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Properties

Label 18.0.93597656065...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{34}\cdot 3^{20}\cdot 5^{6}$
Root discriminant $21.47$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 0, 50, 288, -642, 625, -510, 396, -330, 291, -210, 128, -78, 51, -34, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 34*x^15 + 51*x^14 - 78*x^13 + 128*x^12 - 210*x^11 + 291*x^10 - 330*x^9 + 396*x^8 - 510*x^7 + 625*x^6 - 642*x^5 + 288*x^4 + 50*x^3 + 2)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 34*x^15 + 51*x^14 - 78*x^13 + 128*x^12 - 210*x^11 + 291*x^10 - 330*x^9 + 396*x^8 - 510*x^7 + 625*x^6 - 642*x^5 + 288*x^4 + 50*x^3 + 2, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 34 x^{15} + 51 x^{14} - 78 x^{13} + 128 x^{12} - 210 x^{11} + 291 x^{10} - 330 x^{9} + 396 x^{8} - 510 x^{7} + 625 x^{6} - 642 x^{5} + 288 x^{4} + 50 x^{3} + 2 \)

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:  \(-935976560656121856000000=-\,2^{34}\cdot 3^{20}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.47$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{14} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{3}{16} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{5}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{14} - \frac{1}{8} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{3}{32} a^{8} - \frac{1}{2} a^{7} - \frac{9}{32} a^{6} + \frac{1}{8} a^{5} - \frac{7}{16} a^{4} - \frac{7}{16} a^{3} - \frac{3}{16} a^{2} + \frac{1}{8} a + \frac{5}{16}$, $\frac{1}{3742757482720} a^{17} + \frac{2194012733}{1871378741360} a^{16} - \frac{10379918305}{748551496544} a^{15} - \frac{56882494117}{1871378741360} a^{14} - \frac{35981787243}{935689370680} a^{13} + \frac{184024595979}{1871378741360} a^{12} - \frac{4389145603}{116961171335} a^{11} + \frac{7701111481}{467844685340} a^{10} - \frac{785363521493}{3742757482720} a^{9} + \frac{537435104347}{1871378741360} a^{8} + \frac{1088023094659}{3742757482720} a^{7} + \frac{651841934069}{1871378741360} a^{6} + \frac{199959586913}{1871378741360} a^{5} - \frac{168802056149}{374275748272} a^{4} - \frac{295740317361}{1871378741360} a^{3} - \frac{34000439213}{467844685340} a^{2} + \frac{703499671221}{1871378741360} a - \frac{425908858939}{935689370680}$

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:  \( \frac{3171949203}{116961171335} a^{17} - \frac{16030824792}{116961171335} a^{16} + \frac{14934820547}{46784468534} a^{15} - \frac{362235032031}{935689370680} a^{14} + \frac{150117999471}{467844685340} a^{13} - \frac{233320624459}{467844685340} a^{12} + \frac{474889832133}{467844685340} a^{11} - \frac{786197192469}{467844685340} a^{10} + \frac{302307970947}{233922342670} a^{9} + \frac{212337925443}{467844685340} a^{8} - \frac{16045943613}{116961171335} a^{7} - \frac{986531098963}{935689370680} a^{6} + \frac{142476362907}{467844685340} a^{5} + \frac{272972096043}{93568937068} a^{4} - \frac{6446420081419}{467844685340} a^{3} + \frac{1513495950558}{116961171335} a^{2} + \frac{107275568586}{116961171335} a - \frac{25806574807}{467844685340} \) (order $4$)
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:  \( 489834.7260177265 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.108.1, 3.1.1080.1, 6.0.18662400.2, 6.0.186624.1, 9.1.60466176000.1

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

Sibling fields

Degree 12 siblings: 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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.26.64$x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{6} - 2 x^{4} + 4 x^{3} + 2$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$