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Properties

Label 18.0.69192308370...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{18}\cdot 5^{12}\cdot 11^{15}\cdot 103^{9}$
Root discriminant $1857.34$
Ramified primes $2, 3, 5, 11, 103$
Class number $51732592704$ (GRH)
Class group $[3, 3, 6, 18, 18, 18, 234, 702]$ (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![2708125245357200, -2203214877726000, 597733410996390, 4275378675392, -26838014382981, 1142790467772, 964213213642, -55026746730, -20638244151, 207124472, 362900991, 23306106, 606345, 94968, 13413, -1506, -126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 126*x^16 - 1506*x^15 + 13413*x^14 + 94968*x^13 + 606345*x^12 + 23306106*x^11 + 362900991*x^10 + 207124472*x^9 - 20638244151*x^8 - 55026746730*x^7 + 964213213642*x^6 + 1142790467772*x^5 - 26838014382981*x^4 + 4275378675392*x^3 + 597733410996390*x^2 - 2203214877726000*x + 2708125245357200)
 
gp: K = bnfinit(x^18 - 126*x^16 - 1506*x^15 + 13413*x^14 + 94968*x^13 + 606345*x^12 + 23306106*x^11 + 362900991*x^10 + 207124472*x^9 - 20638244151*x^8 - 55026746730*x^7 + 964213213642*x^6 + 1142790467772*x^5 - 26838014382981*x^4 + 4275378675392*x^3 + 597733410996390*x^2 - 2203214877726000*x + 2708125245357200, 1)
 

Normalized defining polynomial

\( x^{18} - 126 x^{16} - 1506 x^{15} + 13413 x^{14} + 94968 x^{13} + 606345 x^{12} + 23306106 x^{11} + 362900991 x^{10} + 207124472 x^{9} - 20638244151 x^{8} - 55026746730 x^{7} + 964213213642 x^{6} + 1142790467772 x^{5} - 26838014382981 x^{4} + 4275378675392 x^{3} + 597733410996390 x^{2} - 2203214877726000 x + 2708125245357200 \)

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:  \(-69192308370830114650627499075939863765733769216000000000000=-\,2^{27}\cdot 3^{18}\cdot 5^{12}\cdot 11^{15}\cdot 103^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1857.34$
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:  $2, 3, 5, 11, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{110} a^{9} + \frac{7}{55} a^{7} + \frac{14}{55} a^{6} - \frac{3}{11} a^{5} - \frac{27}{55} a^{4} - \frac{53}{110} a^{3} + \frac{14}{55} a^{2} - \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{110} a^{10} + \frac{3}{110} a^{8} + \frac{3}{55} a^{7} - \frac{4}{55} a^{6} + \frac{6}{55} a^{5} - \frac{31}{110} a^{4} + \frac{3}{55} a^{3} - \frac{21}{110} a^{2} + \frac{3}{11} a$, $\frac{1}{220} a^{11} - \frac{1}{220} a^{10} - \frac{1}{220} a^{9} - \frac{2}{55} a^{8} + \frac{9}{110} a^{7} - \frac{7}{22} a^{6} + \frac{3}{20} a^{5} + \frac{1}{4} a^{4} + \frac{53}{220} a^{3} - \frac{18}{55} a^{2} + \frac{1}{22} a + \frac{5}{11}$, $\frac{1}{1100} a^{12} - \frac{1}{550} a^{11} - \frac{1}{275} a^{10} - \frac{1}{220} a^{9} - \frac{2}{275} a^{8} - \frac{32}{275} a^{7} - \frac{17}{44} a^{6} - \frac{87}{550} a^{5} + \frac{139}{550} a^{4} - \frac{299}{1100} a^{3} + \frac{2}{11} a^{2} - \frac{1}{22} a + \frac{4}{11}$, $\frac{1}{1100} a^{13} + \frac{1}{550} a^{11} - \frac{3}{1100} a^{10} + \frac{1}{550} a^{9} - \frac{27}{550} a^{8} + \frac{259}{1100} a^{7} - \frac{41}{275} a^{6} + \frac{2}{55} a^{5} + \frac{547}{1100} a^{4} - \frac{9}{550} a^{3} + \frac{8}{55} a^{2} - \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{11000} a^{14} + \frac{1}{5500} a^{13} + \frac{1}{11000} a^{12} + \frac{9}{5500} a^{11} - \frac{9}{2200} a^{10} + \frac{3}{1100} a^{9} + \frac{169}{11000} a^{8} + \frac{843}{2750} a^{7} - \frac{1893}{11000} a^{6} - \frac{523}{1375} a^{5} - \frac{3847}{11000} a^{4} + \frac{276}{1375} a^{3} - \frac{129}{1100} a^{2} - \frac{4}{55} a + \frac{24}{55}$, $\frac{1}{390159000} a^{15} + \frac{117}{6193000} a^{14} - \frac{16099}{43351000} a^{13} + \frac{43877}{390159000} a^{12} + \frac{62269}{130053000} a^{11} + \frac{2769}{8670200} a^{10} - \frac{73721}{390159000} a^{9} - \frac{15333}{3941000} a^{8} + \frac{167587}{337800} a^{7} - \frac{168470531}{390159000} a^{6} + \frac{5536983}{43351000} a^{5} + \frac{18589}{148632} a^{4} + \frac{3243857}{65026500} a^{3} - \frac{5798147}{13005300} a^{2} - \frac{246544}{650265} a - \frac{678059}{1950795}$, $\frac{1}{390159000} a^{16} + \frac{107}{10837750} a^{14} + \frac{67393}{195079500} a^{13} + \frac{8737}{32513250} a^{12} + \frac{2003}{2167550} a^{11} + \frac{27481}{97539750} a^{10} - \frac{2411}{10837750} a^{9} + \frac{403649}{9289500} a^{8} + \frac{7753271}{195079500} a^{7} + \frac{5248181}{10837750} a^{6} + \frac{1986653}{4644750} a^{5} + \frac{63330913}{130053000} a^{4} + \frac{2344301}{6502650} a^{3} + \frac{37591}{520212} a^{2} - \frac{223073}{1950795} a + \frac{1691}{6193}$, $\frac{1}{1742362388924035384852442464535088613851540346368458761193189475037151846366642619000} a^{17} - \frac{178779347462897139365368245551069179569637423361763078905642131180063573631}{248908912703433626407491780647869801978791478052636965884741353576735978052377517000} a^{16} - \frac{40195849534539001837304110640565770852429418823785284295381547129259602351}{248908912703433626407491780647869801978791478052636965884741353576735978052377517000} a^{15} - \frac{3502667703284627215585758255754258612135972053450405252682584256941859574244293}{871181194462017692426221232267544306925770173184229380596594737518575923183321309500} a^{14} + \frac{589526526993984948953029400923423708853161829617469674597472612239741865660984393}{1742362388924035384852442464535088613851540346368458761193189475037151846366642619000} a^{13} + \frac{3874746681150681528401556044219938624586663004248525042473137977256697325394148}{43559059723100884621311061613377215346288508659211469029829736875928796159166065475} a^{12} - \frac{118843793111530553125643108694679659146394683050755125029738384070066440192461323}{248908912703433626407491780647869801978791478052636965884741353576735978052377517000} a^{11} - \frac{164790075185865242547371384463045182209996126823050938999479198691612101838857497}{871181194462017692426221232267544306925770173184229380596594737518575923183321309500} a^{10} - \frac{642942884996124080831777320485205680006930781125886180280134226574366611291351951}{158396580811275944077494769503189873986503667851678069199380861367013804215149329000} a^{9} + \frac{2658415806624945781721909097766113698330333617396657486001968413899870843709288929}{871181194462017692426221232267544306925770173184229380596594737518575923183321309500} a^{8} + \frac{870669665977932570827739992793376407374708892750672690845946094157688252772037953459}{1742362388924035384852442464535088613851540346368458761193189475037151846366642619000} a^{7} - \frac{67177932603102588125147193701779576489837997446932073189789688073082810677278148647}{871181194462017692426221232267544306925770173184229380596594737518575923183321309500} a^{6} + \frac{969366946410436300623430062433801477332394993859591557775528984282032546216750219}{24199477623944935900728367562987341859049171477339705016572076042182664532870036375} a^{5} + \frac{15639720106254119228406464265369816700241532484915716898707937384328838064081390437}{38719164198311897441165388100779746974478674363743528026515321667492263252592058200} a^{4} - \frac{17659951525434036898518513387286268476331039777146446725038793065011068078821232333}{58078746297467846161748082151169620461718011545615292039772982501238394878888087300} a^{3} + \frac{17117147098740324554576059193757940835192846616443807747732958367158154843064641021}{34847247778480707697048849290701772277030806927369175223863789500743036927332852380} a^{2} - \frac{48073999973789034089620779062123743286362945846041825747840187364463385654712153}{3484724777848070769704884929070177227703080692736917522386378950074303692733285238} a - \frac{25186845115839169744869000113685744480098474753905250972761330700792523300688010}{1742362388924035384852442464535088613851540346368458761193189475037151846366642619}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{18}\times C_{18}\times C_{18}\times C_{234}\times C_{702}$, which has order $51732592704$ (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:  \( 11922805943498.25 \) (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

\(\Q(\sqrt{-2266}) \), 3.1.81675.1, 3.1.9064.1 x3, Deg 6, 6.0.744662854144.1, 9.1.3353063280402836088000000.1 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$11$11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$103$103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.2.1.1$x^{2} - 103$$2$$1$$1$$C_2$$[\ ]_{2}$