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Properties

Label 18.18.1261494321...6401.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{44}\cdot 71^{6}$
Root discriminant $60.73$
Ramified primes $3, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_9$ (as 18T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-26241, -206577, -566379, -421407, 829629, 1650123, 410361, -957456, -512883, 240556, 164547, -30708, -24780, 1944, 1917, -48, -72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241)
 
gp: K = bnfinit(x^18 - 72*x^16 - 48*x^15 + 1917*x^14 + 1944*x^13 - 24780*x^12 - 30708*x^11 + 164547*x^10 + 240556*x^9 - 512883*x^8 - 957456*x^7 + 410361*x^6 + 1650123*x^5 + 829629*x^4 - 421407*x^3 - 566379*x^2 - 206577*x - 26241, 1)
 

Normalized defining polynomial

\( x^{18} - 72 x^{16} - 48 x^{15} + 1917 x^{14} + 1944 x^{13} - 24780 x^{12} - 30708 x^{11} + 164547 x^{10} + 240556 x^{9} - 512883 x^{8} - 957456 x^{7} + 410361 x^{6} + 1650123 x^{5} + 829629 x^{4} - 421407 x^{3} - 566379 x^{2} - 206577 x - 26241 \)

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:  \(126149432166859917805140950806401=3^{44}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.73$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{114058824935167611220226426609989031} a^{17} - \frac{17231572137385694260003632967495694}{114058824935167611220226426609989031} a^{16} + \frac{25261885562601020404103708020190145}{114058824935167611220226426609989031} a^{15} + \frac{31949978374743118760709312896918673}{114058824935167611220226426609989031} a^{14} - \frac{4872862523536539743461765400422462}{114058824935167611220226426609989031} a^{13} + \frac{3931855763948480593350908624808644}{114058824935167611220226426609989031} a^{12} - \frac{35346184579401067376937110632009854}{114058824935167611220226426609989031} a^{11} + \frac{21198427886467217786110920318298885}{114058824935167611220226426609989031} a^{10} - \frac{39897621271875959864540882910834479}{114058824935167611220226426609989031} a^{9} - \frac{31676462387089935361279577590290392}{114058824935167611220226426609989031} a^{8} + \frac{5128975128402939511383520358028009}{114058824935167611220226426609989031} a^{7} + \frac{39322028487078092560340965259183158}{114058824935167611220226426609989031} a^{6} + \frac{18052963998382836047733387523559543}{114058824935167611220226426609989031} a^{5} - \frac{19049183563897358704677297299450167}{114058824935167611220226426609989031} a^{4} - \frac{14655545884455575933053303257977375}{114058824935167611220226426609989031} a^{3} + \frac{39330002490071492937003490716520575}{114058824935167611220226426609989031} a^{2} - \frac{41712540296039668485895468115343300}{114058824935167611220226426609989031} a - \frac{53423775881995547922468912344510647}{114058824935167611220226426609989031}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 25129615009.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_9$ (as 18T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2^2 : C_9$
Character table for $C_2^2 : C_9$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.33074001.1, \(\Q(\zeta_{27})^+\)

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
$71$71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.0.1$x^{6} - 2 x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$
71.6.3.1$x^{6} - 142 x^{4} + 5041 x^{2} - 1431644$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$