↑

Properties

Label 18.0.11911501053...4176.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 19^{9}\cdot 37^{14}$
Root discriminant $114.76$
Ramified primes $2, 19, 37$
Class number $149544$ (GRH)
Class group $[2, 2, 37386]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![66363211, -80448578, 90274691, -61821841, 40163743, -19663227, 9385751, -3591517, 1381097, -436376, 143358, -37201, 9993, -1965, 519, -122, 39, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 39*x^16 - 122*x^15 + 519*x^14 - 1965*x^13 + 9993*x^12 - 37201*x^11 + 143358*x^10 - 436376*x^9 + 1381097*x^8 - 3591517*x^7 + 9385751*x^6 - 19663227*x^5 + 40163743*x^4 - 61821841*x^3 + 90274691*x^2 - 80448578*x + 66363211)
 
gp: K = bnfinit(x^18 - 7*x^17 + 39*x^16 - 122*x^15 + 519*x^14 - 1965*x^13 + 9993*x^12 - 37201*x^11 + 143358*x^10 - 436376*x^9 + 1381097*x^8 - 3591517*x^7 + 9385751*x^6 - 19663227*x^5 + 40163743*x^4 - 61821841*x^3 + 90274691*x^2 - 80448578*x + 66363211, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 39 x^{16} - 122 x^{15} + 519 x^{14} - 1965 x^{13} + 9993 x^{12} - 37201 x^{11} + 143358 x^{10} - 436376 x^{9} + 1381097 x^{8} - 3591517 x^{7} + 9385751 x^{6} - 19663227 x^{5} + 40163743 x^{4} - 61821841 x^{3} + 90274691 x^{2} - 80448578 x + 66363211 \)

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:  \(-11911501053328835794135616273165234176=-\,2^{12}\cdot 19^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.76$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{134757098521903121298653186739229163458255475648983} a^{17} + \frac{21112381010721013029330150098109003239340709201288}{134757098521903121298653186739229163458255475648983} a^{16} - \frac{32412597879612001829437834504694202132230302146320}{134757098521903121298653186739229163458255475648983} a^{15} + \frac{45708582039597405123187870391956913703722677160047}{134757098521903121298653186739229163458255475648983} a^{14} - \frac{47364665144425202526044356444027603758609035850602}{134757098521903121298653186739229163458255475648983} a^{13} + \frac{57531822390392997145390537754289508001804865955234}{134757098521903121298653186739229163458255475648983} a^{12} + \frac{66522085907635071677549782943134030813868150692999}{134757098521903121298653186739229163458255475648983} a^{11} + \frac{25554017965273786275884064681121165630354400147116}{134757098521903121298653186739229163458255475648983} a^{10} - \frac{29295904517059008217026918352525695092001573332715}{134757098521903121298653186739229163458255475648983} a^{9} + \frac{2326178668616546269647909144021647790138895615635}{134757098521903121298653186739229163458255475648983} a^{8} + \frac{27382403772286236651641881106125048335397280258039}{134757098521903121298653186739229163458255475648983} a^{7} - \frac{23382147616781936554081749828595504703809400744963}{134757098521903121298653186739229163458255475648983} a^{6} + \frac{13684159301517258049719238106292760389084945634619}{134757098521903121298653186739229163458255475648983} a^{5} + \frac{57966925743998708972957626954888245737109964206649}{134757098521903121298653186739229163458255475648983} a^{4} - \frac{9370833302393548603232875302641322176005150528727}{134757098521903121298653186739229163458255475648983} a^{3} + \frac{59073394141836600480012869630808215706321641264609}{134757098521903121298653186739229163458255475648983} a^{2} + \frac{66091742932428775816216341191793798044510822106633}{134757098521903121298653186739229163458255475648983} a + \frac{8117831823245415800833367026784459311705419235865}{134757098521903121298653186739229163458255475648983}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{37386}$, which has order $149544$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-19}) \), 3.3.1369.1, 3.3.148.1, 6.0.12854870299.1, 6.0.150239536.1, 9.9.6075640136512.1

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 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$