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Properties

Label 18.0.17464179998...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 5^{12}\cdot 73^{12}$
Root discriminant $421.27$
Ramified primes $2, 3, 5, 73$
Class number $1205308188$ (GRH)
Class group $[3, 3, 9, 9, 9, 9, 54, 378]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7625597484987, 0, 0, -387420489, 0, 0, 1171847088, 0, 0, 118341, 0, 0, 59536, 0, 0, -1, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^15 + 59536*x^12 + 118341*x^9 + 1171847088*x^6 - 387420489*x^3 + 7625597484987)
 
gp: K = bnfinit(x^18 - x^15 + 59536*x^12 + 118341*x^9 + 1171847088*x^6 - 387420489*x^3 + 7625597484987, 1)
 

Normalized defining polynomial

\( x^{18} - x^{15} + 59536 x^{12} + 118341 x^{9} + 1171847088 x^{6} - 387420489 x^{3} + 7625597484987 \)

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:  \(-174641799984367553282969504761435227000000000000=-\,2^{12}\cdot 3^{27}\cdot 5^{12}\cdot 73^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $421.27$
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:  $2, 3, 5, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{5} + \frac{1}{27} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{81} a^{6} + \frac{1}{81} a^{3}$, $\frac{1}{729} a^{10} - \frac{1}{729} a^{7} - \frac{242}{729} a^{4} + \frac{1}{3} a$, $\frac{1}{2187} a^{11} - \frac{1}{2187} a^{8} + \frac{487}{2187} a^{5} + \frac{1}{9} a^{2}$, $\frac{1}{12774267} a^{12} + \frac{52244}{12774267} a^{9} - \frac{4165505}{12774267} a^{6} + \frac{323}{1947} a^{3} + \frac{213}{649}$, $\frac{1}{344905209} a^{13} + \frac{209951}{344905209} a^{10} - \frac{55420280}{344905209} a^{7} + \frac{482491}{1419363} a^{4} + \frac{2809}{17523} a$, $\frac{1}{9312440643} a^{14} + \frac{209951}{9312440643} a^{11} - \frac{55420280}{9312440643} a^{8} - \frac{6614324}{38322801} a^{5} - \frac{224990}{473121} a^{2}$, $\frac{1}{39836749270184757} a^{15} + \frac{38421215}{39836749270184757} a^{12} + \frac{6059237218156}{39836749270184757} a^{9} - \frac{1103381633335}{14903385435909} a^{6} + \frac{187534490140}{2023916540679} a^{3} + \frac{38737603}{102825613}$, $\frac{1}{1075592230294988439} a^{16} + \frac{38421215}{1075592230294988439} a^{13} - \frac{485752482166841}{1075592230294988439} a^{10} + \frac{58694152523090}{402391406769543} a^{7} - \frac{9957034837214}{54645746598333} a^{4} + \frac{655691281}{2776291551} a$, $\frac{1}{29040990217964687853} a^{17} + \frac{38421215}{29040990217964687853} a^{14} - \frac{1961187640321832}{29040990217964687853} a^{11} + \frac{193376598684638}{10864567982777661} a^{8} - \frac{173969234504090}{1475435158154991} a^{5} - \frac{10449474923}{74959871877} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{9}\times C_{9}\times C_{54}\times C_{378}$, which has order $1205308188$ (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:  \( -\frac{237412}{39836749270184757} a^{15} + \frac{257095}{39836749270184757} a^{12} - \frac{15689045440}{39836749270184757} a^{9} - \frac{102805930}{163937239794999} a^{6} - \frac{23509298170}{2023916540679} a^{3} + \frac{51472342}{102825613} \) (order $6$)
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:  \( 66851395.5778903 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1598700.2 x3, 3.3.431649.2, 6.0.7667525070000.2, 6.0.1048907070000.2 x2, 6.0.558962577603.2, Deg 9 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: 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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
5Data not computed
$73$73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} - 73$$3$$1$$2$$C_3$$[\ ]_{3}$