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Properties

Label 18.12.2272148422...5952.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 37^{9}\cdot 139\cdot 89959^{2}\cdot 615997^{2}$
Root discriminant $198.42$
Ramified primes $2, 37, 139, 89959, 615997$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![196195508, 1035235308, 2288404092, 2727089468, 1823754372, 565616724, -65900133, -115852927, -28617940, 4709032, 3317873, 177232, -164531, -21032, 4725, 704, -90, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 90*x^16 + 704*x^15 + 4725*x^14 - 21032*x^13 - 164531*x^12 + 177232*x^11 + 3317873*x^10 + 4709032*x^9 - 28617940*x^8 - 115852927*x^7 - 65900133*x^6 + 565616724*x^5 + 1823754372*x^4 + 2727089468*x^3 + 2288404092*x^2 + 1035235308*x + 196195508)
 
gp: K = bnfinit(x^18 - 9*x^17 - 90*x^16 + 704*x^15 + 4725*x^14 - 21032*x^13 - 164531*x^12 + 177232*x^11 + 3317873*x^10 + 4709032*x^9 - 28617940*x^8 - 115852927*x^7 - 65900133*x^6 + 565616724*x^5 + 1823754372*x^4 + 2727089468*x^3 + 2288404092*x^2 + 1035235308*x + 196195508, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 90 x^{16} + 704 x^{15} + 4725 x^{14} - 21032 x^{13} - 164531 x^{12} + 177232 x^{11} + 3317873 x^{10} + 4709032 x^{9} - 28617940 x^{8} - 115852927 x^{7} - 65900133 x^{6} + 565616724 x^{5} + 1823754372 x^{4} + 2727089468 x^{3} + 2288404092 x^{2} + 1035235308 x + 196195508 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-227214842256824958738151908292991724285952=-\,2^{12}\cdot 37^{9}\cdot 139\cdot 89959^{2}\cdot 615997^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $198.42$
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:  $2, 37, 139, 89959, 615997$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{870031376188962664105612951942644747489091299316} a^{17} - \frac{10334624665379475196249536476424188697679386659}{435015688094481332052806475971322373744545649658} a^{16} - \frac{9405217375416084919787127052681919573367148147}{217507844047240666026403237985661186872272824829} a^{15} + \frac{182910749168052893958853011677699784154005521301}{870031376188962664105612951942644747489091299316} a^{14} + \frac{82252819554203923649198301956594439694735908637}{870031376188962664105612951942644747489091299316} a^{13} - \frac{2831175142463683465893051945105848291430418957}{66925490476074051085047150149434211345314715332} a^{12} - \frac{51243861727111081100234827530722253654644594631}{435015688094481332052806475971322373744545649658} a^{11} - \frac{86069346438738648845362100102360119477796516993}{870031376188962664105612951942644747489091299316} a^{10} - \frac{54869214465210157091057209794734952997040742146}{217507844047240666026403237985661186872272824829} a^{9} - \frac{115550745226042362986770832717866803472200460959}{870031376188962664105612951942644747489091299316} a^{8} + \frac{415020627695484256529159884702843456228692750627}{870031376188962664105612951942644747489091299316} a^{7} - \frac{379978507132821790371459123749402141201225627541}{870031376188962664105612951942644747489091299316} a^{6} - \frac{10852491779192531793539141279925218442197771289}{217507844047240666026403237985661186872272824829} a^{5} - \frac{36018304360402196274164375801563742492644891276}{217507844047240666026403237985661186872272824829} a^{4} - \frac{10708274756142269130664939871061708374772042409}{66925490476074051085047150149434211345314715332} a^{3} + \frac{3105887102125908493501799677135513505181508071}{435015688094481332052806475971322373744545649658} a^{2} + \frac{37662271065365986804688908068145877397593758923}{435015688094481332052806475971322373744545649658} a + \frac{552811561309885446131984292326163168884507803}{2880898596652194252005340900472333600957255958}$

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:  $14$
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:  \( 76170214385300 \) (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 ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
37Data not computed
139Data not computed
89959Data not computed
615997Data not computed