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Properties

Label 18.0.51717558993...5728.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{12}\cdot 71^{12}$
Root discriminant $746.41$
Ramified primes $2, 3, 7, 71$
Class number $19683$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 3, 3, 3]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![888131124, 404543268, -1123085628, 111900570, 1203660630, -884819106, 281242081, 7936227, -900645, -448638, -95403, 92007, -28406, 3843, -849, 18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 + 18*x^15 - 849*x^14 + 3843*x^13 - 28406*x^12 + 92007*x^11 - 95403*x^10 - 448638*x^9 - 900645*x^8 + 7936227*x^7 + 281242081*x^6 - 884819106*x^5 + 1203660630*x^4 + 111900570*x^3 - 1123085628*x^2 + 404543268*x + 888131124)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 + 18*x^15 - 849*x^14 + 3843*x^13 - 28406*x^12 + 92007*x^11 - 95403*x^10 - 448638*x^9 - 900645*x^8 + 7936227*x^7 + 281242081*x^6 - 884819106*x^5 + 1203660630*x^4 + 111900570*x^3 - 1123085628*x^2 + 404543268*x + 888131124, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} + 18 x^{15} - 849 x^{14} + 3843 x^{13} - 28406 x^{12} + 92007 x^{11} - 95403 x^{10} - 448638 x^{9} - 900645 x^{8} + 7936227 x^{7} + 281242081 x^{6} - 884819106 x^{5} + 1203660630 x^{4} + 111900570 x^{3} - 1123085628 x^{2} + 404543268 x + 888131124 \)

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:  \(-5171755899362511287780502990259247846315138714185728=-\,2^{12}\cdot 3^{33}\cdot 7^{12}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $746.41$
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:  $2, 3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{6} a^{7} + \frac{1}{6} a^{4}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{6} a^{5}$, $\frac{1}{10017882} a^{15} - \frac{40315}{1113098} a^{14} + \frac{631651}{10017882} a^{13} - \frac{83548}{1669647} a^{12} - \frac{62033}{10017882} a^{11} + \frac{42475}{477042} a^{10} + \frac{151321}{5008941} a^{9} + \frac{388531}{1113098} a^{8} + \frac{299795}{10017882} a^{7} + \frac{476906}{1669647} a^{6} - \frac{442483}{1431126} a^{5} + \frac{449167}{3339294} a^{4} + \frac{505839}{1113098} a^{3} + \frac{511855}{1669647} a^{2} + \frac{48852}{556549} a - \frac{13053}{556549}$, $\frac{1}{3482195923678397407878942} a^{16} + \frac{3767105571808349}{193455329093244300437719} a^{15} + \frac{67352221473847763626769}{1741097961839198703939471} a^{14} - \frac{952440706374190951291}{89287074966112754048178} a^{13} - \frac{216977188021117692633083}{3482195923678397407878942} a^{12} - \frac{13705737512424933463759}{193455329093244300437719} a^{11} - \frac{456287555675893355834815}{3482195923678397407878942} a^{10} - \frac{27933222878156811184811}{386910658186488600875438} a^{9} + \frac{5675371324063543134787}{1741097961839198703939471} a^{8} + \frac{28556274022350163785771}{386910658186488600875438} a^{7} - \frac{883229967453282596004175}{3482195923678397407878942} a^{6} - \frac{16676833801443768697109}{580365987279732901313157} a^{5} - \frac{44813919301875876864256}{193455329093244300437719} a^{4} + \frac{415230174922301195567771}{1160731974559465802626314} a^{3} + \frac{39721756706447485038926}{193455329093244300437719} a^{2} - \frac{1684773776986008798963}{4498961141703355824133} a - \frac{58958123044395048696640}{193455329093244300437719}$, $\frac{1}{262837584792517692100143476496149382261302658} a^{17} - \frac{369384218783082212}{14602088044028760672230193138674965681183481} a^{16} - \frac{2794976095137333054330712036142564981}{87612528264172564033381158832049794087100886} a^{15} + \frac{3715437889821740274068943068483435908729505}{87612528264172564033381158832049794087100886} a^{14} - \frac{6463230120149513980422566831614656398483045}{87612528264172564033381158832049794087100886} a^{13} + \frac{2007880970436378290767963153668039196954049}{43806264132086282016690579416024897043550443} a^{12} + \frac{4302899854299972053484068159514097472404507}{37548226398931098871449068070878483180186094} a^{11} - \frac{10289272936185803355440240819421141152934239}{87612528264172564033381158832049794087100886} a^{10} + \frac{3470768301006141010552214009727357623331589}{43806264132086282016690579416024897043550443} a^{9} - \frac{7197678020884227980572908060928648779083555}{29204176088057521344460386277349931362366962} a^{8} - \frac{4255288303738760074386784545860782427700403}{12516075466310366290483022690292827726728698} a^{7} - \frac{4011714550983061657920053716432124088607883}{43806264132086282016690579416024897043550443} a^{6} - \frac{12782756406622922892830999211108576131945521}{131418792396258846050071738248074691130651329} a^{5} - \frac{22714363982301390140170023143994210695650537}{87612528264172564033381158832049794087100886} a^{4} + \frac{2475052332808206616299288886601726368588605}{29204176088057521344460386277349931362366962} a^{3} + \frac{2843961593611962161395063587102079581224486}{43806264132086282016690579416024897043550443} a^{2} - \frac{619412201223807900770559811695194477119967}{14602088044028760672230193138674965681183481} a + \frac{4389042087094845782998690672826938715748669}{14602088044028760672230193138674965681183481}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $19683$ (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:  \( -\frac{239977477723216621832}{825749675813235885869674654809} a^{17} + \frac{424460096839221554987}{825749675813235885869674654809} a^{16} + \frac{1537794629269392337180}{275249891937745295289891551603} a^{15} - \frac{58390372520269012920631}{825749675813235885869674654809} a^{14} + \frac{121646667296501363204210}{825749675813235885869674654809} a^{13} + \frac{918108898271961547646531}{1651499351626471771739349309618} a^{12} + \frac{74410373164602428031808}{39321413133963613612841650229} a^{11} + \frac{22084964909963234081739763}{825749675813235885869674654809} a^{10} - \frac{192128421643179466329665741}{1651499351626471771739349309618} a^{9} + \frac{158942186964011548576646539}{825749675813235885869674654809} a^{8} + \frac{142040822035941981085903492}{117964239401890840838524950687} a^{7} + \frac{1069687763405667308971404349}{1651499351626471771739349309618} a^{6} - \frac{78910521803350075292655236860}{825749675813235885869674654809} a^{5} - \frac{95872490829657644606177873352}{275249891937745295289891551603} a^{4} + \frac{1532050724969429727277310097365}{1651499351626471771739349309618} a^{3} - \frac{328279796012238828761538950368}{275249891937745295289891551603} a^{2} - \frac{84213702368230020085704965496}{275249891937745295289891551603} a + \frac{250744900296591004859751454682}{275249891937745295289891551603} \) (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:  \( 3506023773867162.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.15123.1 x3, Deg 6, Deg 6, 6.0.6805279152.7, 6.0.686115387.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
$71$71.6.4.1$x^{6} + 2272 x^{3} + 6709571$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
71.6.4.1$x^{6} + 2272 x^{3} + 6709571$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
71.6.4.1$x^{6} + 2272 x^{3} + 6709571$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$