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Properties

Label 18.0.68835522420...4224.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{6}\cdot 32009^{6}$
Root discriminant $40.00$
Ramified primes $2, 32009$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5696, 400, 6668, -6372, -785, 1253, -429, -4339, 6472, -5266, 3257, -1923, 1227, -614, 250, -72, 22, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 22*x^16 - 72*x^15 + 250*x^14 - 614*x^13 + 1227*x^12 - 1923*x^11 + 3257*x^10 - 5266*x^9 + 6472*x^8 - 4339*x^7 - 429*x^6 + 1253*x^5 - 785*x^4 - 6372*x^3 + 6668*x^2 + 400*x + 5696)
 
gp: K = bnfinit(x^18 - 5*x^17 + 22*x^16 - 72*x^15 + 250*x^14 - 614*x^13 + 1227*x^12 - 1923*x^11 + 3257*x^10 - 5266*x^9 + 6472*x^8 - 4339*x^7 - 429*x^6 + 1253*x^5 - 785*x^4 - 6372*x^3 + 6668*x^2 + 400*x + 5696, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 22 x^{16} - 72 x^{15} + 250 x^{14} - 614 x^{13} + 1227 x^{12} - 1923 x^{11} + 3257 x^{10} - 5266 x^{9} + 6472 x^{8} - 4339 x^{7} - 429 x^{6} + 1253 x^{5} - 785 x^{4} - 6372 x^{3} + 6668 x^{2} + 400 x + 5696 \)

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:  \(-68835522420844686611068124224=-\,2^{6}\cdot 32009^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.00$
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:  $2, 32009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{50} a^{15} + \frac{3}{50} a^{14} + \frac{1}{25} a^{13} - \frac{2}{25} a^{11} + \frac{3}{50} a^{9} - \frac{21}{50} a^{8} + \frac{17}{50} a^{7} - \frac{1}{25} a^{6} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{23}{50} a^{2} - \frac{11}{50} a - \frac{11}{25}$, $\frac{1}{200} a^{16} + \frac{1}{200} a^{15} - \frac{1}{50} a^{14} + \frac{2}{25} a^{13} - \frac{7}{100} a^{12} - \frac{1}{100} a^{11} - \frac{17}{200} a^{10} - \frac{17}{200} a^{9} + \frac{99}{200} a^{8} + \frac{8}{25} a^{7} - \frac{7}{25} a^{6} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{37}{200} a^{3} + \frac{73}{200} a^{2} - \frac{7}{20} a - \frac{12}{25}$, $\frac{1}{774857372236518724075173431200} a^{17} - \frac{822898534399994629701714367}{774857372236518724075173431200} a^{16} + \frac{855894077995963262087543603}{193714343059129681018793357800} a^{15} - \frac{295545767310186877719561651}{9685717152956484050939667890} a^{14} - \frac{3045201333640709539177428387}{387428686118259362037586715600} a^{13} + \frac{58132866706786703543561471}{77485737223651872407517343120} a^{12} - \frac{80173395643487460590473297}{774857372236518724075173431200} a^{11} + \frac{64571301674155242723696137559}{774857372236518724075173431200} a^{10} + \frac{67960558702273836451469991347}{774857372236518724075173431200} a^{9} - \frac{3876483187037217613581750847}{48428585764782420254698339450} a^{8} - \frac{2799753730843089573178086473}{19371434305912968101879335780} a^{7} + \frac{73395727007856680422279594297}{154971474447303744815034686240} a^{6} - \frac{16260594784322884954179775859}{154971474447303744815034686240} a^{5} - \frac{221008987256840841901802152517}{774857372236518724075173431200} a^{4} + \frac{256036779749999710728082932289}{774857372236518724075173431200} a^{3} + \frac{32114372018912672246780210249}{387428686118259362037586715600} a^{2} + \frac{2547051672300002590132953653}{19371434305912968101879335780} a - \frac{50106097852681533603132879}{4842858576478242025469833945}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 4621437.941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.3.32009.3, 6.0.4098304324.1, 9.9.32795655776729.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
32009Data not computed