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Properties

Label 18.4.18357745623...8224.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 17^{4}\cdot 19$
Root discriminant $15.18$
Ramified primes $2, 3, 17, 19$
Class number $1$
Class group Trivial
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 21, -47, 75, -81, 36, 45, -138, 169, -138, 45, 36, -81, 75, -47, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 47*x^15 + 75*x^14 - 81*x^13 + 36*x^12 + 45*x^11 - 138*x^10 + 169*x^9 - 138*x^8 + 45*x^7 + 36*x^6 - 81*x^5 + 75*x^4 - 47*x^3 + 21*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 47*x^15 + 75*x^14 - 81*x^13 + 36*x^12 + 45*x^11 - 138*x^10 + 169*x^9 - 138*x^8 + 45*x^7 + 36*x^6 - 81*x^5 + 75*x^4 - 47*x^3 + 21*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 47 x^{15} + 75 x^{14} - 81 x^{13} + 36 x^{12} + 45 x^{11} - 138 x^{10} + 169 x^{9} - 138 x^{8} + 45 x^{7} + 36 x^{6} - 81 x^{5} + 75 x^{4} - 47 x^{3} + 21 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1835774562353817268224=-\,2^{12}\cdot 3^{24}\cdot 17^{4}\cdot 19\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.18$
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:  $2, 3, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2719} a^{16} + \frac{337}{2719} a^{15} - \frac{1306}{2719} a^{14} + \frac{293}{2719} a^{13} + \frac{1277}{2719} a^{12} - \frac{122}{2719} a^{11} + \frac{417}{2719} a^{10} - \frac{909}{2719} a^{9} + \frac{343}{2719} a^{8} - \frac{909}{2719} a^{7} + \frac{417}{2719} a^{6} - \frac{122}{2719} a^{5} + \frac{1277}{2719} a^{4} + \frac{293}{2719} a^{3} - \frac{1306}{2719} a^{2} + \frac{337}{2719} a + \frac{1}{2719}$, $\frac{1}{2719} a^{17} - \frac{677}{2719} a^{15} - \frac{63}{2719} a^{14} + \frac{420}{2719} a^{13} - \frac{869}{2719} a^{12} + \frac{746}{2719} a^{11} - \frac{50}{2719} a^{10} - \frac{571}{2719} a^{9} + \frac{417}{2719} a^{8} - \frac{497}{2719} a^{7} + \frac{737}{2719} a^{6} - \frac{1113}{2719} a^{5} - \frac{454}{2719} a^{4} + \frac{556}{2719} a^{3} - \frac{19}{2719} a^{2} + \frac{630}{2719} a - \frac{337}{2719}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3117.95057969 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.5.9829532736.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ R R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $18$

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$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$