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Properties

Label 18.4.60424099571...8352.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{6}\cdot 3^{33}\cdot 19^{8}$
Root discriminant $34.95$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 21, -84, -21, 363, 1113, -4722, 6393, -4350, 883, 1032, -969, 228, 177, -159, 42, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 42*x^15 - 159*x^14 + 177*x^13 + 228*x^12 - 969*x^11 + 1032*x^10 + 883*x^9 - 4350*x^8 + 6393*x^7 - 4722*x^6 + 1113*x^5 + 363*x^4 - 21*x^3 - 84*x^2 + 21*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 42*x^15 - 159*x^14 + 177*x^13 + 228*x^12 - 969*x^11 + 1032*x^10 + 883*x^9 - 4350*x^8 + 6393*x^7 - 4722*x^6 + 1113*x^5 + 363*x^4 - 21*x^3 - 84*x^2 + 21*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 42 x^{15} - 159 x^{14} + 177 x^{13} + 228 x^{12} - 969 x^{11} + 1032 x^{10} + 883 x^{9} - 4350 x^{8} + 6393 x^{7} - 4722 x^{6} + 1113 x^{5} + 363 x^{4} - 21 x^{3} - 84 x^{2} + 21 x + 1 \)

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:  \(-6042409957173305670222428352=-\,2^{6}\cdot 3^{33}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.95$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{534626392792644934586206} a^{17} + \frac{3602027027503508980943}{534626392792644934586206} a^{16} - \frac{5467894666145093462789}{534626392792644934586206} a^{15} + \frac{54709641965597037886888}{267313196396322467293103} a^{14} + \frac{102028352891561429407381}{267313196396322467293103} a^{13} + \frac{23863986353221077187479}{267313196396322467293103} a^{12} + \frac{79555922933812716341420}{267313196396322467293103} a^{11} - \frac{66379443248695282066741}{534626392792644934586206} a^{10} + \frac{71225508118144872245967}{534626392792644934586206} a^{9} - \frac{8586803327737568066838}{267313196396322467293103} a^{8} - \frac{233089162170617643144207}{534626392792644934586206} a^{7} - \frac{67982189606368242755691}{534626392792644934586206} a^{6} + \frac{158029462029494762557499}{534626392792644934586206} a^{5} + \frac{182984530309069342890593}{534626392792644934586206} a^{4} - \frac{17090173389681361842835}{534626392792644934586206} a^{3} - \frac{16886773968503141974141}{534626392792644934586206} a^{2} + \frac{130539688319762239957900}{267313196396322467293103} a + \frac{131493750435702138055638}{267313196396322467293103}$

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:  $10$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12239072.2147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{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
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$