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Properties

Label 18.2.28533206322...0048.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{45}\cdot 11^{9}$
Root discriminant $82.07$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3710192, 3425472, 5449824, -4732980, -3175020, 3146436, -227046, -134568, 39096, -72977, 13086, 4662, 177, 468, -234, -33, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 33*x^15 - 234*x^14 + 468*x^13 + 177*x^12 + 4662*x^11 + 13086*x^10 - 72977*x^9 + 39096*x^8 - 134568*x^7 - 227046*x^6 + 3146436*x^5 - 3175020*x^4 - 4732980*x^3 + 5449824*x^2 + 3425472*x - 3710192)
 
gp: K = bnfinit(x^18 - 33*x^15 - 234*x^14 + 468*x^13 + 177*x^12 + 4662*x^11 + 13086*x^10 - 72977*x^9 + 39096*x^8 - 134568*x^7 - 227046*x^6 + 3146436*x^5 - 3175020*x^4 - 4732980*x^3 + 5449824*x^2 + 3425472*x - 3710192, 1)
 

Normalized defining polynomial

\( x^{18} - 33 x^{15} - 234 x^{14} + 468 x^{13} + 177 x^{12} + 4662 x^{11} + 13086 x^{10} - 72977 x^{9} + 39096 x^{8} - 134568 x^{7} - 227046 x^{6} + 3146436 x^{5} - 3175020 x^{4} - 4732980 x^{3} + 5449824 x^{2} + 3425472 x - 3710192 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28533206322804731469169670894850048=2^{12}\cdot 3^{45}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.07$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{6} a^{4} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{36} a^{15} + \frac{1}{36} a^{12} - \frac{1}{6} a^{10} + \frac{1}{36} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{7}{36} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{18} a^{3} - \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{612} a^{16} - \frac{1}{204} a^{15} - \frac{1}{102} a^{14} + \frac{13}{612} a^{13} + \frac{7}{204} a^{12} + \frac{1}{102} a^{11} - \frac{71}{612} a^{10} + \frac{31}{204} a^{9} - \frac{1}{2} a^{8} - \frac{133}{612} a^{7} - \frac{95}{204} a^{6} + \frac{14}{51} a^{5} - \frac{41}{306} a^{4} - \frac{16}{51} a^{3} - \frac{8}{51} a^{2} + \frac{50}{153} a - \frac{1}{51}$, $\frac{1}{3583761781356286859893054864420126054292825448} a^{17} + \frac{296102391260028940770441538373265222623791}{447970222669535857486631858052515756786603181} a^{16} + \frac{12442240300752105310125775681599143406579469}{895940445339071714973263716105031513573206362} a^{15} - \frac{62477150426043943530434542229891742709920593}{3583761781356286859893054864420126054292825448} a^{14} + \frac{99393211186592702275888514951517684022504075}{1791880890678143429946527432210063027146412724} a^{13} - \frac{4769252631434543710079174094434669705436877}{447970222669535857486631858052515756786603181} a^{12} - \frac{237343700883511503968870986510777429528443803}{3583761781356286859893054864420126054292825448} a^{11} + \frac{11545343808249001212096210789454171624381477}{1791880890678143429946527432210063027146412724} a^{10} - \frac{278709720220269009210023507369474178989656225}{1791880890678143429946527432210063027146412724} a^{9} - \frac{434149300829421375201041849747373055677403921}{3583761781356286859893054864420126054292825448} a^{8} + \frac{5265523493942544942523769472493705593155656}{26351189568796226910978344591324456281564893} a^{7} - \frac{345700788490291454761959846603116278861950571}{895940445339071714973263716105031513573206362} a^{6} + \frac{1344926931306949252670006368509456597238211}{105404758275184907643913378365297825126259572} a^{5} + \frac{194716949859361557818206270460544674720022281}{447970222669535857486631858052515756786603181} a^{4} + \frac{88268467758787286108002826149157083970615677}{447970222669535857486631858052515756786603181} a^{3} + \frac{404355699724741097975673559440685995985333841}{895940445339071714973263716105031513573206362} a^{2} - \frac{206145576259828096640509863669974218902668044}{447970222669535857486631858052515756786603181} a + \frac{99891198550851641305603450122406815852213733}{447970222669535857486631858052515756786603181}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $9$
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:  \( 26942088406.02308 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{33}) \), 3.1.243.1, 6.2.235782657.1, 9.1.2008387814976.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{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}$
3Data not computed
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$