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Properties

Label 18.18.1879333044...9521.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{24}\cdot 13^{16}$
Root discriminant $42.30$
Ramified primes $3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_3:S_3.C_2$ (as 18T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -972, -8262, -21483, -7479, 43173, 42927, -30060, -46602, 8945, 23253, -1089, -5953, 57, 756, -1, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27)
 
gp: K = bnfinit(x^18 - 45*x^16 - x^15 + 756*x^14 + 57*x^13 - 5953*x^12 - 1089*x^11 + 23253*x^10 + 8945*x^9 - 46602*x^8 - 30060*x^7 + 42927*x^6 + 43173*x^5 - 7479*x^4 - 21483*x^3 - 8262*x^2 - 972*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 45 x^{16} - x^{15} + 756 x^{14} + 57 x^{13} - 5953 x^{12} - 1089 x^{11} + 23253 x^{10} + 8945 x^{9} - 46602 x^{8} - 30060 x^{7} + 42927 x^{6} + 43173 x^{5} - 7479 x^{4} - 21483 x^{3} - 8262 x^{2} - 972 x - 27 \)

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:  \(187933304498364210515293279521=3^{24}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.30$
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:  $3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{3} a^{10} - \frac{4}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} + \frac{1}{3} a^{11} - \frac{4}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{21690327925000470987} a^{17} - \frac{1164900734750970862}{21690327925000470987} a^{16} - \frac{85560837471631630}{21690327925000470987} a^{15} - \frac{3376903538722448023}{21690327925000470987} a^{14} + \frac{1833216720104258998}{21690327925000470987} a^{13} + \frac{383031098794732201}{21690327925000470987} a^{12} - \frac{6519194142840169444}{21690327925000470987} a^{11} + \frac{948002908251045850}{21690327925000470987} a^{10} - \frac{55213259451987881}{21690327925000470987} a^{9} + \frac{7310550570323231600}{21690327925000470987} a^{8} - \frac{10391988078963612350}{21690327925000470987} a^{7} + \frac{3300478606106847025}{21690327925000470987} a^{6} + \frac{822208543515376456}{7230109308333490329} a^{5} - \frac{1640744348386008434}{7230109308333490329} a^{4} - \frac{647999283816941335}{2410036436111163443} a^{3} + \frac{183901667152804444}{2410036436111163443} a^{2} - \frac{979901791704081518}{2410036436111163443} a - \frac{853674537034239746}{2410036436111163443}$

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:  \( 637998345.319 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3.C_2$ (as 18T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $C_3\times C_3:S_3.C_2$
Character table for $C_3\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\zeta_{13})^+\), 6.6.187388721.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$3.9.15.3$x^{9} + 9 x^{6} + 72 x^{3} + 27$$3$$3$$15$$S_3\times C_3$$[5/2]_{2}^{3}$
3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$