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Properties

Label 18.18.4450636533...4041.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{32}\cdot 19^{8}\cdot 521^{3}$
Root discriminant $74.02$
Ramified primes $3, 19, 521$
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![-271, 2502, 50184, -400341, -182439, 1659249, 1994634, 22716, -923211, -275611, 157617, 69579, -13662, -7515, 774, 402, -36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 36*x^16 + 402*x^15 + 774*x^14 - 7515*x^13 - 13662*x^12 + 69579*x^11 + 157617*x^10 - 275611*x^9 - 923211*x^8 + 22716*x^7 + 1994634*x^6 + 1659249*x^5 - 182439*x^4 - 400341*x^3 + 50184*x^2 + 2502*x - 271)
 
gp: K = bnfinit(x^18 - 9*x^17 - 36*x^16 + 402*x^15 + 774*x^14 - 7515*x^13 - 13662*x^12 + 69579*x^11 + 157617*x^10 - 275611*x^9 - 923211*x^8 + 22716*x^7 + 1994634*x^6 + 1659249*x^5 - 182439*x^4 - 400341*x^3 + 50184*x^2 + 2502*x - 271, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 36 x^{16} + 402 x^{15} + 774 x^{14} - 7515 x^{13} - 13662 x^{12} + 69579 x^{11} + 157617 x^{10} - 275611 x^{9} - 923211 x^{8} + 22716 x^{7} + 1994634 x^{6} + 1659249 x^{5} - 182439 x^{4} - 400341 x^{3} + 50184 x^{2} + 2502 x - 271 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4450636533424095295669119045874041=3^{32}\cdot 19^{8}\cdot 521^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.02$
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:  $3, 19, 521$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{6} a^{11} + \frac{1}{3} a^{9} + \frac{1}{12} a^{8} - \frac{1}{4} a^{7} - \frac{1}{3} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{12} a^{9} + \frac{1}{3} a^{8} - \frac{5}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{241941419627274141556719694044} a^{17} - \frac{362597036409176187291704841}{40323569937879023592786615674} a^{16} - \frac{1712931180902795583510335561}{60485354906818535389179923511} a^{15} - \frac{4514240711873543481610243511}{241941419627274141556719694044} a^{14} + \frac{1044689956558473393270729121}{40323569937879023592786615674} a^{13} + \frac{1719744401398083342550595077}{40323569937879023592786615674} a^{12} + \frac{489934688273169629029452565}{20161784968939511796393307837} a^{11} + \frac{18141012501951686773512816965}{241941419627274141556719694044} a^{10} + \frac{30054185779846067802180221650}{60485354906818535389179923511} a^{9} + \frac{185542873963952301684359915}{241941419627274141556719694044} a^{8} + \frac{112291296778639132153084741333}{241941419627274141556719694044} a^{7} + \frac{8782383608055446806842562627}{60485354906818535389179923511} a^{6} + \frac{26281643639795919115320764381}{60485354906818535389179923511} a^{5} - \frac{8865537170977806766863975770}{60485354906818535389179923511} a^{4} - \frac{723032753583852693092451182}{60485354906818535389179923511} a^{3} + \frac{37144199302053404644658318611}{120970709813637070778359847022} a^{2} - \frac{1543368585542907076506408721}{80647139875758047185573231348} a - \frac{878731004775247030269328163}{60485354906818535389179923511}$

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:  $17$
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:  \( 136568730607 \) (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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$19$19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
521Data not computed