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Properties

Label 18.0.11062565586...8703.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 29^{9}$
Root discriminant $27.98$
Ramified primes $3, 29$
Class number $6$
Class group $[6]$
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![2481, -5850, 7002, -6540, 3819, -990, -296, 2298, -1836, 20, 504, -144, 114, -48, 45, -10, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 10*x^15 + 45*x^14 - 48*x^13 + 114*x^12 - 144*x^11 + 504*x^10 + 20*x^9 - 1836*x^8 + 2298*x^7 - 296*x^6 - 990*x^5 + 3819*x^4 - 6540*x^3 + 7002*x^2 - 5850*x + 2481)
 
gp: K = bnfinit(x^18 + 6*x^16 - 10*x^15 + 45*x^14 - 48*x^13 + 114*x^12 - 144*x^11 + 504*x^10 + 20*x^9 - 1836*x^8 + 2298*x^7 - 296*x^6 - 990*x^5 + 3819*x^4 - 6540*x^3 + 7002*x^2 - 5850*x + 2481, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 10 x^{15} + 45 x^{14} - 48 x^{13} + 114 x^{12} - 144 x^{11} + 504 x^{10} + 20 x^{9} - 1836 x^{8} + 2298 x^{7} - 296 x^{6} - 990 x^{5} + 3819 x^{4} - 6540 x^{3} + 7002 x^{2} - 5850 x + 2481 \)

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:  \(-110625655867925924191778703=-\,3^{27}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.98$
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:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{18} a^{12} + \frac{2}{9} a^{9} - \frac{1}{6} a^{7} - \frac{1}{9} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{18} a^{13} + \frac{2}{9} a^{10} - \frac{1}{6} a^{8} - \frac{1}{9} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a$, $\frac{1}{36} a^{14} + \frac{1}{9} a^{11} + \frac{1}{6} a^{9} - \frac{1}{18} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{144} a^{15} - \frac{1}{144} a^{14} - \frac{1}{72} a^{12} + \frac{7}{72} a^{11} + \frac{1}{6} a^{10} + \frac{11}{72} a^{9} - \frac{1}{9} a^{8} + \frac{5}{24} a^{7} + \frac{5}{12} a^{6} + \frac{5}{24} a^{5} - \frac{1}{8} a^{4} + \frac{5}{12} a^{3} + \frac{11}{24} a^{2} - \frac{5}{16} a - \frac{1}{16}$, $\frac{1}{1192032} a^{16} + \frac{115}{66224} a^{15} - \frac{2413}{397344} a^{14} + \frac{1063}{66224} a^{13} + \frac{2023}{74502} a^{12} + \frac{30803}{198672} a^{11} - \frac{3745}{66224} a^{10} - \frac{42391}{596016} a^{9} + \frac{6521}{198672} a^{8} + \frac{96643}{596016} a^{7} - \frac{246271}{596016} a^{6} - \frac{1610}{12417} a^{5} + \frac{45017}{198672} a^{4} + \frac{59377}{198672} a^{3} + \frac{14001}{132448} a^{2} - \frac{38633}{99336} a - \frac{196189}{397344}$, $\frac{1}{32886601509225742795584} a^{17} + \frac{12643126478771911}{32886601509225742795584} a^{16} - \frac{83456108964488147453}{32886601509225742795584} a^{15} - \frac{99933643364687677295}{10962200503075247598528} a^{14} + \frac{38880212440037414311}{1827033417179207933088} a^{13} + \frac{63419056751627225255}{5481100251537623799264} a^{12} - \frac{10127625350512193281}{456758354294801983272} a^{11} - \frac{149635297152451858135}{685137531442202974908} a^{10} - \frac{94350134210853799667}{685137531442202974908} a^{9} + \frac{1565233026858141372215}{8221650377306435698896} a^{8} - \frac{184663845286514034479}{1027706297163304462362} a^{7} + \frac{7822738751284078870657}{16443300754612871397792} a^{6} + \frac{960480733089545634133}{5481100251537623799264} a^{5} + \frac{82687240670463642863}{2740550125768811899632} a^{4} - \frac{1753165634854939890617}{3654066834358415866176} a^{3} - \frac{21462679705077643427}{576957921214486715712} a^{2} + \frac{3637530214332587249395}{10962200503075247598528} a + \frac{3132175084056518355511}{10962200503075247598528}$

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

Class group and class number

$C_{6}$, which has order $6$

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:  \( 184881.082486 \)
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

\(\Q(\sqrt{-87}) \), 3.1.87.1 x3, \(\Q(\zeta_{9})^+\), 6.0.658503.1, 6.0.480048687.2 x2, 6.0.480048687.1, 9.3.38883943647.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.480048687.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.3$x^{6} + 3 x^{4} + 24$$6$$1$$9$$C_6$$[2]_{2}$
$29$29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$