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Properties

Label 18.16.1320480580...0000.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,2^{24}\cdot 3^{18}\cdot 5^{4}\cdot 37^{9}\cdot 67\cdot 139^{4}$
Root discriminant $248.65$
Ramified primes $2, 3, 5, 37, 67, 139$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-123654400, 1854816000, -1947556800, -4295389120, 175340160, 2106458820, 427284889, -343327668, -103411782, 25851304, 10023315, -982368, -500656, 18036, 13527, -124, -186, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 186*x^16 - 124*x^15 + 13527*x^14 + 18036*x^13 - 500656*x^12 - 982368*x^11 + 10023315*x^10 + 25851304*x^9 - 103411782*x^8 - 343327668*x^7 + 427284889*x^6 + 2106458820*x^5 + 175340160*x^4 - 4295389120*x^3 - 1947556800*x^2 + 1854816000*x - 123654400)
 
gp: K = bnfinit(x^18 - 186*x^16 - 124*x^15 + 13527*x^14 + 18036*x^13 - 500656*x^12 - 982368*x^11 + 10023315*x^10 + 25851304*x^9 - 103411782*x^8 - 343327668*x^7 + 427284889*x^6 + 2106458820*x^5 + 175340160*x^4 - 4295389120*x^3 - 1947556800*x^2 + 1854816000*x - 123654400, 1)
 

Normalized defining polynomial

\( x^{18} - 186 x^{16} - 124 x^{15} + 13527 x^{14} + 18036 x^{13} - 500656 x^{12} - 982368 x^{11} + 10023315 x^{10} + 25851304 x^{9} - 103411782 x^{8} - 343327668 x^{7} + 427284889 x^{6} + 2106458820 x^{5} + 175340160 x^{4} - 4295389120 x^{3} - 1947556800 x^{2} + 1854816000 x - 123654400 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13204805800957263883631911003213434716160000=-\,2^{24}\cdot 3^{18}\cdot 5^{4}\cdot 37^{9}\cdot 67\cdot 139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.65$
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:  $2, 3, 5, 37, 67, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{2} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{11120} a^{15} - \frac{93}{5560} a^{13} - \frac{31}{2780} a^{12} - \frac{373}{11120} a^{11} + \frac{339}{2780} a^{10} - \frac{16}{695} a^{9} - \frac{257}{2780} a^{8} - \frac{273}{2224} a^{7} - \frac{1369}{2780} a^{6} - \frac{671}{5560} a^{5} - \frac{28}{695} a^{4} + \frac{1669}{11120} a^{3}$, $\frac{1}{44480} a^{16} - \frac{93}{22240} a^{14} - \frac{31}{11120} a^{13} + \frac{2407}{44480} a^{12} + \frac{339}{11120} a^{11} - \frac{4}{695} a^{10} - \frac{119}{1390} a^{9} + \frac{839}{8896} a^{8} + \frac{1053}{5560} a^{7} + \frac{7669}{22240} a^{6} - \frac{2197}{11120} a^{5} + \frac{10009}{44480} a^{4} - \frac{7}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{49613931173576671516995321751952179063933123603102037120} a^{17} + \frac{45607823682292211071096809330331382333703831855303}{12403482793394167879248830437988044765983280900775509280} a^{16} - \frac{90991950777739424722169917586027943087987453572121}{4961393117357667151699532175195217906393312360310203712} a^{15} + \frac{447449228565108750023179923220796688198518829440844351}{12403482793394167879248830437988044765983280900775509280} a^{14} - \frac{1594262031284087510339027341937248585414814345431138537}{49613931173576671516995321751952179063933123603102037120} a^{13} - \frac{379399593320416281141854473818322347929745378549103583}{6201741396697083939624415218994022382991640450387754640} a^{12} + \frac{164676656002379120136735652908826371640910421139880683}{3100870698348541969812207609497011191495820225193877320} a^{11} - \frac{289052000047142043666105510064710886616993524895135099}{3100870698348541969812207609497011191495820225193877320} a^{10} - \frac{1632013274833115117476200960410090473781968711016745677}{49613931173576671516995321751952179063933123603102037120} a^{9} - \frac{508778971928476173946536111289607700164237123870008281}{12403482793394167879248830437988044765983280900775509280} a^{8} - \frac{4043221491003182518018776438710293570526021796230215827}{24806965586788335758497660875976089531966561801551018560} a^{7} + \frac{3020503572428412749365898544609136124477416508101209453}{12403482793394167879248830437988044765983280900775509280} a^{6} - \frac{14386801517052582552644572613528434342260986748548500679}{49613931173576671516995321751952179063933123603102037120} a^{5} - \frac{103972584118483797373197056994500635916057116986275745}{620174139669708393962441521899402238299164045038775464} a^{4} + \frac{72605902360512228353308847070090574970245400003251243}{387608837293567746226525951187126398936977528149234665} a^{3} - \frac{808968032456070446896281981306190609867306572170981}{4461684458055456071672241164743900994958014712509176} a^{2} - \frac{343148469072734414270524325054503652418899883221067}{1115421114513864017918060291185975248739503678127294} a - \frac{33628866549419827219455086511341368822923011874445}{557710557256932008959030145592987624369751839063647}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $16$
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:  \( 5329902372580000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1

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 R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$[\ ]_{3}^{2}$
2.12.20.35$x^{12} - 18 x^{10} - 21 x^{8} - 8 x^{6} + 19 x^{4} - 6 x^{2} + 21$$6$$2$$20$12T30$[2, 8/3, 8/3]_{3}^{2}$
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed
67Data not computed
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
139.3.2.3$x^{3} - 2224$$3$$1$$2$$C_3$$[\ ]_{3}$
139.3.2.2$x^{3} + 556$$3$$1$$2$$C_3$$[\ ]_{3}$