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Properties

Label 18.10.1217816758...9257.1
Degree $18$
Signature $[10, 4]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $36.33$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197, -2133, -369, 398, 1017, 873, -283, 1770, -2367, -1878, 2730, 534, -1109, -99, 201, 26, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 26*x^15 + 201*x^14 - 99*x^13 - 1109*x^12 + 534*x^11 + 2730*x^10 - 1878*x^9 - 2367*x^8 + 1770*x^7 - 283*x^6 + 873*x^5 + 1017*x^4 + 398*x^3 - 369*x^2 - 2133*x + 197)
 
gp: K = bnfinit(x^18 - 3*x^17 - 18*x^16 + 26*x^15 + 201*x^14 - 99*x^13 - 1109*x^12 + 534*x^11 + 2730*x^10 - 1878*x^9 - 2367*x^8 + 1770*x^7 - 283*x^6 + 873*x^5 + 1017*x^4 + 398*x^3 - 369*x^2 - 2133*x + 197, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 18 x^{16} + 26 x^{15} + 201 x^{14} - 99 x^{13} - 1109 x^{12} + 534 x^{11} + 2730 x^{10} - 1878 x^{9} - 2367 x^{8} + 1770 x^{7} - 283 x^{6} + 873 x^{5} + 1017 x^{4} + 398 x^{3} - 369 x^{2} - 2133 x + 197 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12178167588536804870357659257=3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.33$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{7}{16} a^{7} - \frac{1}{16} a^{6} + \frac{5}{16} a^{5} + \frac{5}{16} a^{4} - \frac{3}{16} a^{3} + \frac{3}{16} a^{2} + \frac{1}{8} a + \frac{1}{16}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{9} + \frac{3}{32} a^{8} + \frac{1}{4} a^{7} + \frac{3}{16} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{3}{16} a^{3} + \frac{15}{32} a^{2} - \frac{1}{32} a + \frac{15}{32}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{5}{64} a^{9} - \frac{21}{64} a^{8} - \frac{9}{32} a^{7} + \frac{11}{32} a^{6} - \frac{1}{8} a^{5} + \frac{7}{32} a^{4} - \frac{11}{64} a^{3} - \frac{9}{32} a^{2} - \frac{9}{32} a + \frac{15}{64}$, $\frac{1}{128} a^{16} - \frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{3}{128} a^{13} - \frac{1}{32} a^{11} + \frac{3}{128} a^{10} - \frac{1}{8} a^{9} - \frac{61}{128} a^{8} - \frac{3}{16} a^{7} - \frac{15}{64} a^{6} - \frac{21}{64} a^{5} + \frac{39}{128} a^{4} - \frac{7}{128} a^{3} - \frac{1}{2} a^{2} - \frac{31}{128} a - \frac{15}{128}$, $\frac{1}{245079633373620188271872} a^{17} - \frac{230880599921462010165}{61269908343405047067968} a^{16} + \frac{652430605140332669561}{122539816686810094135936} a^{15} - \frac{76126070225195273301}{30634954171702523533984} a^{14} + \frac{998389888651476000593}{245079633373620188271872} a^{13} + \frac{3544050366480064318967}{61269908343405047067968} a^{12} - \frac{8188919499201019205873}{245079633373620188271872} a^{11} - \frac{49560123246869544043433}{245079633373620188271872} a^{10} - \frac{14865456843130606088077}{245079633373620188271872} a^{9} + \frac{83722544922138759992071}{245079633373620188271872} a^{8} - \frac{46507891130549475338619}{122539816686810094135936} a^{7} - \frac{389847560518619179543}{1914684635731407720874} a^{6} + \frac{105871097208421614666597}{245079633373620188271872} a^{5} - \frac{10620835627571656685579}{61269908343405047067968} a^{4} - \frac{11591933178223566883787}{245079633373620188271872} a^{3} + \frac{33981204555901926106649}{245079633373620188271872} a^{2} + \frac{28831380541644203122699}{122539816686810094135936} a - \frac{2135025884485779973035}{245079633373620188271872}$

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:  $13$
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:  \( 31861811.7637 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed