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Properties

Label 18.18.8609132442...2929.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{32}\cdot 11^{12}\cdot 23^{6}$
Root discriminant $99.17$
Ramified primes $3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times A_5$ (as 18T90)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![29217, -102483, -79641, 568215, -128142, -1122255, 597186, 969615, -634638, -387311, 271242, 69291, -48021, -5607, 3375, 210, -99, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 99*x^16 + 210*x^15 + 3375*x^14 - 5607*x^13 - 48021*x^12 + 69291*x^11 + 271242*x^10 - 387311*x^9 - 634638*x^8 + 969615*x^7 + 597186*x^6 - 1122255*x^5 - 128142*x^4 + 568215*x^3 - 79641*x^2 - 102483*x + 29217)
 
gp: K = bnfinit(x^18 - 3*x^17 - 99*x^16 + 210*x^15 + 3375*x^14 - 5607*x^13 - 48021*x^12 + 69291*x^11 + 271242*x^10 - 387311*x^9 - 634638*x^8 + 969615*x^7 + 597186*x^6 - 1122255*x^5 - 128142*x^4 + 568215*x^3 - 79641*x^2 - 102483*x + 29217, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 99 x^{16} + 210 x^{15} + 3375 x^{14} - 5607 x^{13} - 48021 x^{12} + 69291 x^{11} + 271242 x^{10} - 387311 x^{9} - 634638 x^{8} + 969615 x^{7} + 597186 x^{6} - 1122255 x^{5} - 128142 x^{4} + 568215 x^{3} - 79641 x^{2} - 102483 x + 29217 \)

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:  \(860913244245532536433941348923332929=3^{32}\cdot 11^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.17$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} + \frac{6}{19} a^{12} + \frac{1}{19} a^{11} + \frac{9}{19} a^{10} - \frac{6}{19} a^{9} + \frac{7}{19} a^{8} + \frac{6}{19} a^{7} - \frac{4}{19} a^{6} - \frac{2}{19} a^{5} - \frac{6}{19} a^{4} + \frac{1}{19} a^{3} + \frac{8}{19} a^{2} - \frac{7}{19} a + \frac{5}{19}$, $\frac{1}{57} a^{14} - \frac{1}{57} a^{13} + \frac{16}{57} a^{12} - \frac{17}{57} a^{11} + \frac{26}{57} a^{10} - \frac{8}{57} a^{9} - \frac{8}{19} a^{8} - \frac{9}{19} a^{7} - \frac{4}{19} a^{6} + \frac{9}{19} a^{5} + \frac{8}{19} a^{4} - \frac{6}{19} a^{3} - \frac{2}{19} a^{2} - \frac{1}{19} a + \frac{1}{19}$, $\frac{1}{57} a^{15} + \frac{23}{57} a^{12} - \frac{2}{19} a^{11} - \frac{1}{19} a^{10} + \frac{1}{57} a^{9} + \frac{5}{19} a^{8} - \frac{5}{19} a^{7} + \frac{6}{19} a^{6} + \frac{8}{19} a^{5} - \frac{6}{19} a^{4} + \frac{6}{19} a^{3} - \frac{5}{19} a^{2} - \frac{3}{19} a - \frac{5}{19}$, $\frac{1}{2166} a^{16} + \frac{7}{2166} a^{15} - \frac{8}{1083} a^{14} + \frac{9}{722} a^{13} - \frac{629}{2166} a^{12} - \frac{320}{1083} a^{11} - \frac{215}{1083} a^{10} - \frac{325}{722} a^{9} - \frac{30}{361} a^{8} - \frac{173}{361} a^{7} + \frac{179}{361} a^{6} - \frac{143}{722} a^{5} + \frac{44}{361} a^{4} + \frac{93}{361} a^{3} - \frac{1}{19} a^{2} + \frac{113}{722} a + \frac{309}{722}$, $\frac{1}{740383976687164764348672229653966588} a^{17} - \frac{36445052502052059797520110557297}{370191988343582382174336114826983294} a^{16} + \frac{465580874702350320601585610037719}{740383976687164764348672229653966588} a^{15} + \frac{1624106806052922392677614243452155}{246794658895721588116224076551322196} a^{14} - \frac{4484971306226459231020180821046733}{185095994171791191087168057413491647} a^{13} - \frac{277218616776217802505050381331528475}{740383976687164764348672229653966588} a^{12} + \frac{66155860847159585701607146620852296}{185095994171791191087168057413491647} a^{11} + \frac{50381859711471042836909137236006025}{246794658895721588116224076551322196} a^{10} - \frac{41588436278206680041180892327792849}{246794658895721588116224076551322196} a^{9} + \frac{51110501668198244425062245648305681}{123397329447860794058112038275661098} a^{8} + \frac{30528020294952389060429143616852511}{61698664723930397029056019137830549} a^{7} + \frac{101721256550888925765746347395574033}{246794658895721588116224076551322196} a^{6} + \frac{37928514268125862697509369062426567}{246794658895721588116224076551322196} a^{5} - \frac{15196250407186828530045638644991831}{123397329447860794058112038275661098} a^{4} - \frac{18108051883721430840610424114877814}{61698664723930397029056019137830549} a^{3} + \frac{122924880021305160403386741861021449}{246794658895721588116224076551322196} a^{2} + \frac{20974359567650924079021372363063781}{123397329447860794058112038275661098} a - \frac{94994058769103966390238193868670039}{246794658895721588116224076551322196}$

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:  \( 5246499911450 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_5$ (as 18T90):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 180
The 15 conjugacy class representatives for $C_3\times A_5$
Character table for $C_3\times A_5$

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.6.50815528929.1

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

Sibling fields

Degree 15 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 $15{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R $15{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ R $15{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
3.9.16.12$x^{9} + 6 x^{8} + 3$$9$$1$$16$$S_3\times C_3$$[2, 2]^{2}$
$11$11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
11.9.6.1$x^{9} - 121 x^{3} + 3993$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$23$23.6.3.1$x^{6} - 46 x^{4} + 529 x^{2} - 194672$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$