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Properties

Label 18.0.17436035582...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 7^{12}$
Root discriminant $17.20$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -4, 15, 17, -49, 121, -34, 41, 43, 22, 16, -6, -5, -4, -6, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 1)
 
gp: K = bnfinit(x^18 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 6 x^{15} - 4 x^{14} - 5 x^{13} - 6 x^{12} + 16 x^{11} + 22 x^{10} + 43 x^{9} + 41 x^{8} - 34 x^{7} + 121 x^{6} - 49 x^{5} + 17 x^{4} + 15 x^{3} - 4 x^{2} - x + 1 \)

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:  \(-17436035582546112000000=-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.20$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2938} a^{15} + \frac{111}{1469} a^{13} - \frac{5}{2938} a^{12} + \frac{581}{2938} a^{11} - \frac{444}{1469} a^{10} - \frac{441}{1469} a^{9} + \frac{1195}{2938} a^{8} - \frac{411}{2938} a^{7} - \frac{582}{1469} a^{6} - \frac{1455}{2938} a^{5} + \frac{855}{2938} a^{4} - \frac{887}{2938} a^{3} - \frac{581}{2938} a^{2} + \frac{111}{1469} a - \frac{1}{2938}$, $\frac{1}{208598} a^{16} + \frac{29}{208598} a^{15} - \frac{27689}{208598} a^{14} + \frac{9827}{104299} a^{13} - \frac{21599}{208598} a^{12} + \frac{77659}{208598} a^{11} - \frac{10475}{208598} a^{10} - \frac{49651}{104299} a^{9} - \frac{61241}{208598} a^{8} + \frac{57429}{208598} a^{7} + \frac{36013}{104299} a^{6} + \frac{1453}{16046} a^{5} - \frac{14488}{104299} a^{4} - \frac{68905}{208598} a^{3} + \frac{4597}{16046} a^{2} - \frac{454}{104299} a - \frac{24253}{104299}$, $\frac{1}{208598} a^{17} + \frac{6}{104299} a^{15} - \frac{11757}{208598} a^{14} + \frac{4157}{104299} a^{13} + \frac{39825}{208598} a^{12} - \frac{36473}{104299} a^{11} + \frac{99535}{208598} a^{10} + \frac{34379}{104299} a^{9} - \frac{42047}{208598} a^{8} + \frac{26095}{208598} a^{7} - \frac{39691}{208598} a^{6} - \frac{36435}{104299} a^{5} + \frac{19375}{104299} a^{4} - \frac{1}{2938} a^{3} - \frac{32277}{104299} a^{2} - \frac{48089}{208598} a + \frac{22245}{208598}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{14}{113} a^{17} - \frac{305}{1469} a^{16} - \frac{19}{113} a^{15} + \frac{774}{1469} a^{14} + \frac{2513}{1469} a^{13} + \frac{2507}{1469} a^{12} + \frac{2975}{1469} a^{11} - \frac{805}{1469} a^{10} - \frac{8509}{1469} a^{9} - \frac{15565}{1469} a^{8} - \frac{22435}{1469} a^{7} - \frac{9210}{1469} a^{6} - \frac{14568}{1469} a^{5} - \frac{25462}{1469} a^{4} + \frac{5644}{1469} a^{3} - \frac{6389}{1469} a^{2} - \frac{1671}{1469} a + \frac{75}{113} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 13584.372345087919 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.0.25930800.1, 6.0.64827.1, 9.3.941192000.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$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$$[\ ]_{2}^{3}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$