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Properties

Label 18.0.80556206429...1408.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{19}\cdot 31^{9}$
Root discriminant $35.51$
Ramified primes $2, 3, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34061, 78738, 67788, 9544, -36777, -39294, -13782, 7584, 10962, 4148, -1308, -1926, -416, 300, 153, -22, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 22*x^15 + 153*x^14 + 300*x^13 - 416*x^12 - 1926*x^11 - 1308*x^10 + 4148*x^9 + 10962*x^8 + 7584*x^7 - 13782*x^6 - 39294*x^5 - 36777*x^4 + 9544*x^3 + 67788*x^2 + 78738*x + 34061)
 
gp: K = bnfinit(x^18 - 18*x^16 - 22*x^15 + 153*x^14 + 300*x^13 - 416*x^12 - 1926*x^11 - 1308*x^10 + 4148*x^9 + 10962*x^8 + 7584*x^7 - 13782*x^6 - 39294*x^5 - 36777*x^4 + 9544*x^3 + 67788*x^2 + 78738*x + 34061, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 22 x^{15} + 153 x^{14} + 300 x^{13} - 416 x^{12} - 1926 x^{11} - 1308 x^{10} + 4148 x^{9} + 10962 x^{8} + 7584 x^{7} - 13782 x^{6} - 39294 x^{5} - 36777 x^{4} + 9544 x^{3} + 67788 x^{2} + 78738 x + 34061 \)

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:  \(-8055620642901114529869201408=-\,2^{18}\cdot 3^{19}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.51$
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:  $2, 3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{18} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{2}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a + \frac{2}{9}$, $\frac{1}{2412} a^{16} + \frac{4}{201} a^{15} - \frac{37}{804} a^{14} - \frac{191}{402} a^{13} - \frac{29}{67} a^{12} - \frac{15}{134} a^{11} - \frac{224}{603} a^{10} - \frac{62}{201} a^{9} + \frac{16}{201} a^{8} - \frac{62}{201} a^{7} - \frac{9}{134} a^{6} - \frac{25}{67} a^{5} - \frac{20}{67} a^{4} - \frac{29}{134} a^{3} - \frac{91}{804} a^{2} + \frac{5}{18} a - \frac{209}{804}$, $\frac{1}{9098568105463597055888061699156} a^{17} + \frac{81773445500112217118486221}{2274642026365899263972015424789} a^{16} - \frac{125811203646928539480502109591}{9098568105463597055888061699156} a^{15} + \frac{103384245487169715528199782805}{758214008788633087990671808263} a^{14} - \frac{179310538965573525875733092036}{758214008788633087990671808263} a^{13} + \frac{202002442710564932018587725547}{1516428017577266175981343616526} a^{12} - \frac{261219297563350757559071500922}{2274642026365899263972015424789} a^{11} + \frac{848988181466865675759601126822}{2274642026365899263972015424789} a^{10} + \frac{918311044016052565296117608416}{2274642026365899263972015424789} a^{9} + \frac{227764307621701112164744356329}{758214008788633087990671808263} a^{8} - \frac{303786243410599506503197970717}{1516428017577266175981343616526} a^{7} - \frac{277627407900931584816029573042}{758214008788633087990671808263} a^{6} - \frac{56353190223852198755628346906}{252738002929544362663557269421} a^{5} + \frac{183193043731154964348952320317}{505476005859088725327114538842} a^{4} - \frac{87870186522585887208894536587}{3032856035154532351962687233052} a^{3} + \frac{65541171195745833235172627471}{4549284052731798527944030849578} a^{2} - \frac{994292183739737907349231524647}{9098568105463597055888061699156} a - \frac{1203016847210414205677144334925}{4549284052731798527944030849578}$

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:  $8$
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:  \( 1940041.86545 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.1.31.1, 6.0.5719872.3, 9.3.37528080192.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.12.12.6$x^{12} + 24 x^{11} - 3 x^{10} + 81 x^{9} - 18 x^{8} + 54 x^{7} + 108 x^{5} - 54 x^{4} - 27 x^{3} - 81 x - 81$$3$$4$$12$12T39$[3/2, 3/2]_{2}^{4}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.8.6.1$x^{8} - 7471 x^{4} + 19927296$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$