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Properties

Label 18.0.39606063155...7088.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 53^{12}$
Root discriminant $38.79$
Ramified primes $2, 3, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
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, -936, 3658, -8268, 11355, -9171, 4167, -1806, 2244, -2340, 1678, -816, 396, -294, 210, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 210*x^14 - 294*x^13 + 396*x^12 - 816*x^11 + 1678*x^10 - 2340*x^9 + 2244*x^8 - 1806*x^7 + 4167*x^6 - 9171*x^5 + 11355*x^4 - 8268*x^3 + 3658*x^2 - 936*x + 108, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 210 x^{14} - 294 x^{13} + 396 x^{12} - 816 x^{11} + 1678 x^{10} - 2340 x^{9} + 2244 x^{8} - 1806 x^{7} + 4167 x^{6} - 9171 x^{5} + 11355 x^{4} - 8268 x^{3} + 3658 x^{2} - 936 x + 108 \)

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:  \(-39606063155139137133767897088=-\,2^{12}\cdot 3^{9}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.79$
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:  $2, 3, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{12} + \frac{1}{14} a^{11} + \frac{3}{14} a^{10} - \frac{1}{7} a^{9} - \frac{3}{14} a^{8} - \frac{1}{2} a^{6} + \frac{3}{14} a^{5} + \frac{3}{14} a^{4} + \frac{1}{7} a^{3} + \frac{1}{14} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{14} a^{13} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{14} a^{9} + \frac{3}{14} a^{8} - \frac{1}{2} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{4} - \frac{1}{14} a^{3} - \frac{3}{14} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{168} a^{14} + \frac{5}{168} a^{13} - \frac{1}{84} a^{12} - \frac{41}{168} a^{11} + \frac{13}{56} a^{10} - \frac{11}{84} a^{9} + \frac{9}{56} a^{8} - \frac{27}{56} a^{7} - \frac{19}{42} a^{6} - \frac{41}{168} a^{5} - \frac{67}{168} a^{4} - \frac{2}{21} a^{3} - \frac{13}{28} a^{2} + \frac{5}{21} a - \frac{5}{14}$, $\frac{1}{168} a^{15} - \frac{1}{56} a^{13} + \frac{5}{168} a^{12} - \frac{1}{21} a^{11} + \frac{23}{168} a^{10} + \frac{41}{168} a^{9} - \frac{1}{24} a^{7} - \frac{3}{56} a^{6} + \frac{13}{28} a^{5} - \frac{17}{168} a^{4} + \frac{25}{84} a^{3} - \frac{13}{84} a^{2} - \frac{17}{42} a - \frac{1}{14}$, $\frac{1}{99421728} a^{16} - \frac{1}{12427716} a^{15} + \frac{19135}{12427716} a^{14} - \frac{12755}{1183592} a^{13} - \frac{175169}{49710864} a^{12} + \frac{4007531}{24855432} a^{11} - \frac{1038475}{7101552} a^{10} - \frac{60397}{16570288} a^{9} - \frac{9110525}{49710864} a^{8} - \frac{775471}{24855432} a^{7} + \frac{1955825}{7101552} a^{6} + \frac{2749015}{8285144} a^{5} + \frac{26581369}{99421728} a^{4} - \frac{24684571}{49710864} a^{3} + \frac{1043509}{7101552} a^{2} - \frac{2560713}{8285144} a - \frac{3520577}{8285144}$, $\frac{1}{22767575712} a^{17} + \frac{53}{11383787856} a^{16} + \frac{268623}{135521284} a^{15} - \frac{12475363}{5691893928} a^{14} + \frac{80584481}{3794595952} a^{13} + \frac{19815309}{948648988} a^{12} - \frac{2140118065}{11383787856} a^{11} - \frac{1238407073}{11383787856} a^{10} - \frac{846444779}{11383787856} a^{9} + \frac{91016957}{1422973482} a^{8} + \frac{862089}{3794595952} a^{7} - \frac{291489886}{711486741} a^{6} + \frac{430740651}{7589191904} a^{5} + \frac{252354217}{1897297976} a^{4} + \frac{638400499}{1626255408} a^{3} + \frac{317688842}{711486741} a^{2} + \frac{672736397}{1897297976} a + \frac{72145067}{948648988}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{19092757}{203281926} a^{17} + \frac{324576869}{406563852} a^{16} - \frac{331895264}{101640963} a^{15} + \frac{866330135}{101640963} a^{14} - \frac{1576515436}{101640963} a^{13} + \frac{676919711}{33880321} a^{12} - \frac{2786618690}{101640963} a^{11} + \frac{4282069165}{67760642} a^{10} - \frac{12845807312}{101640963} a^{9} + \frac{16008645040}{101640963} a^{8} - \frac{13593525280}{101640963} a^{7} + \frac{10647508315}{101640963} a^{6} - \frac{69237356587}{203281926} a^{5} + \frac{93809561563}{135521284} a^{4} - \frac{73745502826}{101640963} a^{3} + \frac{14329182505}{33880321} a^{2} - \frac{14326293356}{101640963} a + \frac{755635754}{33880321} \) (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:  \( 373923084.8527347 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.3.33708.1, 3.1.8427.1 x3, 6.0.3408687792.2, 6.0.213042987.1, 9.3.114900048092736.1 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{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}$
$53$53.6.4.1$x^{6} + 742 x^{3} + 351125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53.6.4.1$x^{6} + 742 x^{3} + 351125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
53.6.4.1$x^{6} + 742 x^{3} + 351125$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$