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Properties

Label 18.0.12063933521...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 5^{9}$
Root discriminant $14.83$
Ramified primes $3, 5$
Class number $2$
Class group $[2]$
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![1, -9, 45, -132, 273, -384, 354, -120, -156, 298, -234, 90, 48, -102, 87, -51, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 51*x^15 + 87*x^14 - 102*x^13 + 48*x^12 + 90*x^11 - 234*x^10 + 298*x^9 - 156*x^8 - 120*x^7 + 354*x^6 - 384*x^5 + 273*x^4 - 132*x^3 + 45*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 51*x^15 + 87*x^14 - 102*x^13 + 48*x^12 + 90*x^11 - 234*x^10 + 298*x^9 - 156*x^8 - 120*x^7 + 354*x^6 - 384*x^5 + 273*x^4 - 132*x^3 + 45*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 51 x^{15} + 87 x^{14} - 102 x^{13} + 48 x^{12} + 90 x^{11} - 234 x^{10} + 298 x^{9} - 156 x^{8} - 120 x^{7} + 354 x^{6} - 384 x^{5} + 273 x^{4} - 132 x^{3} + 45 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1206393352117083984375=-\,3^{31}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.83$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4046252836} a^{17} + \frac{58387815}{1011563209} a^{16} + \frac{77789381}{1011563209} a^{15} - \frac{186372905}{2023126418} a^{14} + \frac{97561438}{1011563209} a^{13} + \frac{19703872}{1011563209} a^{12} + \frac{206526624}{1011563209} a^{11} - \frac{248942899}{2023126418} a^{10} + \frac{232993710}{1011563209} a^{9} - \frac{194214175}{1011563209} a^{8} + \frac{245183014}{1011563209} a^{7} - \frac{484777585}{1011563209} a^{6} - \frac{248720789}{1011563209} a^{5} - \frac{446805237}{1011563209} a^{4} + \frac{935778149}{4046252836} a^{3} + \frac{265625164}{1011563209} a^{2} + \frac{235191328}{1011563209} a + \frac{959419275}{2023126418}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1012.11392421 \)
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

\(\Q(\sqrt{-15}) \), 3.1.135.1 x3, \(\Q(\zeta_{9})^+\), 6.0.273375.1, 6.0.2460375.2 x2, 6.0.2460375.1, 9.3.1793613375.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.2460375.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$