861*x^13 + 529*x^12 + 2301*x^11 - 3117*x^10 - 1647*x^9 + 4395*x^8 - 453*x^7 - 2235*x^6 + 573*x^5 + 474*x^4 - 100*x^3 - 45*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 24*x^16 + 96*x^15 + 111*x^14 - 861*x^13 + 529*x^12 + 2301*x^11 - 3117*x^10 - 1647*x^9 + 4395*x^8 - 453*x^7 - 2235*x^6 + 573*x^5 + 474*x^4 - 100*x^3 - 45*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 24 x^{16} + 96 x^{15} + 111 x^{14} - 861 x^{13} + 529 x^{12} + 2301 x^{11} - 3117 x^{10} - 1647 x^{9} + 4395 x^{8} - 453 x^{7} - 2235 x^{6} + 573 x^{5} + 474 x^{4} - 100 x^{3} - 45 x^{2} + 3 x + 1 \)

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:  \(21106009685505727678262841=3^{24}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.52$
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:  $3, 73, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{8}{19} a^{15} - \frac{4}{19} a^{13} - \frac{2}{19} a^{12} - \frac{8}{19} a^{11} - \frac{7}{19} a^{10} - \frac{6}{19} a^{9} + \frac{8}{19} a^{8} + \frac{5}{19} a^{7} + \frac{5}{19} a^{6} - \frac{7}{19} a^{5} + \frac{1}{19} a^{3} - \frac{6}{19} a^{2} + \frac{9}{19} a + \frac{6}{19}$, $\frac{1}{107035765631} a^{17} - \frac{1432046486}{107035765631} a^{16} + \frac{19106459966}{107035765631} a^{15} - \frac{10033949898}{107035765631} a^{14} + \frac{49756588256}{107035765631} a^{13} + \frac{5237285288}{107035765631} a^{12} - \frac{39545211447}{107035765631} a^{11} + \frac{30456511159}{107035765631} a^{10} - \frac{91082085}{5633461349} a^{9} - \frac{18076676814}{107035765631} a^{8} + \frac{24982318776}{107035765631} a^{7} + \frac{50164719758}{107035765631} a^{6} - \frac{38801599515}{107035765631} a^{5} + \frac{11509219905}{107035765631} a^{4} - \frac{30311083488}{107035765631} a^{3} + \frac{52626601300}{107035765631} a^{2} + \frac{23590228334}{107035765631} a - \frac{18330658055}{107035765631}$

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:  \( 4705474.71742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.276355881.1, 9.9.22384826361.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed