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Properties

Label 18.6.60719765548...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{32}\cdot 5^{9}$
Root discriminant $39.73$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3\times A_4$ (as 18T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2699, 3204, -2529, -7212, -8595, -2952, 5634, 7542, 3294, -1444, -2331, -738, 351, 342, 81, -30, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 30*x^15 + 81*x^14 + 342*x^13 + 351*x^12 - 738*x^11 - 2331*x^10 - 1444*x^9 + 3294*x^8 + 7542*x^7 + 5634*x^6 - 2952*x^5 - 8595*x^4 - 7212*x^3 - 2529*x^2 + 3204*x + 2699)
 
gp: K = bnfinit(x^18 - 18*x^16 - 30*x^15 + 81*x^14 + 342*x^13 + 351*x^12 - 738*x^11 - 2331*x^10 - 1444*x^9 + 3294*x^8 + 7542*x^7 + 5634*x^6 - 2952*x^5 - 8595*x^4 - 7212*x^3 - 2529*x^2 + 3204*x + 2699, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 30 x^{15} + 81 x^{14} + 342 x^{13} + 351 x^{12} - 738 x^{11} - 2331 x^{10} - 1444 x^{9} + 3294 x^{8} + 7542 x^{7} + 5634 x^{6} - 2952 x^{5} - 8595 x^{4} - 7212 x^{3} - 2529 x^{2} + 3204 x + 2699 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60719765548297125888000000000=2^{24}\cdot 3^{32}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{13} + \frac{2}{15} a^{12} + \frac{2}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{5} a^{9} + \frac{7}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{15} a^{6} + \frac{4}{15} a^{5} + \frac{1}{5} a^{4} + \frac{4}{15} a^{3} - \frac{1}{3} a^{2} - \frac{7}{15} a - \frac{2}{5}$, $\frac{1}{15} a^{15} + \frac{1}{15} a^{13} - \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{4}{15} a^{8} + \frac{1}{5} a^{7} + \frac{1}{3} a^{6} + \frac{4}{15} a^{5} + \frac{1}{15} a^{4} - \frac{4}{15} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{15}$, $\frac{1}{75} a^{16} - \frac{2}{75} a^{15} - \frac{8}{75} a^{13} - \frac{1}{25} a^{12} + \frac{11}{75} a^{11} + \frac{4}{25} a^{10} - \frac{8}{75} a^{9} + \frac{1}{25} a^{8} + \frac{13}{75} a^{7} + \frac{7}{15} a^{6} + \frac{29}{75} a^{5} - \frac{3}{25} a^{4} - \frac{1}{25} a^{3} + \frac{2}{15} a^{2} - \frac{8}{25} a + \frac{3}{25}$, $\frac{1}{4935807115250729814387167025} a^{17} + \frac{6135259645990156787553}{1645269038416909938129055675} a^{16} - \frac{48220396542676908507409942}{4935807115250729814387167025} a^{15} + \frac{123773637660807439466084192}{4935807115250729814387167025} a^{14} - \frac{54504939042780786535016111}{4935807115250729814387167025} a^{13} + \frac{55408757400686610877314326}{1645269038416909938129055675} a^{12} - \frac{468645999837613071604413497}{4935807115250729814387167025} a^{11} + \frac{215474826484190084122986618}{1645269038416909938129055675} a^{10} + \frac{450122453098776113613556378}{987161423050145962877433405} a^{9} - \frac{386503121904435966276863159}{4935807115250729814387167025} a^{8} + \frac{66960327093893270402556031}{1645269038416909938129055675} a^{7} - \frac{523606399628694266687514911}{4935807115250729814387167025} a^{6} + \frac{318662269372849947788063021}{987161423050145962877433405} a^{5} - \frac{134218664916759553461213399}{1645269038416909938129055675} a^{4} + \frac{541589344342532023352627819}{1645269038416909938129055675} a^{3} - \frac{445748135437108804605238008}{1645269038416909938129055675} a^{2} - \frac{1637528119471119189892099}{11613663800589952504440393} a + \frac{391135614736006742876142443}{1645269038416909938129055675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17244009.6923 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.1620.1, 6.2.52488000.5, 9.9.344373768000.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$