↑

Properties

Label 18.0.32923515554...7136.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 7^{12}\cdot 197^{8}$
Root discriminant $82.73$
Ramified primes $2, 7, 197$
Class number $2160$ (GRH)
Class group $[2, 2, 6, 90]$ (GRH)
Galois group $C_2\times C_3:S_4$ (as 18T66)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2055213, 3140424, 3388775, 3055670, 2414142, 1154974, 593654, 255754, 124796, 30598, 27604, 210, 5142, -934, 734, -146, 51, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 51*x^16 - 146*x^15 + 734*x^14 - 934*x^13 + 5142*x^12 + 210*x^11 + 27604*x^10 + 30598*x^9 + 124796*x^8 + 255754*x^7 + 593654*x^6 + 1154974*x^5 + 2414142*x^4 + 3055670*x^3 + 3388775*x^2 + 3140424*x + 2055213)
 
gp: K = bnfinit(x^18 - 6*x^17 + 51*x^16 - 146*x^15 + 734*x^14 - 934*x^13 + 5142*x^12 + 210*x^11 + 27604*x^10 + 30598*x^9 + 124796*x^8 + 255754*x^7 + 593654*x^6 + 1154974*x^5 + 2414142*x^4 + 3055670*x^3 + 3388775*x^2 + 3140424*x + 2055213, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 51 x^{16} - 146 x^{15} + 734 x^{14} - 934 x^{13} + 5142 x^{12} + 210 x^{11} + 27604 x^{10} + 30598 x^{9} + 124796 x^{8} + 255754 x^{7} + 593654 x^{6} + 1154974 x^{5} + 2414142 x^{4} + 3055670 x^{3} + 3388775 x^{2} + 3140424 x + 2055213 \)

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:  \(-32923515554977373578036580514267136=-\,2^{20}\cdot 7^{12}\cdot 197^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.73$
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:  $2, 7, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{44} a^{12} - \frac{1}{44} a^{10} - \frac{1}{11} a^{9} - \frac{3}{22} a^{8} - \frac{1}{11} a^{7} + \frac{1}{22} a^{6} - \frac{2}{11} a^{5} + \frac{19}{44} a^{4} - \frac{1}{11} a^{3} - \frac{21}{44} a^{2} - \frac{4}{11} a + \frac{5}{22}$, $\frac{1}{44} a^{13} - \frac{1}{44} a^{11} - \frac{1}{11} a^{10} + \frac{5}{44} a^{9} + \frac{7}{44} a^{8} + \frac{1}{22} a^{7} - \frac{2}{11} a^{6} + \frac{19}{44} a^{5} - \frac{1}{11} a^{4} + \frac{1}{44} a^{3} + \frac{3}{22} a^{2} + \frac{21}{44} a + \frac{1}{4}$, $\frac{1}{88} a^{14} - \frac{1}{88} a^{12} - \frac{1}{22} a^{11} + \frac{5}{88} a^{10} - \frac{1}{22} a^{9} - \frac{9}{88} a^{8} - \frac{1}{11} a^{7} + \frac{19}{88} a^{6} + \frac{5}{11} a^{5} - \frac{43}{88} a^{4} + \frac{7}{22} a^{3} + \frac{43}{88} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{2288} a^{15} - \frac{5}{2288} a^{14} + \frac{3}{2288} a^{13} + \frac{15}{2288} a^{12} + \frac{21}{2288} a^{11} - \frac{13}{176} a^{10} - \frac{113}{2288} a^{9} + \frac{135}{2288} a^{8} - \frac{23}{208} a^{7} - \frac{323}{2288} a^{6} + \frac{777}{2288} a^{5} + \frac{713}{2288} a^{4} - \frac{589}{2288} a^{3} + \frac{21}{2288} a^{2} - \frac{327}{2288} a + \frac{877}{2288}$, $\frac{1}{13728} a^{16} - \frac{1}{6864} a^{15} + \frac{5}{1716} a^{14} - \frac{7}{3432} a^{13} + \frac{59}{6864} a^{12} - \frac{139}{2288} a^{11} - \frac{69}{1144} a^{10} + \frac{25}{286} a^{9} + \frac{133}{858} a^{8} - \frac{437}{6864} a^{7} - \frac{503}{3432} a^{6} + \frac{127}{1716} a^{5} + \frac{749}{6864} a^{4} - \frac{2381}{6864} a^{3} - \frac{283}{858} a^{2} + \frac{5}{24} a + \frac{1605}{4576}$, $\frac{1}{23306861537418278149605588826328587200} a^{17} + \frac{40436192519959010080584507520649}{7768953845806092716535196275442862400} a^{16} + \frac{275069996669109720788639713760269}{1294825640967682119422532712573810400} a^{15} + \frac{439186122700099673294999654622733}{145667884608864238435034930164553670} a^{14} - \frac{4525943112411948230304062788374701}{896417751439164544215599570243407200} a^{13} + \frac{18488369313404640553021333092960961}{2913357692177284768700698603291073400} a^{12} + \frac{13763527919930506320666587764761077}{298805917146388181405199856747802400} a^{11} + \frac{25313527050661862488789561381814761}{485559615362880794783449767215178900} a^{10} + \frac{629316324723415641481755358204038397}{5826715384354569537401397206582146800} a^{9} + \frac{38038256686799114968774776613738471}{1059402797155376279527526764833117600} a^{8} - \frac{216020035841432264818086470443628993}{896417751439164544215599570243407200} a^{7} - \frac{10073052421583627755688410371586201}{116534307687091390748027944131642936} a^{6} - \frac{24270974621699968451419221256630843}{1059402797155376279527526764833117600} a^{5} + \frac{480570521603758792231800668074783159}{5826715384354569537401397206582146800} a^{4} + \frac{4498254082348981686358747933043623}{11952236685855527256207994269912096} a^{3} + \frac{26663436792950048241654678595740677}{145667884608864238435034930164553670} a^{2} - \frac{78995977517507255057136651999897413}{4661372307483655629921117765265717440} a + \frac{5497003738661254013608998327676021}{235422843812305839895005947740692800}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{90}$, which has order $2160$ (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:  \( -1 \) (order $2$)
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:  \( 27216227.2532 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_4$ (as 18T66):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_3:S_4$
Character table for $C_2\times C_3:S_4$

Intermediate fields

3.3.9653.1, 3.3.788.1, 3.3.38612.2, 3.3.38612.1, 6.0.1490886544.1, 9.9.11340523913674816.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.2$x^{12} - 108 x^{10} - 171 x^{8} + 344 x^{6} - 61 x^{4} + 468 x^{2} + 359$$6$$2$$16$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
7Data not computed
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$