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Properties

Label 18.0.14572743731...1312.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{33}$
Root discriminant $14.99$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $C_2\times He_3:C_2$ (as 18T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 36, 201, 672, 1314, 1521, 924, 27, -404, -276, -27, 69, 48, 9, -9, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 9*x^15 + 9*x^14 + 48*x^13 + 69*x^12 - 27*x^11 - 276*x^10 - 404*x^9 + 27*x^8 + 924*x^7 + 1521*x^6 + 1314*x^5 + 672*x^4 + 201*x^3 + 36*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^16 - 9*x^15 + 9*x^14 + 48*x^13 + 69*x^12 - 27*x^11 - 276*x^10 - 404*x^9 + 27*x^8 + 924*x^7 + 1521*x^6 + 1314*x^5 + 672*x^4 + 201*x^3 + 36*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 9 x^{15} + 9 x^{14} + 48 x^{13} + 69 x^{12} - 27 x^{11} - 276 x^{10} - 404 x^{9} + 27 x^{8} + 924 x^{7} + 1521 x^{6} + 1314 x^{5} + 672 x^{4} + 201 x^{3} + 36 x^{2} + 6 x + 1 \)

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:  \(-1457274373159131021312=-\,2^{18}\cdot 3^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.99$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1481050478} a^{17} - \frac{31261233}{1481050478} a^{16} + \frac{97749567}{740525239} a^{15} - \frac{144841908}{740525239} a^{14} - \frac{126298878}{740525239} a^{13} - \frac{268484643}{1481050478} a^{12} - \frac{211221647}{1481050478} a^{11} + \frac{158833517}{740525239} a^{10} - \frac{257916227}{740525239} a^{9} - \frac{293485191}{740525239} a^{8} + \frac{739588675}{1481050478} a^{7} - \frac{950007}{1481050478} a^{6} - \frac{374529943}{1481050478} a^{5} + \frac{122652077}{740525239} a^{4} + \frac{139726884}{740525239} a^{3} - \frac{272018934}{740525239} a^{2} - \frac{282439695}{740525239} a - \frac{26915491}{1481050478}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{4985312902}{740525239} a^{17} + \frac{6831279713}{1481050478} a^{16} + \frac{55072260117}{1481050478} a^{15} + \frac{52175950657}{1481050478} a^{14} - \frac{62704608945}{740525239} a^{13} - \frac{392548941733}{1481050478} a^{12} - \frac{419661398379}{1481050478} a^{11} + \frac{555377275393}{1481050478} a^{10} + \frac{1185140586231}{740525239} a^{9} + \frac{2407827495941}{1481050478} a^{8} - \frac{955634521790}{740525239} a^{7} - \frac{7898361176723}{1481050478} a^{6} - \frac{4881001352711}{740525239} a^{5} - \frac{6437909378711}{1481050478} a^{4} - \frac{2324026446311}{1481050478} a^{3} - \frac{221592868740}{740525239} a^{2} - \frac{36535619730}{740525239} a - \frac{7576530537}{740525239} \) (order $18$)
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:  \( 14503.1383261 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.3.22039921152.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed