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Properties

Label 18.0.57892131675...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{14}\cdot 7^{6}$
Root discriminant $18.39$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 12, 40, 72, 76, -314, 619, -597, 353, -108, -23, 43, -54, 41, -21, 2, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 2*x^15 - 21*x^14 + 41*x^13 - 54*x^12 + 43*x^11 - 23*x^10 - 108*x^9 + 353*x^8 - 597*x^7 + 619*x^6 - 314*x^5 + 76*x^4 + 72*x^3 + 40*x^2 + 12*x + 4)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 2*x^15 - 21*x^14 + 41*x^13 - 54*x^12 + 43*x^11 - 23*x^10 - 108*x^9 + 353*x^8 - 597*x^7 + 619*x^6 - 314*x^5 + 76*x^4 + 72*x^3 + 40*x^2 + 12*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 2 x^{15} - 21 x^{14} + 41 x^{13} - 54 x^{12} + 43 x^{11} - 23 x^{10} - 108 x^{9} + 353 x^{8} - 597 x^{7} + 619 x^{6} - 314 x^{5} + 76 x^{4} + 72 x^{3} + 40 x^{2} + 12 x + 4 \)

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:  \(-57892131675000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{14}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{30} a^{12} + \frac{4}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{2}{5} a^{7} - \frac{11}{30} a^{6} + \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{5} a^{2} - \frac{4}{15} a - \frac{7}{15}$, $\frac{1}{60} a^{13} + \frac{1}{3} a^{11} + \frac{23}{60} a^{10} - \frac{1}{3} a^{9} + \frac{2}{15} a^{8} - \frac{1}{12} a^{7} - \frac{1}{3} a^{6} - \frac{4}{15} a^{5} - \frac{1}{4} a^{4} + \frac{1}{15} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{2}{15}$, $\frac{1}{60} a^{14} - \frac{17}{60} a^{11} - \frac{1}{3} a^{10} + \frac{7}{15} a^{9} + \frac{1}{4} a^{8} - \frac{1}{3} a^{7} + \frac{2}{5} a^{6} - \frac{1}{4} a^{5} + \frac{2}{5} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{7}{15} a - \frac{1}{3}$, $\frac{1}{60} a^{15} - \frac{1}{60} a^{12} - \frac{1}{5} a^{11} - \frac{2}{15} a^{10} - \frac{5}{12} a^{9} + \frac{1}{5} a^{7} - \frac{11}{60} a^{6} - \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{15} a^{2} - \frac{7}{15} a + \frac{4}{15}$, $\frac{1}{60} a^{16} - \frac{1}{5} a^{11} - \frac{7}{30} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a + \frac{1}{15}$, $\frac{1}{1204364220} a^{17} - \frac{754433}{602182110} a^{16} + \frac{362681}{200727370} a^{15} + \frac{198108}{100363685} a^{14} + \frac{350014}{100363685} a^{13} - \frac{6599809}{602182110} a^{12} + \frac{47618059}{200727370} a^{11} - \frac{1312056}{20072737} a^{10} - \frac{34215907}{200727370} a^{9} - \frac{55620469}{301091055} a^{8} + \frac{131411024}{301091055} a^{7} - \frac{47442683}{200727370} a^{6} - \frac{1228891}{80290948} a^{5} - \frac{192841553}{602182110} a^{4} + \frac{120025993}{301091055} a^{3} - \frac{48802839}{200727370} a^{2} - \frac{66470023}{301091055} a - \frac{25005376}{100363685}$

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:  \( \frac{13738835}{240872844} a^{17} - \frac{42238045}{240872844} a^{16} + \frac{35133161}{120436422} a^{15} + \frac{23844295}{240872844} a^{14} - \frac{145692715}{120436422} a^{13} + \frac{95261825}{40145474} a^{12} - \frac{765568805}{240872844} a^{11} + \frac{158192696}{60218211} a^{10} - \frac{61207425}{40145474} a^{9} - \frac{475019125}{80290948} a^{8} + \frac{2459458435}{120436422} a^{7} - \frac{4191303455}{120436422} a^{6} + \frac{2229388946}{60218211} a^{5} - \frac{1610479195}{80290948} a^{4} + \frac{418422565}{60218211} a^{3} + \frac{97100075}{60218211} a^{2} + \frac{122494065}{40145474} a + \frac{18268881}{20072737} \) (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:  \( 45128.715573030706 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.1.140.1, 6.0.529200.1, 6.0.270000.1, 9.1.46305000000.1 x3

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 6 sibling: data not computed
Degree 9 sibling: data not computed
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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$