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Properties

Label 18.0.51329217772...7887.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{30}\cdot 7^{12}\cdot 23^{9}$
Root discriminant $109.51$
Ramified primes $3, 7, 23$
Class number $165564$ (GRH)
Class group $[3, 6, 9198]$ (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![2010480704, -2293113888, 2278390032, -1811464128, 1239478116, -715801902, 339321753, -126914076, 36860769, -7444440, 769713, 114906, -67656, 15876, -1611, 42, 51, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 51*x^16 + 42*x^15 - 1611*x^14 + 15876*x^13 - 67656*x^12 + 114906*x^11 + 769713*x^10 - 7444440*x^9 + 36860769*x^8 - 126914076*x^7 + 339321753*x^6 - 715801902*x^5 + 1239478116*x^4 - 1811464128*x^3 + 2278390032*x^2 - 2293113888*x + 2010480704)
 
gp: K = bnfinit(x^18 - 6*x^17 + 51*x^16 + 42*x^15 - 1611*x^14 + 15876*x^13 - 67656*x^12 + 114906*x^11 + 769713*x^10 - 7444440*x^9 + 36860769*x^8 - 126914076*x^7 + 339321753*x^6 - 715801902*x^5 + 1239478116*x^4 - 1811464128*x^3 + 2278390032*x^2 - 2293113888*x + 2010480704, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 51 x^{16} + 42 x^{15} - 1611 x^{14} + 15876 x^{13} - 67656 x^{12} + 114906 x^{11} + 769713 x^{10} - 7444440 x^{9} + 36860769 x^{8} - 126914076 x^{7} + 339321753 x^{6} - 715801902 x^{5} + 1239478116 x^{4} - 1811464128 x^{3} + 2278390032 x^{2} - 2293113888 x + 2010480704 \)

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:  \(-5132921777297808355912172232996877887=-\,3^{30}\cdot 7^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.51$
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:  $3, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{28} a^{9} - \frac{3}{28} a^{8} - \frac{1}{7} a^{5} + \frac{1}{14} a^{4} - \frac{5}{28} a^{3} + \frac{3}{28} a^{2} - \frac{3}{14} a - \frac{2}{7}$, $\frac{1}{56} a^{10} - \frac{1}{56} a^{9} - \frac{3}{28} a^{8} - \frac{1}{4} a^{7} - \frac{1}{14} a^{6} - \frac{3}{28} a^{5} + \frac{13}{56} a^{4} + \frac{3}{8} a^{3} - \frac{5}{14} a - \frac{2}{7}$, $\frac{1}{112} a^{11} + \frac{1}{112} a^{9} + \frac{3}{28} a^{8} + \frac{5}{56} a^{7} + \frac{9}{56} a^{6} + \frac{3}{112} a^{5} - \frac{3}{56} a^{4} + \frac{9}{112} a^{3} - \frac{3}{14} a^{2} - \frac{3}{14}$, $\frac{1}{448} a^{12} - \frac{1}{448} a^{10} + \frac{3}{224} a^{9} - \frac{5}{224} a^{8} + \frac{51}{224} a^{7} - \frac{45}{448} a^{6} + \frac{47}{224} a^{5} - \frac{33}{448} a^{4} + \frac{15}{224} a^{3} - \frac{3}{56} a^{2} + \frac{15}{56} a + \frac{13}{28}$, $\frac{1}{448} a^{13} - \frac{1}{448} a^{11} - \frac{1}{224} a^{10} - \frac{1}{224} a^{9} + \frac{19}{224} a^{8} + \frac{67}{448} a^{7} - \frac{7}{32} a^{6} + \frac{15}{448} a^{5} - \frac{37}{224} a^{4} + \frac{1}{14} a^{3} - \frac{27}{56} a^{2} - \frac{5}{28} a + \frac{2}{7}$, $\frac{1}{2688} a^{14} - \frac{1}{2688} a^{13} + \frac{1}{2688} a^{12} + \frac{1}{384} a^{11} + \frac{1}{192} a^{10} + \frac{1}{96} a^{9} + \frac{19}{896} a^{8} - \frac{5}{128} a^{7} + \frac{109}{896} a^{6} + \frac{77}{384} a^{5} + \frac{11}{48} a^{4} - \frac{205}{672} a^{3} - \frac{131}{336} a^{2} + \frac{5}{12} a + \frac{13}{84}$, $\frac{1}{666624} a^{15} - \frac{15}{111104} a^{14} + \frac{1}{111104} a^{13} + \frac{241}{333312} a^{12} + \frac{65}{222208} a^{11} - \frac{461}{111104} a^{10} + \frac{5293}{666624} a^{9} - \frac{1791}{111104} a^{8} + \frac{3155}{55552} a^{7} - \frac{9487}{41664} a^{6} - \frac{16501}{222208} a^{5} + \frac{6857}{27776} a^{4} + \frac{13561}{166656} a^{3} - \frac{339}{27776} a^{2} - \frac{2725}{6944} a + \frac{121}{20832}$, $\frac{1}{1333248} a^{16} - \frac{79}{666624} a^{14} + \frac{263}{666624} a^{13} + \frac{919}{1333248} a^{12} + \frac{211}{83328} a^{11} - \frac{533}{444416} a^{10} - \frac{2929}{166656} a^{9} - \frac{969}{13888} a^{8} + \frac{2947}{23808} a^{7} + \frac{104043}{444416} a^{6} + \frac{105745}{666624} a^{5} - \frac{1675}{333312} a^{4} + \frac{12745}{41664} a^{3} - \frac{30619}{83328} a^{2} + \frac{5601}{13888} a + \frac{8669}{20832}$, $\frac{1}{9122380906498417152085860391027799331654875480064} a^{17} - \frac{1164407069021372365433639944635303748365}{9242533846502955574555076384020060113125507072} a^{16} + \frac{1888028008440222912321981645274355036177821}{4561190453249208576042930195513899665827437740032} a^{15} + \frac{371799640241016384000665818055202774766097}{23756200277339628000223594768301560759517904896} a^{14} - \frac{4309963705577985021267157824435555483550712779}{9122380906498417152085860391027799331654875480064} a^{13} - \frac{8323919985879471356854954238169761744661307945}{9122380906498417152085860391027799331654875480064} a^{12} + \frac{3926748789225090805478565475105135700216432825}{9122380906498417152085860391027799331654875480064} a^{11} + \frac{5093145124265229932984540548924212875267374025}{9122380906498417152085860391027799331654875480064} a^{10} + \frac{2632134068027023388409345014243372481985904027}{190049602218717024001788758146412486076143239168} a^{9} - \frac{23905412843207674945669338907285097962385071565}{1140297613312302144010732548878474916456859435008} a^{8} + \frac{267537887757750147488815282160924552943206459011}{3040793635499472384028620130342599777218291826688} a^{7} + \frac{1400003294212623433946303087061784700169370070659}{9122380906498417152085860391027799331654875480064} a^{6} - \frac{7462555779815555944921266137421855484241939181}{49045058637088264258526131134558060922875674624} a^{5} + \frac{58065755996510446341007486509851889483332179435}{325799318089229184003066442536707118987674124288} a^{4} + \frac{40898758440458321963052756736065838936761056845}{570148806656151072005366274439237458228429717504} a^{3} + \frac{112211108066000520695575847712388019801297379199}{570148806656151072005366274439237458228429717504} a^{2} + \frac{9656582322809003510898875639507575981455907733}{285074403328075536002683137219618729114214858752} a - \frac{4498661987612811825210419425810550391670101405}{47512400554679256000447189536603121519035809792}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{9198}$, which has order $165564$ (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:  \( 237885673.5279126 \) (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

\(\Q(\sqrt{-23}) \), 3.3.3969.1, 3.3.621.1, 6.0.191666276487.2, 6.0.8869743.1, 9.9.20539533187176381.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$3.9.15.28$x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 6$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.28$x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 6$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$7$7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$