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Properties

Label 18.0.18295286285...8528.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{31}\cdot 7^{10}$
Root discriminant $42.24$
Ramified primes $2, 3, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5061, 21546, 51030, 77742, 76230, 43218, 11037, -5742, -13158, -7740, 2754, 3312, 27, -720, -48, 78, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 78*x^15 - 48*x^14 - 720*x^13 + 27*x^12 + 3312*x^11 + 2754*x^10 - 7740*x^9 - 13158*x^8 - 5742*x^7 + 11037*x^6 + 43218*x^5 + 76230*x^4 + 77742*x^3 + 51030*x^2 + 21546*x + 5061)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 78*x^15 - 48*x^14 - 720*x^13 + 27*x^12 + 3312*x^11 + 2754*x^10 - 7740*x^9 - 13158*x^8 - 5742*x^7 + 11037*x^6 + 43218*x^5 + 76230*x^4 + 77742*x^3 + 51030*x^2 + 21546*x + 5061, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 78 x^{15} - 48 x^{14} - 720 x^{13} + 27 x^{12} + 3312 x^{11} + 2754 x^{10} - 7740 x^{9} - 13158 x^{8} - 5742 x^{7} + 11037 x^{6} + 43218 x^{5} + 76230 x^{4} + 77742 x^{3} + 51030 x^{2} + 21546 x + 5061 \)

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:  \(-182952862853086423052769558528=-\,2^{20}\cdot 3^{31}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.24$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{292005953060905039089750839425190278717582} a^{17} - \frac{3442915785440476765541220251357380696363}{292005953060905039089750839425190278717582} a^{16} - \frac{3119358719340192191619286818404660189921}{146002976530452519544875419712595139358791} a^{15} + \frac{60847074567673230560488874707236794219373}{146002976530452519544875419712595139358791} a^{14} - \frac{41780439761135742730555966093059270589621}{292005953060905039089750839425190278717582} a^{13} - \frac{72513707508872092097141266533503773521133}{292005953060905039089750839425190278717582} a^{12} - \frac{102233882121401411283266016940208807489495}{292005953060905039089750839425190278717582} a^{11} + \frac{113935097934632346899483557524719744627781}{292005953060905039089750839425190278717582} a^{10} + \frac{74064671370863461948271829621398778356373}{292005953060905039089750839425190278717582} a^{9} - \frac{143834269944856101978399002344280905180201}{292005953060905039089750839425190278717582} a^{8} - \frac{21571181326533260719043343536767554906359}{292005953060905039089750839425190278717582} a^{7} + \frac{9523074231859776402242155188090847536703}{292005953060905039089750839425190278717582} a^{6} + \frac{30834168899518439595901469482999116478745}{146002976530452519544875419712595139358791} a^{5} - \frac{22530850873780576148076738207223338670383}{146002976530452519544875419712595139358791} a^{4} + \frac{107587660807271430811721993089033381138105}{292005953060905039089750839425190278717582} a^{3} - \frac{8290260020138919600934887950511461309245}{22461996389300387622288526109630021439814} a^{2} - \frac{14827178708086484806636045630379935749706}{146002976530452519544875419712595139358791} a + \frac{25810004676749623701839325145625023522189}{146002976530452519544875419712595139358791}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( \frac{37788197852763681964107}{130546882060232385397969061} a^{17} - \frac{228144505759412027259246}{130546882060232385397969061} a^{16} + \frac{428526571198505758332269}{261093764120464770795938122} a^{15} + \frac{6212952493854431328700323}{261093764120464770795938122} a^{14} - \frac{2358457900098925832280984}{130546882060232385397969061} a^{13} - \frac{28111607660195548546333125}{130546882060232385397969061} a^{12} + \frac{11029614023889098390001723}{261093764120464770795938122} a^{11} + \frac{268868555816519667905362305}{261093764120464770795938122} a^{10} + \frac{151791722264373367200446357}{261093764120464770795938122} a^{9} - \frac{659479855899016189591974807}{261093764120464770795938122} a^{8} - \frac{893874208852113207988879263}{261093764120464770795938122} a^{7} - \frac{154553389294434056353051041}{261093764120464770795938122} a^{6} + \frac{790378537270642905808958721}{261093764120464770795938122} a^{5} + \frac{3311496328347994351403924631}{261093764120464770795938122} a^{4} + \frac{2626476280716153042156425922}{130546882060232385397969061} a^{3} + \frac{2319943808730944660401422873}{130546882060232385397969061} a^{2} + \frac{2703569894929037826216617463}{261093764120464770795938122} a + \frac{1034645245312121388255797441}{261093764120464770795938122} \) (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:  \( 12168177.008988433 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.756.1, 6.0.15431472.1, 6.0.1714608.3

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[\ ]_{3}^{6}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$