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Properties

Label 18.0.30038232568...4672.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{9}\cdot 127^{14}$
Root discriminant $229.02$
Ramified primes $2, 7, 127$
Class number $23333184$ (GRH)
Class group $[6, 6, 648144]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![242280448, -521534464, 517853824, -242844992, 14364176, 37023896, -6028004, -10349906, 5579540, -156909, -651061, 144874, 31102, -14596, 296, 520, -50, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 50*x^16 + 520*x^15 + 296*x^14 - 14596*x^13 + 31102*x^12 + 144874*x^11 - 651061*x^10 - 156909*x^9 + 5579540*x^8 - 10349906*x^7 - 6028004*x^6 + 37023896*x^5 + 14364176*x^4 - 242844992*x^3 + 517853824*x^2 - 521534464*x + 242280448)
 
gp: K = bnfinit(x^18 - 7*x^17 - 50*x^16 + 520*x^15 + 296*x^14 - 14596*x^13 + 31102*x^12 + 144874*x^11 - 651061*x^10 - 156909*x^9 + 5579540*x^8 - 10349906*x^7 - 6028004*x^6 + 37023896*x^5 + 14364176*x^4 - 242844992*x^3 + 517853824*x^2 - 521534464*x + 242280448, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 50 x^{16} + 520 x^{15} + 296 x^{14} - 14596 x^{13} + 31102 x^{12} + 144874 x^{11} - 651061 x^{10} - 156909 x^{9} + 5579540 x^{8} - 10349906 x^{7} - 6028004 x^{6} + 37023896 x^{5} + 14364176 x^{4} - 242844992 x^{3} + 517853824 x^{2} - 521534464 x + 242280448 \)

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:  \(-3003823256884925694987438506048666589724672=-\,2^{18}\cdot 7^{9}\cdot 127^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $229.02$
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:  $2, 7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{64} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} + \frac{5}{64} a^{7} - \frac{1}{32} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} + \frac{3}{128} a^{10} + \frac{3}{128} a^{9} + \frac{5}{128} a^{8} - \frac{3}{128} a^{7} - \frac{1}{32} a^{6} + \frac{11}{64} a^{5} + \frac{3}{32} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} - \frac{1}{16} a^{8} - \frac{1}{128} a^{7} + \frac{3}{32} a^{6} - \frac{1}{64} a^{5} - \frac{1}{4} a^{4} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1024} a^{16} - \frac{1}{256} a^{15} - \frac{1}{512} a^{14} + \frac{1}{512} a^{13} + \frac{3}{512} a^{12} - \frac{5}{512} a^{11} + \frac{3}{128} a^{10} - \frac{27}{512} a^{9} - \frac{39}{1024} a^{8} + \frac{3}{512} a^{7} + \frac{25}{512} a^{6} - \frac{45}{256} a^{5} + \frac{11}{128} a^{4} + \frac{7}{64} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{335360194648085423020297631572698100439163904} a^{17} + \frac{24506031606334418479955855789510301237659}{83840048662021355755074407893174525109790976} a^{16} + \frac{469374838285088631415978369174518675269403}{167680097324042711510148815786349050219581952} a^{15} + \frac{344196359373891422557119859407546021771149}{167680097324042711510148815786349050219581952} a^{14} + \frac{306580782958333804783290741208710170881871}{167680097324042711510148815786349050219581952} a^{13} - \frac{609807029372582430706059872138451700371889}{167680097324042711510148815786349050219581952} a^{12} - \frac{603774435432667561689180733947338028381269}{20960012165505338938768601973293631277447744} a^{11} + \frac{1083674296809954800194287189062516407544953}{167680097324042711510148815786349050219581952} a^{10} + \frac{2025041121806988437538064347451845760025921}{335360194648085423020297631572698100439163904} a^{9} - \frac{2732928610225666981135593449584703359129081}{167680097324042711510148815786349050219581952} a^{8} - \frac{16559295319935316230154635675058621076161263}{167680097324042711510148815786349050219581952} a^{7} + \frac{5042041547848567751712930716978185693218467}{83840048662021355755074407893174525109790976} a^{6} + \frac{385827046590123212642574480974727751507379}{41920024331010677877537203946587262554895488} a^{5} + \frac{4939525187957689802942226150611493470698773}{20960012165505338938768601973293631277447744} a^{4} + \frac{1139215337644150167839559572643540400910523}{2620001520688167367346075246661703909680968} a^{3} + \frac{204286784504321324386620698978855529481065}{1310000760344083683673037623330851954840484} a^{2} - \frac{68911400931170724378919277074849407844277}{655000380172041841836518811665425977420242} a + \frac{52236940540667075875826020940199208058836}{327500190086020920918259405832712988710121}$

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

Class group and class number

$C_{6}\times C_{6}\times C_{648144}$, which has order $23333184$ (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:  \( -1 \) (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:  \( 5546046730.2947445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.16129.1, 3.3.1016.1, 6.0.89229611863.2, 6.0.354063808.1, 9.9.272832440404737536.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$