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Properties

Label 18.0.50396986113...488.13
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 19^{12}$
Root discriminant $84.71$
Ramified primes $2, 3, 19$
Class number $243$ (GRH)
Class group $[3, 3, 3, 9]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8796, -56916, 174384, -315030, 408726, -339732, 160075, -43155, 6465, 12342, -10119, 3825, -1100, 225, 69, -90, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 1100*x^12 + 3825*x^11 - 10119*x^10 + 12342*x^9 + 6465*x^8 - 43155*x^7 + 160075*x^6 - 339732*x^5 + 408726*x^4 - 315030*x^3 + 174384*x^2 - 56916*x + 8796)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 1100*x^12 + 3825*x^11 - 10119*x^10 + 12342*x^9 + 6465*x^8 - 43155*x^7 + 160075*x^6 - 339732*x^5 + 408726*x^4 - 315030*x^3 + 174384*x^2 - 56916*x + 8796, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 90 x^{15} + 69 x^{14} + 225 x^{13} - 1100 x^{12} + 3825 x^{11} - 10119 x^{10} + 12342 x^{9} + 6465 x^{8} - 43155 x^{7} + 160075 x^{6} - 339732 x^{5} + 408726 x^{4} - 315030 x^{3} + 174384 x^{2} - 56916 x + 8796 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-50396986113842071567939653456703488=-\,2^{12}\cdot 3^{33}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{39026} a^{15} - \frac{3033}{39026} a^{14} - \frac{3443}{39026} a^{13} - \frac{4337}{39026} a^{12} + \frac{18101}{39026} a^{11} - \frac{7471}{39026} a^{10} - \frac{11363}{39026} a^{9} - \frac{16807}{39026} a^{8} + \frac{12531}{39026} a^{7} - \frac{14273}{39026} a^{6} - \frac{10509}{39026} a^{5} + \frac{18185}{39026} a^{4} - \frac{302}{19513} a^{3} - \frac{5125}{19513} a^{2} + \frac{1231}{19513} a - \frac{8009}{19513}$, $\frac{1}{10473932012362} a^{16} + \frac{6063944}{5236966006181} a^{15} + \frac{1457743935883}{10473932012362} a^{14} - \frac{38120378059}{616113647786} a^{13} + \frac{625791630857}{10473932012362} a^{12} + \frac{2529946653145}{10473932012362} a^{11} + \frac{98690445041}{805687077874} a^{10} + \frac{3071804915535}{10473932012362} a^{9} - \frac{2492364704343}{10473932012362} a^{8} - \frac{4358642436651}{10473932012362} a^{7} + \frac{1898185612419}{10473932012362} a^{6} - \frac{4284694191}{47393357522} a^{5} - \frac{135899429604}{402843538937} a^{4} - \frac{3581490924467}{10473932012362} a^{3} + \frac{84023637163}{308056823893} a^{2} + \frac{186956750431}{748138000883} a + \frac{1535334557045}{5236966006181}$, $\frac{1}{254117011436139765325183954} a^{17} + \frac{2927286008091}{254117011436139765325183954} a^{16} + \frac{1246728385845424712}{18151215102581411808941711} a^{15} + \frac{1535376001950655021054283}{254117011436139765325183954} a^{14} - \frac{23083690456172811588175634}{127058505718069882662591977} a^{13} + \frac{2261166526156568418485711}{127058505718069882662591977} a^{12} - \frac{64579616925516088676389505}{254117011436139765325183954} a^{11} - \frac{5412306566044056903286785}{18151215102581411808941711} a^{10} - \frac{8863795744488834086196569}{127058505718069882662591977} a^{9} + \frac{124154118458765970931060353}{254117011436139765325183954} a^{8} + \frac{24815952273939465234068021}{127058505718069882662591977} a^{7} + \frac{27818598364371403612692785}{127058505718069882662591977} a^{6} - \frac{1194779762062147839754504}{9773731209082298666353229} a^{5} + \frac{116313301597633133999825793}{254117011436139765325183954} a^{4} + \frac{2882789567927235418361293}{9773731209082298666353229} a^{3} - \frac{41984463560771382263500008}{127058505718069882662591977} a^{2} + \frac{3099865042224130738311672}{127058505718069882662591977} a - \frac{36179003722130243379063629}{127058505718069882662591977}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $243$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{171578082553251}{1051586639671072499} a^{17} - \frac{3038562993680841}{2103173279342144998} a^{16} + \frac{6435757265676459}{1051586639671072499} a^{15} - \frac{28448656216769415}{2103173279342144998} a^{14} + \frac{17337216209744769}{2103173279342144998} a^{13} + \frac{83968395693862869}{2103173279342144998} a^{12} - \frac{19262764358632593}{110693330491691842} a^{11} + \frac{1242091745224671657}{2103173279342144998} a^{10} - \frac{3228751054004764381}{2103173279342144998} a^{9} + \frac{209134996081133055}{123716075255420294} a^{8} + \frac{190635628778343399}{123716075255420294} a^{7} - \frac{14711521067150097735}{2103173279342144998} a^{6} + \frac{51948218355646448865}{2103173279342144998} a^{5} - \frac{53326581289556269446}{1051586639671072499} a^{4} + \frac{115521991623939045451}{2103173279342144998} a^{3} - \frac{38192438306715275709}{1051586639671072499} a^{2} + \frac{17436843972730930896}{1051586639671072499} a - \frac{2974682479746422357}{1051586639671072499} \) (order $6$)
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:  \( 480855346.25948936 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 6.0.369375586992.13, 6.0.369375586992.7, 6.0.2834352.3, 6.0.3518667.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$