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Properties

Label 18.0.10301979323...2448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 23^{9}$
Root discriminant $215.80$
Ramified primes $2, 3, 7, 23$
Class number $15186528$ (GRH)
Class group $[2, 6, 6, 18, 11718]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![619938747, -738392301, 800021124, -527232024, 317659023, -141091389, 59042850, -19367253, 6375792, -1697075, 481848, -100113, 20406, -1401, -39, 90, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 90*x^15 - 39*x^14 - 1401*x^13 + 20406*x^12 - 100113*x^11 + 481848*x^10 - 1697075*x^9 + 6375792*x^8 - 19367253*x^7 + 59042850*x^6 - 141091389*x^5 + 317659023*x^4 - 527232024*x^3 + 800021124*x^2 - 738392301*x + 619938747)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 + 90*x^15 - 39*x^14 - 1401*x^13 + 20406*x^12 - 100113*x^11 + 481848*x^10 - 1697075*x^9 + 6375792*x^8 - 19367253*x^7 + 59042850*x^6 - 141091389*x^5 + 317659023*x^4 - 527232024*x^3 + 800021124*x^2 - 738392301*x + 619938747, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} + 90 x^{15} - 39 x^{14} - 1401 x^{13} + 20406 x^{12} - 100113 x^{11} + 481848 x^{10} - 1697075 x^{9} + 6375792 x^{8} - 19367253 x^{7} + 59042850 x^{6} - 141091389 x^{5} + 317659023 x^{4} - 527232024 x^{3} + 800021124 x^{2} - 738392301 x + 619938747 \)

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:  \(-1030197932390779328264996615851405379432448=-\,2^{12}\cdot 3^{30}\cdot 7^{14}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $215.80$
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:  $2, 3, 7, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{7}$, $\frac{1}{7091111092767018310572894719537007879049455708175008813} a^{17} - \frac{620232110578703479615745749643173714903396864323877960}{7091111092767018310572894719537007879049455708175008813} a^{16} + \frac{839476668428073899632699051432926941984818722012146129}{7091111092767018310572894719537007879049455708175008813} a^{15} - \frac{240225260662819579967522080578656560386922572552642422}{7091111092767018310572894719537007879049455708175008813} a^{14} - \frac{468745692176334447191467144110586841763923367013781}{23403006906821842609151467721244250425905794416419171} a^{13} + \frac{30567320768187086040090702764718687507780662906499822}{214882154326273282138572567258697208456044112368939661} a^{12} - \frac{206735160057452696556525881670163233081532824044491472}{2363703697589006103524298239845669293016485236058336271} a^{11} - \frac{743935568126575926113811676689566374204116416304716092}{7091111092767018310572894719537007879049455708175008813} a^{10} - \frac{2263801661473170983617490303734041588244556937867835}{70209020720465527827454403163732751277717383249257513} a^{9} - \frac{107108695373364469062063774850647054155149363308903903}{2363703697589006103524298239845669293016485236058336271} a^{8} - \frac{924619676511207344746991324881042572494698344609032405}{7091111092767018310572894719537007879049455708175008813} a^{7} - \frac{393765666166235900739830239406947649961282213935590381}{2363703697589006103524298239845669293016485236058336271} a^{6} + \frac{21634682801673101231523639870549346559668423044017583}{2363703697589006103524298239845669293016485236058336271} a^{5} - \frac{633469643710795544706203934418845558978619181414088529}{2363703697589006103524298239845669293016485236058336271} a^{4} + \frac{1019381324281133964423878449196232788084885720155686530}{2363703697589006103524298239845669293016485236058336271} a^{3} + \frac{215754518219037136236637911803314713956762719772329270}{2363703697589006103524298239845669293016485236058336271} a^{2} - \frac{860599295787888354064092740233257644200551567346699703}{2363703697589006103524298239845669293016485236058336271} a - \frac{861732370863556097232772621876434319116652470313489278}{2363703697589006103524298239845669293016485236058336271}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{18}\times C_{11718}$, which has order $15186528$ (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:  \( -1 \) (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:  \( 4695974.091249611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.3.756.1, 3.3.3969.2, 6.0.6953878512.6, 6.0.191666276487.3, 9.9.756284282720064.2

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.14$x^{9} + 3 x^{7} + 3 x^{6} + 6 x^{3} + 15$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$7$7.6.4.2$x^{6} - 7 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$