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Properties

Label 18.6.30133614172...2033.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 37^{5}\cdot 109^{5}$
Root discriminant $43.42$
Ramified primes $3, 37, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-373248, 186624, 373248, -20736, -106272, -58320, 20736, 14616, -2808, -755, 669, 36, -202, -180, -69, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 69*x^14 - 180*x^13 - 202*x^12 + 36*x^11 + 669*x^10 - 755*x^9 - 2808*x^8 + 14616*x^7 + 20736*x^6 - 58320*x^5 - 106272*x^4 - 20736*x^3 + 373248*x^2 + 186624*x - 373248)
 
gp: K = bnfinit(x^18 - 3*x^16 - 69*x^14 - 180*x^13 - 202*x^12 + 36*x^11 + 669*x^10 - 755*x^9 - 2808*x^8 + 14616*x^7 + 20736*x^6 - 58320*x^5 - 106272*x^4 - 20736*x^3 + 373248*x^2 + 186624*x - 373248, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 69 x^{14} - 180 x^{13} - 202 x^{12} + 36 x^{11} + 669 x^{10} - 755 x^{9} - 2808 x^{8} + 14616 x^{7} + 20736 x^{6} - 58320 x^{5} - 106272 x^{4} - 20736 x^{3} + 373248 x^{2} + 186624 x - 373248 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(301336141724322998158191792033=3^{24}\cdot 37^{5}\cdot 109^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.42$
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:  $3, 37, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{6} a^{5} - \frac{1}{4} a^{3} + \frac{1}{12} a^{2}$, $\frac{1}{72} a^{12} - \frac{1}{24} a^{10} + \frac{1}{24} a^{8} - \frac{1}{2} a^{7} + \frac{7}{36} a^{6} - \frac{1}{2} a^{5} + \frac{7}{24} a^{4} - \frac{35}{72} a^{3}$, $\frac{1}{144} a^{13} - \frac{1}{48} a^{11} - \frac{23}{48} a^{9} - \frac{1}{4} a^{8} - \frac{29}{72} a^{7} + \frac{1}{4} a^{6} - \frac{17}{48} a^{5} - \frac{35}{144} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{864} a^{14} - \frac{1}{288} a^{12} - \frac{23}{288} a^{10} - \frac{5}{24} a^{9} - \frac{101}{432} a^{8} + \frac{1}{24} a^{7} - \frac{65}{288} a^{6} + \frac{109}{864} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{1}{2} a$, $\frac{1}{5184} a^{15} - \frac{1}{1728} a^{13} - \frac{23}{1728} a^{11} - \frac{5}{144} a^{10} - \frac{101}{2592} a^{9} + \frac{1}{144} a^{8} + \frac{223}{1728} a^{7} - \frac{755}{5184} a^{6} + \frac{11}{24} a^{5} - \frac{13}{72} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{10368} a^{16} - \frac{1}{3456} a^{14} - \frac{23}{3456} a^{12} - \frac{5}{288} a^{11} - \frac{101}{5184} a^{10} + \frac{1}{288} a^{9} + \frac{223}{3456} a^{8} - \frac{755}{10368} a^{7} - \frac{13}{48} a^{6} + \frac{59}{144} a^{5} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4212109145604459298018309784172288} a^{17} - \frac{11378639599637414316949746613}{702018190934076549669718297362048} a^{16} + \frac{18293294637162141103464143831}{1404036381868153099339436594724096} a^{15} - \frac{115489685839966067198964982103}{234006063644692183223239432454016} a^{14} - \frac{860629145736362580696841067759}{1404036381868153099339436594724096} a^{13} - \frac{795478250322136755934457672915}{234006063644692183223239432454016} a^{12} - \frac{45201913036881230395947575970821}{2106054572802229649009154892086144} a^{11} + \frac{8116627617687291555264075769507}{175504547733519137417429574340512} a^{10} + \frac{287844393320399769083893869871543}{1404036381868153099339436594724096} a^{9} - \frac{1604018098731782892820611250820681}{4212109145604459298018309784172288} a^{8} + \frac{245533946695134189751581570251639}{702018190934076549669718297362048} a^{7} + \frac{21417107572239135452186851216139}{58501515911173045805809858113504} a^{6} - \frac{990308214623686317610481163301}{19500505303724348601936619371168} a^{5} - \frac{2314124769823841013317826940997}{4875126325931087150484154842792} a^{4} + \frac{355147551541042005497645340551}{4875126325931087150484154842792} a^{3} + \frac{78377932236749599090752814195}{270840351440615952804675269044} a^{2} + \frac{5124698581506165519936635672}{67710087860153988201168817261} a + \frac{17354013936789024541357672424}{67710087860153988201168817261}$

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:  $11$
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:  \( 66577194.5022 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.26460513.1, 9.5.2143301553.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
37Data not computed
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$