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Properties

Label 18.0.30062027722...5872.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 17^{12}$
Root discriminant $22.90$
Ramified primes $2, 3, 17$
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![108, -234, 813, -537, 1639, 4, 1501, 673, 640, 581, 54, 197, -40, 7, -3, -14, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 14*x^15 - 3*x^14 + 7*x^13 - 40*x^12 + 197*x^11 + 54*x^10 + 581*x^9 + 640*x^8 + 673*x^7 + 1501*x^6 + 4*x^5 + 1639*x^4 - 537*x^3 + 813*x^2 - 234*x + 108)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 14*x^15 - 3*x^14 + 7*x^13 - 40*x^12 + 197*x^11 + 54*x^10 + 581*x^9 + 640*x^8 + 673*x^7 + 1501*x^6 + 4*x^5 + 1639*x^4 - 537*x^3 + 813*x^2 - 234*x + 108, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 14 x^{15} - 3 x^{14} + 7 x^{13} - 40 x^{12} + 197 x^{11} + 54 x^{10} + 581 x^{9} + 640 x^{8} + 673 x^{7} + 1501 x^{6} + 4 x^{5} + 1639 x^{4} - 537 x^{3} + 813 x^{2} - 234 x + 108 \)

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:  \(-3006202772296403717455872=-\,2^{18}\cdot 3^{9}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.90$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{4} a^{7} + \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{1}{3} a^{3} - \frac{5}{12} a^{2}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{10} + \frac{5}{24} a^{9} - \frac{1}{4} a^{7} - \frac{1}{24} a^{6} + \frac{1}{24} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{5}{24} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{12} a^{8} + \frac{5}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{4} a^{5} + \frac{1}{3} a^{4} - \frac{1}{8} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{14} - \frac{1}{48} a^{13} + \frac{1}{48} a^{12} - \frac{1}{24} a^{10} + \frac{5}{48} a^{9} - \frac{11}{48} a^{8} - \frac{1}{12} a^{7} + \frac{1}{24} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{7}{16} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{48} a^{16} + \frac{1}{48} a^{12} - \frac{1}{16} a^{10} + \frac{1}{24} a^{9} + \frac{5}{48} a^{8} - \frac{1}{12} a^{7} + \frac{1}{48} a^{6} + \frac{1}{24} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{23}{48} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{2984976513072} a^{17} + \frac{685124087}{82916014252} a^{16} - \frac{9514982189}{1492488256536} a^{15} - \frac{8404308895}{1492488256536} a^{14} + \frac{8536104841}{994992171024} a^{13} - \frac{15841126615}{1492488256536} a^{12} - \frac{102999995959}{2984976513072} a^{11} + \frac{33676113175}{1492488256536} a^{10} - \frac{38154126007}{994992171024} a^{9} + \frac{137373184051}{1492488256536} a^{8} + \frac{463340713933}{2984976513072} a^{7} - \frac{309561367957}{1492488256536} a^{6} + \frac{421494220583}{1492488256536} a^{5} + \frac{620835693455}{1492488256536} a^{4} + \frac{132790420999}{2984976513072} a^{3} + \frac{8066961823}{41458007126} a^{2} + \frac{55593820313}{124374021378} a - \frac{5725945607}{41458007126}$

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:  \( -\frac{1643045}{396043056} a^{17} + \frac{2838407}{132014352} a^{16} - \frac{17898445}{396043056} a^{15} + \frac{33584797}{396043056} a^{14} - \frac{2448251}{33003588} a^{13} - \frac{12268823}{99010764} a^{12} + \frac{130002035}{396043056} a^{11} - \frac{416825587}{396043056} a^{10} + \frac{43079857}{33003588} a^{9} - \frac{55613573}{49505382} a^{8} + \frac{375970687}{396043056} a^{7} + \frac{1149551581}{396043056} a^{6} - \frac{1216920683}{396043056} a^{5} + \frac{3186297583}{396043056} a^{4} - \frac{767550395}{99010764} a^{3} + \frac{212294633}{33003588} a^{2} - \frac{75469561}{16501794} a + \frac{4558807}{2750299} \) (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:  \( 2179489.4273506934 \)
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

\(\Q(\sqrt{-3}) \), 3.1.6936.1, 3.1.867.1 x3, 6.0.144324288.3, 6.0.2255067.2, 9.1.1001033261568.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$