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Properties

Label 18.0.11494707829...4688.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 103^{12}$
Root discriminant $60.41$
Ramified primes $2, 3, 103$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![986049, -2713869, 5648127, -4964658, 3734884, -2280246, 1244342, -578710, 242241, -89021, 30915, -7806, 2244, -524, 228, -40, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 40*x^15 + 228*x^14 - 524*x^13 + 2244*x^12 - 7806*x^11 + 30915*x^10 - 89021*x^9 + 242241*x^8 - 578710*x^7 + 1244342*x^6 - 2280246*x^5 + 3734884*x^4 - 4964658*x^3 + 5648127*x^2 - 2713869*x + 986049)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 40*x^15 + 228*x^14 - 524*x^13 + 2244*x^12 - 7806*x^11 + 30915*x^10 - 89021*x^9 + 242241*x^8 - 578710*x^7 + 1244342*x^6 - 2280246*x^5 + 3734884*x^4 - 4964658*x^3 + 5648127*x^2 - 2713869*x + 986049, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 40 x^{15} + 228 x^{14} - 524 x^{13} + 2244 x^{12} - 7806 x^{11} + 30915 x^{10} - 89021 x^{9} + 242241 x^{8} - 578710 x^{7} + 1244342 x^{6} - 2280246 x^{5} + 3734884 x^{4} - 4964658 x^{3} + 5648127 x^{2} - 2713869 x + 986049 \)

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:  \(-114947078290609521390591403634688=-\,2^{12}\cdot 3^{9}\cdot 103^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.41$
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:  $2, 3, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{9} - \frac{1}{6} a^{5} - \frac{2}{9} a^{3} - \frac{1}{6} a$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{6} + \frac{1}{12} a^{5} - \frac{13}{36} a^{4} + \frac{1}{9} a^{3} - \frac{1}{4} a^{2} + \frac{1}{12} a - \frac{1}{4}$, $\frac{1}{36} a^{11} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{7}{18} a^{5} - \frac{1}{4} a^{4} + \frac{5}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{9} - \frac{1}{12} a^{8} + \frac{1}{18} a^{6} - \frac{5}{12} a^{5} + \frac{1}{12} a^{4} - \frac{7}{18} a^{3} + \frac{1}{6} a^{2} - \frac{5}{12} a$, $\frac{1}{108} a^{13} + \frac{1}{108} a^{12} - \frac{1}{108} a^{11} - \frac{1}{108} a^{10} + \frac{1}{18} a^{8} - \frac{7}{108} a^{7} - \frac{7}{108} a^{6} - \frac{8}{27} a^{5} + \frac{1}{27} a^{4} - \frac{11}{36} a^{3} - \frac{1}{36} a^{2} - \frac{1}{4} a + \frac{1}{12}$, $\frac{1}{648} a^{14} - \frac{1}{648} a^{13} - \frac{1}{108} a^{12} + \frac{1}{648} a^{11} - \frac{1}{648} a^{10} - \frac{1}{36} a^{9} - \frac{19}{648} a^{8} + \frac{25}{648} a^{7} - \frac{11}{216} a^{6} + \frac{79}{324} a^{5} - \frac{299}{648} a^{4} + \frac{59}{216} a^{3} + \frac{7}{108} a^{2} - \frac{7}{24} a + \frac{7}{72}$, $\frac{1}{104328} a^{15} + \frac{67}{104328} a^{14} - \frac{11}{26082} a^{13} + \frac{775}{104328} a^{12} + \frac{29}{4536} a^{11} - \frac{517}{52164} a^{10} - \frac{2305}{104328} a^{9} + \frac{1181}{104328} a^{8} - \frac{7453}{104328} a^{7} - \frac{347}{52164} a^{6} - \frac{2191}{14904} a^{5} + \frac{6827}{14904} a^{4} + \frac{533}{1932} a^{3} - \frac{3515}{34776} a^{2} + \frac{4567}{11592} a - \frac{1151}{5796}$, $\frac{1}{142239647592} a^{16} - \frac{148987}{71119823796} a^{15} + \frac{15800579}{142239647592} a^{14} + \frac{143300251}{47413215864} a^{13} + \frac{581686619}{71119823796} a^{12} - \frac{914868607}{142239647592} a^{11} - \frac{606295031}{142239647592} a^{10} + \frac{489972253}{71119823796} a^{9} + \frac{98995049}{17779955949} a^{8} + \frac{3597032239}{47413215864} a^{7} + \frac{762910745}{142239647592} a^{6} + \frac{3368242793}{10159974828} a^{5} - \frac{38302808525}{142239647592} a^{4} + \frac{4670876963}{15804405288} a^{3} + \frac{6794350621}{23706607932} a^{2} + \frac{7523249813}{15804405288} a + \frac{3653975}{23873724}$, $\frac{1}{15512329067307745757663414808} a^{17} - \frac{33697813800415771}{15512329067307745757663414808} a^{16} - \frac{17554056521132555369105}{5170776355769248585887804936} a^{15} + \frac{995408393546160091512127}{7756164533653872878831707404} a^{14} + \frac{8609699773723948305340615}{15512329067307745757663414808} a^{13} - \frac{21575379345594662055085031}{5170776355769248585887804936} a^{12} + \frac{5011470157432078314423907}{861796059294874764314634156} a^{11} - \frac{31108709036358854653271497}{5170776355769248585887804936} a^{10} - \frac{17938890286723634252981371}{2585388177884624292943902468} a^{9} - \frac{238658037569685878535931847}{15512329067307745757663414808} a^{8} - \frac{201576988898035009553386217}{7756164533653872878831707404} a^{7} - \frac{133706844588879493347031495}{1723592118589749528629268312} a^{6} + \frac{5071333932078814901802818081}{15512329067307745757663414808} a^{5} + \frac{996080431493560844489648827}{3878082266826936439415853702} a^{4} - \frac{1148589938201032004461601941}{5170776355769248585887804936} a^{3} + \frac{5063045981348107160221325}{5170776355769248585887804936} a^{2} + \frac{94220816216814044078214497}{574530706196583176209756104} a - \frac{81824618085612757250827}{2603613472189953970739076}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( -\frac{774788943560307398}{1412265938392912031833887} a^{17} + \frac{1931392913744924873}{1412265938392912031833887} a^{16} - \frac{1162951228678831880}{470755312797637343944629} a^{15} + \frac{31276966541299555952}{1412265938392912031833887} a^{14} - \frac{165057961801452065300}{1412265938392912031833887} a^{13} + \frac{16408992659688200368}{67250758971091049134947} a^{12} - \frac{188687105793317707816}{156918437599212447981543} a^{11} + \frac{264242396058348695228}{67250758971091049134947} a^{10} - \frac{7528966825705778819822}{470755312797637343944629} a^{9} + \frac{62061743732791205178418}{1412265938392912031833887} a^{8} - \frac{175110231727180964279560}{1412265938392912031833887} a^{7} + \frac{46096227692409339973316}{156918437599212447981543} a^{6} - \frac{907393086513301655877676}{1412265938392912031833887} a^{5} + \frac{1632310883323505449620424}{1412265938392912031833887} a^{4} - \frac{901609885116059617320040}{470755312797637343944629} a^{3} + \frac{1130627757557595648300980}{470755312797637343944629} a^{2} - \frac{156357622902741378565882}{52306145866404149327181} a + \frac{683796006242875408339}{474073829604871444053} \) (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:  \( 1689544065.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.127308.1 x3, 3.3.10609.1, 6.0.48621980592.1, 6.0.4583088.2 x2, 6.0.3038873787.1, 9.3.2063322368402112.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.4583088.2
Degree 9 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$103$103.9.6.1$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
103.9.6.1$x^{9} + 8961 x^{6} + 26755898 x^{3} + 26650518803$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$