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Properties

Label 18.0.35246033837...1591.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 71^{3}$
Root discriminant $29.84$
Ramified primes $3, 71$
Class number $21$
Class group $[21]$
Galois group $C_2\times C_2^2:C_9$ (as 18T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, -243, 1215, -3267, 7452, -10530, 14742, -11961, 12636, -5939, 6264, -1404, 1473, -99, 162, -9, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 9*x^15 + 162*x^14 - 99*x^13 + 1473*x^12 - 1404*x^11 + 6264*x^10 - 5939*x^9 + 12636*x^8 - 11961*x^7 + 14742*x^6 - 10530*x^5 + 7452*x^4 - 3267*x^3 + 1215*x^2 - 243*x + 27)
 
gp: K = bnfinit(x^18 + 9*x^16 - 9*x^15 + 162*x^14 - 99*x^13 + 1473*x^12 - 1404*x^11 + 6264*x^10 - 5939*x^9 + 12636*x^8 - 11961*x^7 + 14742*x^6 - 10530*x^5 + 7452*x^4 - 3267*x^3 + 1215*x^2 - 243*x + 27, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 9 x^{15} + 162 x^{14} - 99 x^{13} + 1473 x^{12} - 1404 x^{11} + 6264 x^{10} - 5939 x^{9} + 12636 x^{8} - 11961 x^{7} + 14742 x^{6} - 10530 x^{5} + 7452 x^{4} - 3267 x^{3} + 1215 x^{2} - 243 x + 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-352460338371438479971671591=-\,3^{44}\cdot 71^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.84$
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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{9} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{9} + \frac{1}{9} a^{6}$, $\frac{1}{27} a^{16} - \frac{1}{9} a^{13} + \frac{1}{3} a^{11} - \frac{1}{9} a^{10} + \frac{10}{27} a^{7} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a$, $\frac{1}{28413622886531926918619835459} a^{17} - \frac{11074204946674273893928424}{3157069209614658546513315051} a^{16} + \frac{12630236375693410591634719}{1052356403204886182171105017} a^{15} + \frac{262508912144760625746747458}{9471207628843975639539945153} a^{14} - \frac{361163591630663115991111016}{3157069209614658546513315051} a^{13} + \frac{228862743062356682673995540}{3157069209614658546513315051} a^{12} - \frac{4161026043950666516493754897}{9471207628843975639539945153} a^{11} + \frac{391895485187855328363257073}{1052356403204886182171105017} a^{10} + \frac{204892188700272501761736024}{1052356403204886182171105017} a^{9} + \frac{5286395397800835724409138800}{28413622886531926918619835459} a^{8} + \frac{155932127252992541935486036}{3157069209614658546513315051} a^{7} - \frac{1211626597706022261554427304}{3157069209614658546513315051} a^{6} - \frac{926428296245398845781170016}{9471207628843975639539945153} a^{5} - \frac{6991817188266597766852352}{3157069209614658546513315051} a^{4} + \frac{283649692125368924357868634}{3157069209614658546513315051} a^{3} - \frac{1311104624227171381282885133}{3157069209614658546513315051} a^{2} - \frac{102778970724412649629453482}{1052356403204886182171105017} a + \frac{338509866103359584776018440}{1052356403204886182171105017}$

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

Class group and class number

$C_{21}$, which has order $21$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 40934.0329443 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_9$ (as 18T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 24 conjugacy class representatives for $C_2\times C_2^2:C_9$
Character table for $C_2\times C_2^2:C_9$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.0.465831.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ $18$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ $18$

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$71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
71.6.3.1$x^{6} - 142 x^{4} + 5041 x^{2} - 1431644$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.0.1$x^{6} - 2 x + 13$$1$$6$$0$$C_6$$[\ ]^{6}$