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Properties

Label 18.10.5973287043...6384.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 181$
Root discriminant $58.26$
Ramified primes $2, 3, 7, 41, 181$
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![-27, 234, 7128, -67385, 213196, -290598, 135322, 93136, -148329, 70943, -3122, -11744, 5442, -723, -343, 164, -13, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 13*x^16 + 164*x^15 - 343*x^14 - 723*x^13 + 5442*x^12 - 11744*x^11 - 3122*x^10 + 70943*x^9 - 148329*x^8 + 93136*x^7 + 135322*x^6 - 290598*x^5 + 213196*x^4 - 67385*x^3 + 7128*x^2 + 234*x - 27)
 
gp: K = bnfinit(x^18 - 5*x^17 - 13*x^16 + 164*x^15 - 343*x^14 - 723*x^13 + 5442*x^12 - 11744*x^11 - 3122*x^10 + 70943*x^9 - 148329*x^8 + 93136*x^7 + 135322*x^6 - 290598*x^5 + 213196*x^4 - 67385*x^3 + 7128*x^2 + 234*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 13 x^{16} + 164 x^{15} - 343 x^{14} - 723 x^{13} + 5442 x^{12} - 11744 x^{11} - 3122 x^{10} + 70943 x^{9} - 148329 x^{8} + 93136 x^{7} + 135322 x^{6} - 290598 x^{5} + 213196 x^{4} - 67385 x^{3} + 7128 x^{2} + 234 x - 27 \)

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:  \(59732870437511656046688565776384=2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 41^{8}\cdot 181\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.26$
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:  $2, 3, 7, 41, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{2}{9} a^{11} - \frac{4}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} + \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{117} a^{16} + \frac{2}{39} a^{15} - \frac{1}{9} a^{14} + \frac{5}{39} a^{13} + \frac{17}{117} a^{12} - \frac{44}{117} a^{11} + \frac{14}{117} a^{10} + \frac{11}{117} a^{9} - \frac{25}{117} a^{8} + \frac{6}{13} a^{7} + \frac{16}{39} a^{6} + \frac{10}{117} a^{5} + \frac{6}{13} a^{4} - \frac{4}{13} a^{3} - \frac{38}{117} a^{2} + \frac{5}{39} a - \frac{3}{13}$, $\frac{1}{21788520993499700531616292193241270429} a^{17} - \frac{15222399936341114063931451643046971}{21788520993499700531616292193241270429} a^{16} + \frac{922960482120503550597003834970172185}{21788520993499700531616292193241270429} a^{15} + \frac{77917211671502257130768999983271081}{7262840331166566843872097397747090143} a^{14} + \frac{1457002943348538017825502723553514477}{21788520993499700531616292193241270429} a^{13} + \frac{195062239254667212539775059343564712}{7262840331166566843872097397747090143} a^{12} - \frac{335642669583801446524223339303109265}{751328310120679328676423868732457601} a^{11} + \frac{4115702391567941315533808836117462673}{21788520993499700531616292193241270429} a^{10} - \frac{3758109400707996728101130152005358382}{21788520993499700531616292193241270429} a^{9} + \frac{1621428886768907626463433982713635603}{7262840331166566843872097397747090143} a^{8} - \frac{103102079938863152618050544709754418}{2420946777055522281290699132582363381} a^{7} + \frac{258565271631598167197012360982378517}{21788520993499700531616292193241270429} a^{6} - \frac{1975517695743820365714464235734942024}{21788520993499700531616292193241270429} a^{5} - \frac{7826577735165864389049444347002906747}{21788520993499700531616292193241270429} a^{4} - \frac{7948899993969601860415012839243915430}{21788520993499700531616292193241270429} a^{3} + \frac{3045160972256523431341936007783456257}{7262840331166566843872097397747090143} a^{2} - \frac{3533058416574695637322705591767532193}{7262840331166566843872097397747090143} a + \frac{286513540136017923518582659743674964}{2420946777055522281290699132582363381}$

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:  \( 3588912073.22 \) (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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
181Data not computed