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Properties

Label 18.0.12957224433...5923.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 37^{12}$
Root discriminant $19.23$
Ramified primes $3, 37$
Class number $1$
Class group Trivial
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![1, -1, 10, 41, 72, 135, 210, 192, 188, 192, -52, -65, 109, -47, 3, 0, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 3*x^14 - 47*x^13 + 109*x^12 - 65*x^11 - 52*x^10 + 192*x^9 + 188*x^8 + 192*x^7 + 210*x^6 + 135*x^5 + 72*x^4 + 41*x^3 + 10*x^2 - x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 3*x^14 - 47*x^13 + 109*x^12 - 65*x^11 - 52*x^10 + 192*x^9 + 188*x^8 + 192*x^7 + 210*x^6 + 135*x^5 + 72*x^4 + 41*x^3 + 10*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 3 x^{14} - 47 x^{13} + 109 x^{12} - 65 x^{11} - 52 x^{10} + 192 x^{9} + 188 x^{8} + 192 x^{7} + 210 x^{6} + 135 x^{5} + 72 x^{4} + 41 x^{3} + 10 x^{2} - x + 1 \)

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:  \(-129572244330949414435923=-\,3^{9}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.23$
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:  $3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{588} a^{15} - \frac{1}{294} a^{14} + \frac{19}{588} a^{13} - \frac{43}{294} a^{12} + \frac{57}{196} a^{11} - \frac{13}{84} a^{10} - \frac{15}{49} a^{9} - \frac{6}{49} a^{8} - \frac{281}{588} a^{7} - \frac{253}{588} a^{6} + \frac{107}{588} a^{5} - \frac{23}{147} a^{4} + \frac{37}{196} a^{3} + \frac{25}{84} a^{2} - \frac{27}{196} a + \frac{145}{588}$, $\frac{1}{59388} a^{16} - \frac{5}{8484} a^{15} + \frac{61}{588} a^{14} + \frac{9479}{59388} a^{13} + \frac{6341}{59388} a^{12} + \frac{6443}{29694} a^{11} - \frac{7291}{19796} a^{10} - \frac{99}{4949} a^{9} + \frac{9453}{19796} a^{8} - \frac{2890}{14847} a^{7} - \frac{377}{2121} a^{6} + \frac{4217}{59388} a^{5} + \frac{6283}{59388} a^{4} - \frac{3910}{14847} a^{3} + \frac{986}{14847} a^{2} - \frac{11527}{29694} a - \frac{11449}{59388}$, $\frac{1}{17843183988} a^{17} - \frac{715}{424837714} a^{16} + \frac{2117498}{4460795997} a^{15} + \frac{228872738}{4460795997} a^{14} - \frac{1268878277}{8921591994} a^{13} - \frac{1483651201}{17843183988} a^{12} + \frac{7623597019}{17843183988} a^{11} - \frac{1740906487}{17843183988} a^{10} + \frac{1924255813}{5947727996} a^{9} + \frac{1379548507}{17843183988} a^{8} - \frac{1012413621}{2973863998} a^{7} - \frac{5517480649}{17843183988} a^{6} + \frac{42981637}{182073306} a^{5} - \frac{32019277}{849675428} a^{4} + \frac{4402083755}{8921591994} a^{3} + \frac{1807770413}{4460795997} a^{2} + \frac{5192447383}{17843183988} a - \frac{5935077515}{17843183988}$

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:  \( -\frac{2520941}{176665188} a^{17} - \frac{4401374}{44166297} a^{16} + \frac{23339391}{58888396} a^{15} - \frac{37722481}{44166297} a^{14} + \frac{42424189}{176665188} a^{13} + \frac{12680585}{176665188} a^{12} + \frac{476642459}{88332594} a^{11} - \frac{735120485}{44166297} a^{10} + \frac{2738471089}{176665188} a^{9} - \frac{70185443}{58888396} a^{8} - \frac{710653079}{25237884} a^{7} - \frac{694073969}{29444198} a^{6} - \frac{1514997517}{58888396} a^{5} - \frac{4475985025}{176665188} a^{4} - \frac{2426246005}{176665188} a^{3} - \frac{1283012159}{176665188} a^{2} - \frac{11983771}{4206314} a + \frac{6116627}{14722099} \) (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:  \( 25740.6797756 \)
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

\(\Q(\sqrt{-3}) \), 3.1.4107.1 x3, 3.3.1369.1, 6.0.50602347.2, 6.0.50602347.1, 6.0.36963.1 x2, 9.3.69274613043.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.36963.1
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$