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Properties

Label 18.18.8674894461...2608.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{16}\cdot 3^{33}\cdot 47^{8}$
Root discriminant $76.82$
Ramified primes $2, 3, 47$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-597, -38430, 54270, 256128, -128340, -430524, 143671, 327990, -91158, -132374, 33048, 29718, -6681, -3708, 738, 238, -42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 42*x^16 + 238*x^15 + 738*x^14 - 3708*x^13 - 6681*x^12 + 29718*x^11 + 33048*x^10 - 132374*x^9 - 91158*x^8 + 327990*x^7 + 143671*x^6 - 430524*x^5 - 128340*x^4 + 256128*x^3 + 54270*x^2 - 38430*x - 597)
 
gp: K = bnfinit(x^18 - 6*x^17 - 42*x^16 + 238*x^15 + 738*x^14 - 3708*x^13 - 6681*x^12 + 29718*x^11 + 33048*x^10 - 132374*x^9 - 91158*x^8 + 327990*x^7 + 143671*x^6 - 430524*x^5 - 128340*x^4 + 256128*x^3 + 54270*x^2 - 38430*x - 597, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 42 x^{16} + 238 x^{15} + 738 x^{14} - 3708 x^{13} - 6681 x^{12} + 29718 x^{11} + 33048 x^{10} - 132374 x^{9} - 91158 x^{8} + 327990 x^{7} + 143671 x^{6} - 430524 x^{5} - 128340 x^{4} + 256128 x^{3} + 54270 x^{2} - 38430 x - 597 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8674894461032534679022957756612608=2^{16}\cdot 3^{33}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.82$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{5}{18} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{36} a^{16} - \frac{1}{36} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{2} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{2}{9} a^{7} - \frac{4}{9} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{12} a + \frac{1}{12}$, $\frac{1}{633652221844107846601932204} a^{17} - \frac{3826614722925089641299011}{633652221844107846601932204} a^{16} + \frac{337012995877656950444279}{12185619650848227819267927} a^{15} + \frac{10229921272941928860465979}{211217407281369282200644068} a^{14} - \frac{12020199009034762406803253}{211217407281369282200644068} a^{13} + \frac{836253329260327038879343}{52804351820342320550161017} a^{12} + \frac{723711134528149678825061}{4061873216949409273089309} a^{11} - \frac{1149398032571108597801819}{8123746433898818546178618} a^{10} + \frac{8083330304780052860362472}{52804351820342320550161017} a^{9} + \frac{67934837323092022272330304}{158413055461026961650483051} a^{8} + \frac{71024112808423893939498817}{158413055461026961650483051} a^{7} + \frac{45086485372981890650904449}{158413055461026961650483051} a^{6} - \frac{19960261854311846960559881}{70405802427123094066881356} a^{5} - \frac{4983788829555511661249985}{70405802427123094066881356} a^{4} - \frac{553270082200618904024957}{1353957738983136424363103} a^{3} - \frac{58004953698132598073361193}{211217407281369282200644068} a^{2} + \frac{71444488178356002876357797}{211217407281369282200644068} a - \frac{11851333698320149758339319}{105608703640684641100322034}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $17$
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:  \( 138586384159 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.45684.1, 3.3.45684.2, 3.3.564.1, 3.3.11421.1, 6.6.1565270892.1, 9.9.13443473060864064.2

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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}$$[2]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.6.11.1$x^{6} + 24$$6$$1$$11$$D_{6}$$[5/2]_{2}^{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
47Data not computed