↑

Properties

Label 18.0.70410449619...3303.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 23^{9}$
Root discriminant $75.93$
Ramified primes $3, 7, 23$
Class number $70956$ (GRH)
Class group $[3, 6, 3942]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![258264152, -208331724, 257711352, -147782137, 102647091, -45561114, 22683008, -8208228, 3218604, -978324, 315630, -81192, 22294, -5004, 1272, -268, 60, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 268*x^15 + 1272*x^14 - 5004*x^13 + 22294*x^12 - 81192*x^11 + 315630*x^10 - 978324*x^9 + 3218604*x^8 - 8208228*x^7 + 22683008*x^6 - 45561114*x^5 + 102647091*x^4 - 147782137*x^3 + 257711352*x^2 - 208331724*x + 258264152)
 
gp: K = bnfinit(x^18 - 9*x^17 + 60*x^16 - 268*x^15 + 1272*x^14 - 5004*x^13 + 22294*x^12 - 81192*x^11 + 315630*x^10 - 978324*x^9 + 3218604*x^8 - 8208228*x^7 + 22683008*x^6 - 45561114*x^5 + 102647091*x^4 - 147782137*x^3 + 257711352*x^2 - 208331724*x + 258264152, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 60 x^{16} - 268 x^{15} + 1272 x^{14} - 5004 x^{13} + 22294 x^{12} - 81192 x^{11} + 315630 x^{10} - 978324 x^{9} + 3218604 x^{8} - 8208228 x^{7} + 22683008 x^{6} - 45561114 x^{5} + 102647091 x^{4} - 147782137 x^{3} + 257711352 x^{2} - 208331724 x + 258264152 \)

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:  \(-7041044961999737113734118289433303=-\,3^{24}\cdot 7^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $75.93$
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 Galois and abelian over $\Q$.
Conductor:  \(1449=3^{2}\cdot 7\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1449}(1,·)$, $\chi_{1449}(898,·)$, $\chi_{1449}(967,·)$, $\chi_{1449}(781,·)$, $\chi_{1449}(277,·)$, $\chi_{1449}(22,·)$, $\chi_{1449}(919,·)$, $\chi_{1449}(1243,·)$, $\chi_{1449}(988,·)$, $\chi_{1449}(415,·)$, $\chi_{1449}(484,·)$, $\chi_{1449}(1381,·)$, $\chi_{1449}(298,·)$, $\chi_{1449}(1264,·)$, $\chi_{1449}(436,·)$, $\chi_{1449}(760,·)$, $\chi_{1449}(505,·)$, $\chi_{1449}(1402,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{508} a^{16} - \frac{31}{127} a^{15} + \frac{43}{254} a^{14} - \frac{7}{127} a^{13} + \frac{19}{127} a^{12} - \frac{1}{254} a^{11} - \frac{20}{127} a^{10} - \frac{45}{254} a^{9} - \frac{59}{254} a^{8} - \frac{89}{254} a^{7} - \frac{6}{127} a^{6} + \frac{1}{127} a^{5} - \frac{87}{254} a^{4} + \frac{10}{127} a^{3} + \frac{37}{508} a^{2} - \frac{97}{254} a$, $\frac{1}{655339075270377545057067299748321148925610824446442372} a^{17} + \frac{72794240482453252881892076882424603260161343699677}{163834768817594386264266824937080287231402706111610593} a^{16} + \frac{77656360983461315844646289864437987017548517738757623}{327669537635188772528533649874160574462805412223221186} a^{15} + \frac{5456155039402531012509910006210546445601416412496969}{163834768817594386264266824937080287231402706111610593} a^{14} - \frac{10566856397539410520299474096772711367531274316212915}{327669537635188772528533649874160574462805412223221186} a^{13} - \frac{264166910673261171582937700986113148507488690786675}{327669537635188772528533649874160574462805412223221186} a^{12} + \frac{77152093841148929908035469686469254639794895294210899}{327669537635188772528533649874160574462805412223221186} a^{11} + \frac{15187457485761688265069513003343788300575663372550934}{163834768817594386264266824937080287231402706111610593} a^{10} + \frac{5818424889251718903437484607819122639776795643952227}{163834768817594386264266824937080287231402706111610593} a^{9} - \frac{14221507295301025366475906549274161408131820364011038}{163834768817594386264266824937080287231402706111610593} a^{8} - \frac{48368112484063274260869033195842501687158258905294812}{163834768817594386264266824937080287231402706111610593} a^{7} - \frac{161956713095422372162579670125521040544759755256393773}{327669537635188772528533649874160574462805412223221186} a^{6} + \frac{89654539960765361723726329269787067943672299001545191}{327669537635188772528533649874160574462805412223221186} a^{5} - \frac{113951746908162470487956501860959884112756771299142931}{327669537635188772528533649874160574462805412223221186} a^{4} - \frac{169695659319046430339380520163771167495802222725443581}{655339075270377545057067299748321148925610824446442372} a^{3} + \frac{19590830193091615510796981687565269019218749738688972}{163834768817594386264266824937080287231402706111610593} a^{2} + \frac{116225555424272345381763811285183106310355429659093127}{327669537635188772528533649874160574462805412223221186} a + \frac{56789695129976933695975865024667100845999013045501}{1290037549744837687120211219977010135680336268595359}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{3942}$, which has order $70956$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.79827687.1, 6.0.191666276487.3, 6.0.29212967.1, 6.0.191666276487.2, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ 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 ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$