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Properties

Label 18.0.34652609452...9443.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 127^{12}$
Root discriminant $43.76$
Ramified primes $3, 127$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1728, 5616, 972, -16614, 15117, -3567, -4961, 1110, 7506, -9414, 6516, -3120, 1144, -462, 234, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 234*x^14 - 462*x^13 + 1144*x^12 - 3120*x^11 + 6516*x^10 - 9414*x^9 + 7506*x^8 + 1110*x^7 - 4961*x^6 - 3567*x^5 + 15117*x^4 - 16614*x^3 + 972*x^2 + 5616*x + 1728)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 234*x^14 - 462*x^13 + 1144*x^12 - 3120*x^11 + 6516*x^10 - 9414*x^9 + 7506*x^8 + 1110*x^7 - 4961*x^6 - 3567*x^5 + 15117*x^4 - 16614*x^3 + 972*x^2 + 5616*x + 1728, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 234 x^{14} - 462 x^{13} + 1144 x^{12} - 3120 x^{11} + 6516 x^{10} - 9414 x^{9} + 7506 x^{8} + 1110 x^{7} - 4961 x^{6} - 3567 x^{5} + 15117 x^{4} - 16614 x^{3} + 972 x^{2} + 5616 x + 1728 \)

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:  \(-346526094527773303165120969443=-\,3^{9}\cdot 127^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.76$
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:  $3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{5} + \frac{5}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} - \frac{1}{24} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{144} a^{12} + \frac{1}{72} a^{10} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{11}{144} a^{6} + \frac{1}{6} a^{5} - \frac{13}{72} a^{4} - \frac{5}{48} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{720} a^{13} + \frac{1}{720} a^{12} + \frac{1}{90} a^{11} + \frac{1}{144} a^{10} - \frac{7}{240} a^{9} + \frac{1}{120} a^{8} + \frac{59}{720} a^{7} - \frac{49}{720} a^{6} - \frac{8}{45} a^{5} + \frac{163}{720} a^{4} + \frac{29}{80} a^{3} + \frac{9}{40} a^{2} - \frac{1}{4} a - \frac{1}{5}$, $\frac{1}{1440} a^{14} - \frac{1}{1440} a^{13} - \frac{1}{360} a^{12} + \frac{19}{1440} a^{11} + \frac{1}{160} a^{10} + \frac{1}{30} a^{9} + \frac{47}{1440} a^{8} + \frac{13}{1440} a^{7} - \frac{1}{18} a^{6} - \frac{331}{1440} a^{5} - \frac{11}{96} a^{4} - \frac{1}{8} a^{3} + \frac{29}{60} a^{2} + \frac{3}{20} a + \frac{1}{5}$, $\frac{1}{1440} a^{15} - \frac{1}{1440} a^{13} - \frac{1}{1440} a^{12} - \frac{23}{1440} a^{10} - \frac{49}{1440} a^{9} - \frac{1}{40} a^{8} - \frac{71}{1440} a^{7} + \frac{13}{1440} a^{6} - \frac{3}{40} a^{5} - \frac{73}{1440} a^{4} + \frac{5}{24} a^{3} + \frac{1}{12} a^{2} + \frac{7}{20} a - \frac{1}{5}$, $\frac{1}{72604800} a^{16} - \frac{1}{9075600} a^{15} + \frac{4871}{18151200} a^{14} - \frac{2213}{4537800} a^{13} - \frac{85661}{36302400} a^{12} + \frac{148169}{9075600} a^{11} + \frac{6851}{806720} a^{10} + \frac{1099963}{36302400} a^{9} + \frac{62981}{2268900} a^{8} + \frac{663637}{9075600} a^{7} - \frac{2263531}{36302400} a^{6} - \frac{1077863}{9075600} a^{5} - \frac{1119031}{72604800} a^{4} - \frac{279827}{806720} a^{3} + \frac{141809}{2016800} a^{2} - \frac{28269}{100840} a - \frac{11311}{126050}$, $\frac{1}{84439382400} a^{17} + \frac{191}{28146460800} a^{16} + \frac{161641}{5277461400} a^{15} + \frac{18418}{73298075} a^{14} - \frac{5210461}{8443938240} a^{13} + \frac{9208361}{8443938240} a^{12} + \frac{622240361}{42219691200} a^{11} + \frac{25950229}{21109845600} a^{10} - \frac{1672149791}{42219691200} a^{9} - \frac{59820427}{2345538400} a^{8} + \frac{2802394397}{42219691200} a^{7} + \frac{2078859827}{42219691200} a^{6} + \frac{91878497}{16887876480} a^{5} + \frac{2282799959}{84439382400} a^{4} - \frac{540244977}{4691076800} a^{3} - \frac{149715013}{7036615200} a^{2} + \frac{8790991}{586384600} a - \frac{4983461}{146596150}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{3082697}{8443938240} a^{17} - \frac{52405849}{16887876480} a^{16} + \frac{1519711}{117276920} a^{15} - \frac{29658545}{844393824} a^{14} + \frac{35824051}{469107680} a^{13} - \frac{256569079}{1688787648} a^{12} + \frac{1630219579}{4221969120} a^{11} - \frac{8721159601}{8443938240} a^{10} + \frac{5909363503}{2814646080} a^{9} - \frac{6392463559}{2110984560} a^{8} + \frac{1143294423}{469107680} a^{7} + \frac{681156587}{8443938240} a^{6} - \frac{7572937711}{8443938240} a^{5} - \frac{2127653781}{1876430720} a^{4} + \frac{13844535881}{2814646080} a^{3} - \frac{2509404233}{469107680} a^{2} + \frac{73307541}{117276920} a + \frac{29234671}{29319230} \) (order $6$)
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:  \( 534990510.048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.48387.1 x3, 3.3.16129.1, 6.0.7023905307.1, 6.0.435483.1 x2, 6.0.7023905307.2, 9.3.113288568696603.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.435483.1
Degree 9 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$