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Properties

Label 18.0.63426874804...4832.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 23^{6}$
Root discriminant $16.26$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -9, 12, 30, -75, 10, 108, -30, -95, -9, 78, 55, -33, -15, 17, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 17*x^15 - 15*x^14 - 33*x^13 + 55*x^12 + 78*x^11 - 9*x^10 - 95*x^9 - 30*x^8 + 108*x^7 + 10*x^6 - 75*x^5 + 30*x^4 + 12*x^3 - 9*x^2 + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 17*x^15 - 15*x^14 - 33*x^13 + 55*x^12 + 78*x^11 - 9*x^10 - 95*x^9 - 30*x^8 + 108*x^7 + 10*x^6 - 75*x^5 + 30*x^4 + 12*x^3 - 9*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 17 x^{15} - 15 x^{14} - 33 x^{13} + 55 x^{12} + 78 x^{11} - 9 x^{10} - 95 x^{9} - 30 x^{8} + 108 x^{7} + 10 x^{6} - 75 x^{5} + 30 x^{4} + 12 x^{3} - 9 x^{2} + 1 \)

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:  \(-6342687480463779704832=-\,2^{12}\cdot 3^{21}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.26$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{1735} a^{16} - \frac{11}{347} a^{15} + \frac{41}{1735} a^{14} - \frac{97}{1735} a^{13} + \frac{143}{1735} a^{12} + \frac{866}{1735} a^{11} - \frac{117}{1735} a^{10} + \frac{154}{1735} a^{9} - \frac{557}{1735} a^{8} + \frac{733}{1735} a^{7} + \frac{32}{1735} a^{6} + \frac{237}{1735} a^{5} - \frac{157}{1735} a^{4} - \frac{714}{1735} a^{3} - \frac{32}{347} a^{2} + \frac{864}{1735} a + \frac{807}{1735}$, $\frac{1}{261934685} a^{17} + \frac{13007}{52386937} a^{16} + \frac{1652504}{261934685} a^{15} - \frac{22892468}{261934685} a^{14} + \frac{1066069}{52386937} a^{13} + \frac{18387297}{261934685} a^{12} - \frac{35752696}{261934685} a^{11} + \frac{70838548}{261934685} a^{10} + \frac{26043155}{52386937} a^{9} - \frac{117418044}{261934685} a^{8} - \frac{92657091}{261934685} a^{7} + \frac{13088994}{52386937} a^{6} - \frac{23398083}{52386937} a^{5} - \frac{115311}{754855} a^{4} + \frac{29238551}{261934685} a^{3} - \frac{14391977}{52386937} a^{2} + \frac{92058683}{261934685} a + \frac{34807381}{261934685}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{119254}{150971} a^{17} + \frac{1652291}{754855} a^{16} + \frac{242316}{754855} a^{15} - \frac{9647388}{754855} a^{14} + \frac{6657566}{754855} a^{13} + \frac{18907477}{754855} a^{12} - \frac{26051242}{754855} a^{11} - \frac{48337746}{754855} a^{10} - \frac{13768522}{754855} a^{9} + \frac{43919171}{754855} a^{8} + \frac{30813514}{754855} a^{7} - \frac{8736406}{150971} a^{6} - \frac{2694832}{150971} a^{5} + \frac{5147247}{150971} a^{4} - \frac{12794734}{754855} a^{3} - \frac{24180}{150971} a^{2} + \frac{1218606}{754855} a - \frac{152726}{754855} \) (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:  \( 7152.852400477029 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 3.1.23.1, 6.0.14283.1, 6.0.34992.1, 9.1.15326915904.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$