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Properties

Label 18.0.13899900815...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{14}\cdot 7^{10}$
Root discriminant $28.34$
Ramified primes $2, 3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 90, -336, 769, -1052, 759, -204, 127, -294, 72, 196, -86, -82, 69, -16, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 5*x^16 - 16*x^15 + 69*x^14 - 82*x^13 - 86*x^12 + 196*x^11 + 72*x^10 - 294*x^9 + 127*x^8 - 204*x^7 + 759*x^6 - 1052*x^5 + 769*x^4 - 336*x^3 + 90*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 5*x^16 - 16*x^15 + 69*x^14 - 82*x^13 - 86*x^12 + 196*x^11 + 72*x^10 - 294*x^9 + 127*x^8 - 204*x^7 + 759*x^6 - 1052*x^5 + 769*x^4 - 336*x^3 + 90*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 5 x^{16} - 16 x^{15} + 69 x^{14} - 82 x^{13} - 86 x^{12} + 196 x^{11} + 72 x^{10} - 294 x^{9} + 127 x^{8} - 204 x^{7} + 759 x^{6} - 1052 x^{5} + 769 x^{4} - 336 x^{3} + 90 x^{2} - 14 x + 1 \)

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:  \(-138999008151675000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.34$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} - \frac{2}{5}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{11} - \frac{1}{10} a^{9} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} + \frac{1}{10} a^{4} - \frac{2}{5} a$, $\frac{1}{140} a^{15} + \frac{3}{140} a^{14} - \frac{3}{70} a^{13} - \frac{1}{140} a^{12} - \frac{3}{28} a^{11} - \frac{9}{140} a^{10} - \frac{9}{70} a^{9} - \frac{19}{140} a^{8} + \frac{3}{70} a^{7} - \frac{3}{7} a^{6} + \frac{6}{35} a^{5} - \frac{3}{35} a^{4} + \frac{9}{140} a^{3} - \frac{41}{140} a^{2} - \frac{3}{7} a + \frac{57}{140}$, $\frac{1}{140} a^{16} - \frac{1}{140} a^{14} + \frac{3}{140} a^{13} + \frac{1}{70} a^{12} - \frac{17}{70} a^{11} + \frac{9}{140} a^{10} + \frac{3}{20} a^{9} + \frac{1}{20} a^{8} + \frac{12}{35} a^{7} - \frac{3}{70} a^{6} - \frac{1}{10} a^{5} - \frac{11}{140} a^{4} + \frac{29}{70} a^{3} - \frac{9}{20} a^{2} + \frac{27}{140} a - \frac{31}{140}$, $\frac{1}{140} a^{17} + \frac{3}{70} a^{14} - \frac{1}{35} a^{13} - \frac{1}{20} a^{12} - \frac{17}{70} a^{11} - \frac{3}{14} a^{10} - \frac{11}{140} a^{9} + \frac{29}{140} a^{8} + \frac{3}{10} a^{7} - \frac{23}{70} a^{6} + \frac{11}{28} a^{5} + \frac{23}{70} a^{4} - \frac{27}{70} a^{3} - \frac{2}{5} a^{2} - \frac{7}{20} a + \frac{3}{28}$

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:  \( -\frac{35}{2} a^{17} + \frac{249}{4} a^{16} - \frac{597}{10} a^{15} + \frac{5057}{20} a^{14} - \frac{21903}{20} a^{13} + 947 a^{12} + \frac{9687}{5} a^{11} - \frac{10309}{4} a^{10} - \frac{48657}{20} a^{9} + \frac{81723}{20} a^{8} - \frac{727}{2} a^{7} + \frac{33781}{10} a^{6} - 11809 a^{5} + \frac{262547}{20} a^{4} - \frac{75099}{10} a^{3} + \frac{9849}{4} a^{2} - \frac{9157}{20} a + \frac{801}{20} \) (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:  \( 560964.0932099912 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.140.1, 6.0.826875.2, 6.0.529200.1

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

Sibling fields

Degree 12 sibling: 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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$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$$[\ ]_{2}^{3}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$