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Properties

Label 18.8.13991415038...6391.2
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 71^{3}$
Root discriminant $32.22$
Ramified primes $3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 129, -690, 1399, -1920, 2064, -732, -459, 1890, -2113, 1551, -723, 36, 174, -138, 47, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 47*x^15 - 138*x^14 + 174*x^13 + 36*x^12 - 723*x^11 + 1551*x^10 - 2113*x^9 + 1890*x^8 - 459*x^7 - 732*x^6 + 2064*x^5 - 1920*x^4 + 1399*x^3 - 690*x^2 + 129*x - 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 47*x^15 - 138*x^14 + 174*x^13 + 36*x^12 - 723*x^11 + 1551*x^10 - 2113*x^9 + 1890*x^8 - 459*x^7 - 732*x^6 + 2064*x^5 - 1920*x^4 + 1399*x^3 - 690*x^2 + 129*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 47 x^{15} - 138 x^{14} + 174 x^{13} + 36 x^{12} - 723 x^{11} + 1551 x^{10} - 2113 x^{9} + 1890 x^{8} - 459 x^{7} - 732 x^{6} + 2064 x^{5} - 1920 x^{4} + 1399 x^{3} - 690 x^{2} + 129 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1399141503834185765244506391=-\,3^{24}\cdot 7^{12}\cdot 71^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.22$
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:  $3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{3}{16} a^{10} - \frac{1}{8} a^{9} - \frac{3}{16} a^{7} - \frac{1}{16} a^{6} + \frac{5}{16} a^{5} - \frac{3}{8} a^{4} - \frac{5}{16} a^{3} - \frac{1}{4} a - \frac{3}{16}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} + \frac{3}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{3}{32} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{32} a^{5} - \frac{11}{32} a^{4} + \frac{11}{32} a^{3} - \frac{1}{8} a^{2} + \frac{9}{32} a - \frac{3}{32}$, $\frac{1}{128} a^{15} + \frac{1}{128} a^{14} + \frac{3}{128} a^{13} + \frac{3}{64} a^{12} - \frac{3}{32} a^{11} + \frac{11}{128} a^{10} + \frac{27}{128} a^{9} + \frac{17}{128} a^{8} + \frac{3}{32} a^{7} + \frac{15}{128} a^{6} + \frac{3}{8} a^{5} + \frac{1}{16} a^{4} - \frac{45}{128} a^{3} - \frac{19}{128} a^{2} + \frac{15}{64} a - \frac{39}{128}$, $\frac{1}{80137984} a^{16} + \frac{151997}{40068992} a^{15} - \frac{218875}{20034496} a^{14} + \frac{2164369}{80137984} a^{13} + \frac{2440121}{40068992} a^{12} + \frac{8534503}{80137984} a^{11} - \frac{4282845}{40068992} a^{10} + \frac{3580049}{20034496} a^{9} - \frac{12680595}{80137984} a^{8} - \frac{11585381}{80137984} a^{7} + \frac{28866503}{80137984} a^{6} - \frac{674963}{2504312} a^{5} + \frac{22754387}{80137984} a^{4} - \frac{1928583}{5008624} a^{3} + \frac{11389347}{80137984} a^{2} - \frac{942529}{80137984} a + \frac{33961545}{80137984}$, $\frac{1}{160275968} a^{17} - \frac{1}{160275968} a^{16} + \frac{294947}{80137984} a^{15} + \frac{1524757}{160275968} a^{14} - \frac{4599401}{160275968} a^{13} - \frac{1194863}{160275968} a^{12} - \frac{2599047}{160275968} a^{11} + \frac{13877441}{80137984} a^{10} + \frac{29998097}{160275968} a^{9} - \frac{5375241}{40068992} a^{8} - \frac{13721737}{80137984} a^{7} + \frac{10406035}{160275968} a^{6} - \frac{55730877}{160275968} a^{5} + \frac{74726399}{160275968} a^{4} - \frac{26833421}{160275968} a^{3} - \frac{14515841}{80137984} a^{2} - \frac{2589315}{40068992} a + \frac{36606445}{160275968}$

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:  $12$
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:  \( 10625039.2594 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 48 conjugacy class representatives for t18n263
Character table for t18n263 is not computed

Intermediate fields

3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
7Data not computed
71Data not computed