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Properties

Label 18.0.25528356304...4448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 7^{12}$
Root discriminant $55.57$
Ramified primes $2, 3, 7$
Class number $48$ (GRH)
Class group $[48]$ (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![1764, 5292, 31752, -49392, 124362, -111132, 164619, -137592, 142317, -80304, 46620, -14364, 5961, -1188, 540, -60, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 - 60*x^15 + 540*x^14 - 1188*x^13 + 5961*x^12 - 14364*x^11 + 46620*x^10 - 80304*x^9 + 142317*x^8 - 137592*x^7 + 164619*x^6 - 111132*x^5 + 124362*x^4 - 49392*x^3 + 31752*x^2 + 5292*x + 1764)
 
gp: K = bnfinit(x^18 + 27*x^16 - 60*x^15 + 540*x^14 - 1188*x^13 + 5961*x^12 - 14364*x^11 + 46620*x^10 - 80304*x^9 + 142317*x^8 - 137592*x^7 + 164619*x^6 - 111132*x^5 + 124362*x^4 - 49392*x^3 + 31752*x^2 + 5292*x + 1764, 1)
 

Normalized defining polynomial

\( x^{18} + 27 x^{16} - 60 x^{15} + 540 x^{14} - 1188 x^{13} + 5961 x^{12} - 14364 x^{11} + 46620 x^{10} - 80304 x^{9} + 142317 x^{8} - 137592 x^{7} + 164619 x^{6} - 111132 x^{5} + 124362 x^{4} - 49392 x^{3} + 31752 x^{2} + 5292 x + 1764 \)

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:  \(-25528356304590234835421802344448=-\,2^{12}\cdot 3^{37}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.57$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{18} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3}$, $\frac{1}{18} a^{10} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{11} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{126} a^{12} - \frac{1}{126} a^{10} + \frac{1}{42} a^{9} - \frac{3}{14} a^{8} - \frac{11}{42} a^{7} + \frac{10}{21} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{378} a^{13} + \frac{1}{63} a^{11} + \frac{1}{126} a^{10} - \frac{1}{63} a^{9} - \frac{10}{21} a^{8} + \frac{41}{126} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{5}{18} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{378} a^{14} + \frac{1}{126} a^{11} - \frac{1}{42} a^{9} + \frac{16}{63} a^{8} - \frac{1}{7} a^{7} - \frac{5}{42} a^{6} + \frac{5}{18} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{756} a^{15} - \frac{1}{756} a^{14} - \frac{1}{756} a^{13} - \frac{1}{84} a^{11} - \frac{1}{84} a^{10} - \frac{1}{252} a^{9} - \frac{13}{126} a^{8} + \frac{121}{252} a^{7} + \frac{53}{252} a^{6} + \frac{7}{36} a^{5} - \frac{2}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{70847028} a^{16} - \frac{241}{7871892} a^{15} + \frac{48395}{70847028} a^{14} + \frac{4889}{35423514} a^{13} - \frac{9851}{3373668} a^{12} + \frac{87391}{3373668} a^{11} + \frac{364579}{23615676} a^{10} + \frac{41957}{1686834} a^{9} + \frac{714733}{23615676} a^{8} - \frac{80207}{23615676} a^{7} + \frac{1212039}{2623964} a^{6} - \frac{799057}{1686834} a^{5} - \frac{378424}{843417} a^{4} - \frac{139154}{281139} a^{3} + \frac{30570}{93713} a^{2} - \frac{6415}{93713} a - \frac{16648}{281139}$, $\frac{1}{65167891463955337324092} a^{17} - \frac{40881476422375}{9309698780565048189156} a^{16} + \frac{15188030850429725581}{65167891463955337324092} a^{15} - \frac{2806091200160891305}{5430657621996278110341} a^{14} + \frac{13529405972750199467}{21722630487985112441364} a^{13} - \frac{17056822591622654245}{7240876829328370813788} a^{12} + \frac{385309094633540683465}{21722630487985112441364} a^{11} - \frac{19523532488353175494}{775808231713754015763} a^{10} - \frac{39449184852492087817}{3103232926855016063052} a^{9} + \frac{805980675615497608151}{3103232926855016063052} a^{8} + \frac{65434142635211557549}{443318989550716580436} a^{7} + \frac{41148807254874000620}{110829747387679145109} a^{6} + \frac{81910088471174939698}{258602743904584671921} a^{5} - \frac{7794529470776218277}{36943249129226381703} a^{4} + \frac{2594748486040215776}{36943249129226381703} a^{3} + \frac{5862719067383463826}{36943249129226381703} a^{2} - \frac{3506505427470240391}{36943249129226381703} a - \frac{1649055301865163068}{12314416376408793901}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  \( -\frac{2325190665193535}{21557357414474144004} a^{17} + \frac{1753681715771}{57030046070037418} a^{16} - \frac{2311560989844931}{798420644980523852} a^{15} + \frac{13109157512305780}{1796446451206178667} a^{14} - \frac{11902529402732243}{199605161245130963} a^{13} + \frac{171986341320356987}{1197630967470785778} a^{12} - \frac{4814735685459545189}{7185785804824714668} a^{11} + \frac{13933423833653157}{8147149438576774} a^{10} - \frac{152966526185653177}{28515023035018709} a^{9} + \frac{5045563357765921279}{513270414630336762} a^{8} - \frac{1936737477356035083}{114060092140074836} a^{7} + \frac{1010433748731686365}{57030046070037418} a^{6} - \frac{6642200328446986099}{342180276420224508} a^{5} + \frac{59736978664021506}{4073574719288387} a^{4} - \frac{117502091927488145}{8147149438576774} a^{3} + \frac{198309135031242791}{24441448315730322} a^{2} - \frac{13377240741357000}{4073574719288387} a + \frac{1924239635074046}{4073574719288387} \) (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:  \( 65376984.0719 \) (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

\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 9.9.2917096519063104.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$