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Properties

Label 18.18.2561749146...6672.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 113^{15}$
Root discriminant $81.58$
Ramified primes $2, 113$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $F_9$ (as 18T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4096, -36864, -86784, 58864, 492272, 556336, -156488, -630088, -218908, 205707, 130119, -19944, -23886, 138, 1964, 44, -74, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 74*x^16 + 44*x^15 + 1964*x^14 + 138*x^13 - 23886*x^12 - 19944*x^11 + 130119*x^10 + 205707*x^9 - 218908*x^8 - 630088*x^7 - 156488*x^6 + 556336*x^5 + 492272*x^4 + 58864*x^3 - 86784*x^2 - 36864*x - 4096)
 
gp: K = bnfinit(x^18 - x^17 - 74*x^16 + 44*x^15 + 1964*x^14 + 138*x^13 - 23886*x^12 - 19944*x^11 + 130119*x^10 + 205707*x^9 - 218908*x^8 - 630088*x^7 - 156488*x^6 + 556336*x^5 + 492272*x^4 + 58864*x^3 - 86784*x^2 - 36864*x - 4096, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 74 x^{16} + 44 x^{15} + 1964 x^{14} + 138 x^{13} - 23886 x^{12} - 19944 x^{11} + 130119 x^{10} + 205707 x^{9} - 218908 x^{8} - 630088 x^{7} - 156488 x^{6} + 556336 x^{5} + 492272 x^{4} + 58864 x^{3} - 86784 x^{2} - 36864 x - 4096 \)

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:  \(25617491467048076163656841929756672=2^{12}\cdot 113^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.58$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} + \frac{3}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{5}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{32} a^{7} + \frac{7}{32} a^{6} - \frac{1}{16} a^{5} + \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{32} a^{14} + \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{64} a^{8} - \frac{13}{64} a^{7} + \frac{3}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{9853072264746735089627361792} a^{17} + \frac{27527885240611626164602711}{9853072264746735089627361792} a^{16} - \frac{7571593251421371164994659}{1642178710791122514937893632} a^{15} + \frac{60720337520090402699665763}{2463268066186683772406840448} a^{14} + \frac{3146245523533941835139257}{821089355395561257468946816} a^{13} - \frac{51418960661840203430521065}{1642178710791122514937893632} a^{12} + \frac{52902373545399419073100763}{1642178710791122514937893632} a^{11} - \frac{39995159333235575186586073}{410544677697780628734473408} a^{10} + \frac{171508732696645660762525965}{3284357421582245029875787264} a^{9} - \frac{596364090340924870054255}{3284357421582245029875787264} a^{8} + \frac{313750369435617548649231767}{2463268066186683772406840448} a^{7} - \frac{69431637473530118271063209}{410544677697780628734473408} a^{6} + \frac{211094717067262731488469815}{1231634033093341886203420224} a^{5} + \frac{627566316176708998485827}{205272338848890314367236704} a^{4} + \frac{306963762807766729143176291}{615817016546670943101710112} a^{3} + \frac{271790518508430951579606071}{615817016546670943101710112} a^{2} - \frac{6563766051035632473756370}{19244281767083466971928441} a - \frac{677130664482592460558918}{19244281767083466971928441}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 692727849211 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_9$ (as 18T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $F_9$
Character table for $F_9$

Intermediate fields

\(\Q(\sqrt{113}) \), 9.9.15056675074868288.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$113$113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.8.7.1$x^{8} - 113$$8$$1$$7$$C_8$$[\ ]_{8}$
113.8.7.1$x^{8} - 113$$8$$1$$7$$C_8$$[\ ]_{8}$