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Properties

Label 18.0.40525551530...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{8}\cdot 3^{39}\cdot 5^{8}$
Root discriminant $30.08$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 54, -201, 567, -1017, 1494, -1386, 1647, -1217, 1116, -459, 282, -27, 45, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 + 45*x^14 - 27*x^13 + 282*x^12 - 459*x^11 + 1116*x^10 - 1217*x^9 + 1647*x^8 - 1386*x^7 + 1494*x^6 - 1017*x^5 + 567*x^4 - 201*x^3 + 54*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 + 45*x^14 - 27*x^13 + 282*x^12 - 459*x^11 + 1116*x^10 - 1217*x^9 + 1647*x^8 - 1386*x^7 + 1494*x^6 - 1017*x^5 + 567*x^4 - 201*x^3 + 54*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} + 45 x^{14} - 27 x^{13} + 282 x^{12} - 459 x^{11} + 1116 x^{10} - 1217 x^{9} + 1647 x^{8} - 1386 x^{7} + 1494 x^{6} - 1017 x^{5} + 567 x^{4} - 201 x^{3} + 54 x^{2} - 9 x + 1 \)

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:  \(-405255515301897626700000000=-\,2^{8}\cdot 3^{39}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.08$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{3} - \frac{1}{9}$, $\frac{1}{9} a^{10} - \frac{1}{3} a^{4} - \frac{1}{9} a$, $\frac{1}{9} a^{11} - \frac{1}{3} a^{5} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{6} - \frac{1}{9} a^{3}$, $\frac{1}{27} a^{13} - \frac{1}{27} a^{12} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{9} a^{7} + \frac{4}{9} a^{6} + \frac{2}{27} a^{4} + \frac{7}{27} a^{3} + \frac{1}{27} a + \frac{8}{27}$, $\frac{1}{27} a^{14} - \frac{1}{27} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} + \frac{4}{9} a^{6} + \frac{2}{27} a^{5} + \frac{1}{3} a^{4} + \frac{7}{27} a^{3} + \frac{1}{27} a^{2} + \frac{1}{3} a + \frac{8}{27}$, $\frac{1}{2970} a^{15} + \frac{1}{90} a^{14} - \frac{7}{594} a^{13} - \frac{41}{990} a^{12} + \frac{8}{165} a^{11} - \frac{1}{2970} a^{10} + \frac{2}{135} a^{9} - \frac{16}{33} a^{8} + \frac{281}{990} a^{7} - \frac{482}{1485} a^{6} + \frac{221}{495} a^{5} - \frac{223}{2970} a^{4} + \frac{9}{110} a^{3} + \frac{1}{110} a^{2} + \frac{248}{1485} a - \frac{731}{2970}$, $\frac{1}{2970} a^{16} - \frac{4}{495} a^{14} + \frac{7}{495} a^{13} + \frac{133}{2970} a^{12} + \frac{29}{990} a^{11} + \frac{7}{270} a^{10} - \frac{71}{1485} a^{9} + \frac{19}{110} a^{8} - \frac{1063}{2970} a^{7} - \frac{197}{495} a^{6} - \frac{67}{990} a^{5} + \frac{112}{495} a^{4} + \frac{349}{1485} a^{3} + \frac{47}{198} a^{2} - \frac{1259}{2970} a - \frac{127}{270}$, $\frac{1}{1713690} a^{17} + \frac{118}{856845} a^{16} - \frac{4}{25965} a^{15} + \frac{4618}{285615} a^{14} - \frac{13}{38082} a^{13} + \frac{443}{10386} a^{12} + \frac{68219}{1713690} a^{11} + \frac{958}{171369} a^{10} - \frac{1597}{63470} a^{9} + \frac{117673}{342738} a^{8} - \frac{62300}{171369} a^{7} + \frac{113869}{571230} a^{6} - \frac{90974}{285615} a^{5} - \frac{16648}{57123} a^{4} - \frac{251579}{571230} a^{3} + \frac{429901}{1713690} a^{2} - \frac{500941}{1713690} a - \frac{3589}{95205}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{191977}{856845} a^{17} - \frac{65111}{1713690} a^{16} - \frac{3475313}{1713690} a^{15} - \frac{590951}{1713690} a^{14} - \frac{17459759}{1713690} a^{13} + \frac{3695513}{856845} a^{12} - \frac{107442821}{1713690} a^{11} + \frac{15817060}{171369} a^{10} - \frac{202112518}{856845} a^{9} + \frac{402311819}{1713690} a^{8} - \frac{287019143}{856845} a^{7} + \frac{222470161}{856845} a^{6} - \frac{513937441}{1713690} a^{5} + \frac{315514313}{1713690} a^{4} - \frac{35698175}{342738} a^{3} + \frac{27240304}{856845} a^{2} - \frac{17310397}{1713690} a + \frac{1451878}{856845} \) (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:  \( 1060591.9742133385 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.3.11622614670000.2

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

Sibling fields

Degree 9 siblings: data not computed
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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$