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Properties

Label 18.12.6061731293...9952.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 7^{13}\cdot 41^{9}$
Root discriminant $75.30$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3683, 6872, 25776, -30696, -3586, 89742, -42441, -107848, 1365, 36468, 26824, 4554, -9201, -2700, 972, 254, -51, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 51*x^16 + 254*x^15 + 972*x^14 - 2700*x^13 - 9201*x^12 + 4554*x^11 + 26824*x^10 + 36468*x^9 + 1365*x^8 - 107848*x^7 - 42441*x^6 + 89742*x^5 - 3586*x^4 - 30696*x^3 + 25776*x^2 + 6872*x - 3683)
 
gp: K = bnfinit(x^18 - 6*x^17 - 51*x^16 + 254*x^15 + 972*x^14 - 2700*x^13 - 9201*x^12 + 4554*x^11 + 26824*x^10 + 36468*x^9 + 1365*x^8 - 107848*x^7 - 42441*x^6 + 89742*x^5 - 3586*x^4 - 30696*x^3 + 25776*x^2 + 6872*x - 3683, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 51 x^{16} + 254 x^{15} + 972 x^{14} - 2700 x^{13} - 9201 x^{12} + 4554 x^{11} + 26824 x^{10} + 36468 x^{9} + 1365 x^{8} - 107848 x^{7} - 42441 x^{6} + 89742 x^{5} - 3586 x^{4} - 30696 x^{3} + 25776 x^{2} + 6872 x - 3683 \)

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:  \(-6061731293901735349533566719229952=-\,2^{18}\cdot 3^{6}\cdot 7^{13}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.30$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{6} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a + \frac{1}{12}$, $\frac{1}{24} a^{16} - \frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{3}{8} a^{10} - \frac{1}{4} a^{9} + \frac{5}{24} a^{8} + \frac{1}{6} a^{7} + \frac{1}{12} a^{6} + \frac{1}{24} a^{4} + \frac{1}{6} a^{3} - \frac{3}{8} a^{2} - \frac{1}{12} a + \frac{1}{8}$, $\frac{1}{14150719720912987403068381663840189701246744} a^{17} - \frac{8257821240424272956226687810379554275527}{1768839965114123425383547707980023712655843} a^{16} + \frac{98257942808726660823249419171420266468123}{7075359860456493701534190831920094850623372} a^{15} + \frac{162717112570173010501101771130641359217455}{7075359860456493701534190831920094850623372} a^{14} + \frac{1168519608770179961970782813311440683237683}{7075359860456493701534190831920094850623372} a^{13} - \frac{807971554787360985032721672673011083470621}{7075359860456493701534190831920094850623372} a^{12} + \frac{6867367783876425374049194675211105555962713}{14150719720912987403068381663840189701246744} a^{11} + \frac{1895841744120264589183238584288726299610559}{7075359860456493701534190831920094850623372} a^{10} + \frac{29404266467811778302313653062485645882031}{115046501796040547992425867185692599197128} a^{9} + \frac{1132423571773350844990373220187118053019587}{3537679930228246850767095415960047425311686} a^{8} - \frac{565673188723471031058639736210765157435165}{2358453286818831233844730277306698283541124} a^{7} - \frac{149925766078863269899976277601475295594351}{1179226643409415616922365138653349141770562} a^{6} + \frac{3984646682262667769744606351989922642589577}{14150719720912987403068381663840189701246744} a^{5} + \frac{851784136686164494734731727481987618536841}{1768839965114123425383547707980023712655843} a^{4} - \frac{3992068147851689954379352730894867512164169}{14150719720912987403068381663840189701246744} a^{3} + \frac{601021546342318213449314297854688553823233}{2358453286818831233844730277306698283541124} a^{2} - \frac{336063043662560945817993447066172768506823}{4716906573637662467689460554613396567082248} a + \frac{124999981602449099487899659917566634806967}{1179226643409415616922365138653349141770562}$

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:  $14$
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:  \( 59572327682.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
41Data not computed