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Properties

Label 18.0.26821542628...1547.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 71^{4}$
Root discriminant $49.03$
Ramified primes $3, 7, 71$
Class number $960$ (GRH)
Class group $[2, 2, 2, 120]$ (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![187741, -77649, 485196, 99481, 328221, 69153, 117034, -21567, 39402, -15787, 12342, -5793, 3364, -1293, 507, -144, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 36*x^16 - 144*x^15 + 507*x^14 - 1293*x^13 + 3364*x^12 - 5793*x^11 + 12342*x^10 - 15787*x^9 + 39402*x^8 - 21567*x^7 + 117034*x^6 + 69153*x^5 + 328221*x^4 + 99481*x^3 + 485196*x^2 - 77649*x + 187741)
 
gp: K = bnfinit(x^18 - 6*x^17 + 36*x^16 - 144*x^15 + 507*x^14 - 1293*x^13 + 3364*x^12 - 5793*x^11 + 12342*x^10 - 15787*x^9 + 39402*x^8 - 21567*x^7 + 117034*x^6 + 69153*x^5 + 328221*x^4 + 99481*x^3 + 485196*x^2 - 77649*x + 187741, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 36 x^{16} - 144 x^{15} + 507 x^{14} - 1293 x^{13} + 3364 x^{12} - 5793 x^{11} + 12342 x^{10} - 15787 x^{9} + 39402 x^{8} - 21567 x^{7} + 117034 x^{6} + 69153 x^{5} + 328221 x^{4} + 99481 x^{3} + 485196 x^{2} - 77649 x + 187741 \)

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:  \(-2682154262850134111973718751547=-\,3^{27}\cdot 7^{12}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.03$
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:  $3, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{204236824679891277333762384186489531082128357434} a^{17} + \frac{27524381608961569844016268118393349555257114735}{204236824679891277333762384186489531082128357434} a^{16} - \frac{9532703287904976121897056884500100969607865378}{102118412339945638666881192093244765541064178717} a^{15} + \frac{23039174313630542761518974453692711387121253407}{204236824679891277333762384186489531082128357434} a^{14} + \frac{3518053185034745013198302509059592446757758213}{204236824679891277333762384186489531082128357434} a^{13} - \frac{35554689870639701711356431339844623273300140663}{204236824679891277333762384186489531082128357434} a^{12} + \frac{23730027368840049988935505651318133469111337727}{204236824679891277333762384186489531082128357434} a^{11} + \frac{99077322933266548775093100767738267045868003369}{204236824679891277333762384186489531082128357434} a^{10} + \frac{24257442136742904164692839571754146134108456455}{204236824679891277333762384186489531082128357434} a^{9} - \frac{2203955334121244008110289249829764243165417294}{102118412339945638666881192093244765541064178717} a^{8} - \frac{47274488410509179789088222542127572516461020857}{102118412339945638666881192093244765541064178717} a^{7} - \frac{20863106488122992723524682671250664849373092251}{102118412339945638666881192093244765541064178717} a^{6} + \frac{48446047301584324489025527115960389332577355561}{102118412339945638666881192093244765541064178717} a^{5} - \frac{17866577376408483669772341695617413055085306530}{102118412339945638666881192093244765541064178717} a^{4} + \frac{65932178978241077956831003282438629411904271561}{204236824679891277333762384186489531082128357434} a^{3} + \frac{36446354561283731195856688161387210979571975555}{204236824679891277333762384186489531082128357434} a^{2} - \frac{12994420816620390643599812988591793360441157087}{204236824679891277333762384186489531082128357434} a + \frac{14009624351250016844062506109611321706954703542}{102118412339945638666881192093244765541064178717}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{120}$, which has order $960$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
71Data not computed