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Properties

Label 18.0.59046319790...2347.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{39}\cdot 19^{12}\cdot 37^{12}$
Root discriminant $854.54$
Ramified primes $3, 19, 37$
Class number $57395628$ (GRH)
Class group $[3, 3, 3, 3, 9, 9, 9, 18, 54]$ (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![40353607, 0, 0, 51364578, 0, 0, 1020101190, 0, 0, 3025762, 0, 0, 2229, 0, 0, -60, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^15 + 2229*x^12 + 3025762*x^9 + 1020101190*x^6 + 51364578*x^3 + 40353607)
 
gp: K = bnfinit(x^18 - 60*x^15 + 2229*x^12 + 3025762*x^9 + 1020101190*x^6 + 51364578*x^3 + 40353607, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{15} + 2229 x^{12} + 3025762 x^{9} + 1020101190 x^{6} + 51364578 x^{3} + 40353607 \)

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:  \(-59046319790637747019476233422195951296270284280622347=-\,3^{39}\cdot 19^{12}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $854.54$
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:  $3, 19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{4} - \frac{2}{9} a$, $\frac{1}{27} a^{8} + \frac{1}{27} a^{7} + \frac{1}{27} a^{6} - \frac{2}{27} a^{5} - \frac{2}{27} a^{4} - \frac{2}{27} a^{3} + \frac{1}{27} a^{2} + \frac{1}{27} a + \frac{1}{27}$, $\frac{1}{54} a^{9} - \frac{1}{18} a^{6} - \frac{1}{9} a^{3} - \frac{19}{54}$, $\frac{1}{54} a^{10} - \frac{1}{18} a^{7} - \frac{1}{9} a^{4} - \frac{19}{54} a$, $\frac{1}{162} a^{11} + \frac{1}{162} a^{10} + \frac{1}{162} a^{9} - \frac{1}{54} a^{8} - \frac{1}{54} a^{7} - \frac{1}{54} a^{6} - \frac{4}{27} a^{5} - \frac{4}{27} a^{4} - \frac{4}{27} a^{3} - \frac{55}{162} a^{2} - \frac{55}{162} a - \frac{55}{162}$, $\frac{1}{162} a^{12} - \frac{1}{162} a^{9} + \frac{1}{27} a^{6} - \frac{13}{162} a^{3} - \frac{37}{81}$, $\frac{1}{486} a^{13} - \frac{1}{486} a^{12} - \frac{2}{243} a^{10} + \frac{2}{243} a^{9} - \frac{7}{162} a^{7} + \frac{7}{162} a^{6} - \frac{31}{486} a^{4} + \frac{31}{486} a^{3} + \frac{55}{486} a - \frac{55}{486}$, $\frac{1}{486} a^{14} - \frac{1}{486} a^{12} - \frac{1}{486} a^{11} + \frac{1}{162} a^{10} - \frac{1}{243} a^{9} + \frac{1}{81} a^{8} - \frac{1}{18} a^{7} + \frac{7}{162} a^{6} - \frac{13}{486} a^{5} - \frac{2}{27} a^{4} + \frac{49}{486} a^{3} - \frac{118}{243} a^{2} - \frac{61}{162} a - \frac{67}{486}$, $\frac{1}{4641477349620861558} a^{15} + \frac{1510272570286664}{2320738674810430779} a^{12} - \frac{371833170129080}{43787522166234543} a^{9} - \frac{117448719223261982}{2320738674810430779} a^{6} + \frac{191603071986524921}{4641477349620861558} a^{3} + \frac{31270911931957079}{4641477349620861558}$, $\frac{1}{4776080192759866543182} a^{16} - \frac{1}{13924432048862584674} a^{15} - \frac{227698485435681808}{2388040096379933271591} a^{13} + \frac{25630549610172731}{13924432048862584674} a^{12} + \frac{106833085648392137}{90114720618110689494} a^{10} - \frac{709340216691526}{131362566498703629} a^{9} + \frac{207170377462204197137}{4776080192759866543182} a^{7} + \frac{664663859707714849}{13924432048862584674} a^{6} + \frac{389330771976619498259}{4776080192759866543182} a^{4} + \frac{749405759153746280}{6962216024431292337} a^{3} + \frac{408107503538217469906}{2388040096379933271591} a - \frac{3353487994427894413}{6962216024431292337}$, $\frac{1}{1638195506116634224311426} a^{17} - \frac{1}{13924432048862584674} a^{15} + \frac{540047505501953039419}{1638195506116634224311426} a^{14} - \frac{1510272570286664}{6962216024431292337} a^{12} - \frac{36196435063843787051}{15454674586005983248221} a^{11} + \frac{371833170129080}{131362566498703629} a^{9} - \frac{17953727145624695744839}{1638195506116634224311426} a^{8} - \frac{1}{27} a^{7} + \frac{375308571979976513}{6962216024431292337} a^{6} + \frac{58858884955288769274901}{819097753058317112155713} a^{5} + \frac{2}{27} a^{4} - \frac{1223042483013383045}{13924432048862584674} a^{3} + \frac{388292644969036020884296}{819097753058317112155713} a^{2} - \frac{1}{27} a - \frac{4157028556039389575}{13924432048862584674}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{9}\times C_{18}\times C_{54}$, which has order $57395628$ (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:  \( \frac{54059}{12563514470838} a^{15} - \frac{29211697}{113071630237542} a^{12} + \frac{20565517}{2133426985614} a^{9} + \frac{27201571619}{2093919078473} a^{6} + \frac{248175292981436}{56535815118771} a^{3} + \frac{69061039644049}{113071630237542} \) (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:  \( 2535693926127.3457 \) (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

\(\Q(\sqrt{-3}) \), 3.1.120092787.3 x3, Deg 6, 6.0.36889110963.5, 6.0.23085974187.4, 6.0.43266832468282107.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$