↑

Properties

Label 18.0.65814407127...3904.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 7^{14}\cdot 67^{14}$
Root discriminant $585.71$
Ramified primes $2, 3, 7, 67$
Class number $1959987456$ (GRH)
Class group $[2, 2, 2, 6, 6, 6, 42, 27006]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3405745481641, -1171630181320, 333518260810, 4438727658, 48699532177, -19415573440, 3550027525, 706102334, 48529363, -100422380, 14933513, 4506790, -825357, -118856, 24231, 1138, -256, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 256*x^16 + 1138*x^15 + 24231*x^14 - 118856*x^13 - 825357*x^12 + 4506790*x^11 + 14933513*x^10 - 100422380*x^9 + 48529363*x^8 + 706102334*x^7 + 3550027525*x^6 - 19415573440*x^5 + 48699532177*x^4 + 4438727658*x^3 + 333518260810*x^2 - 1171630181320*x + 3405745481641)
 
gp: K = bnfinit(x^18 - 4*x^17 - 256*x^16 + 1138*x^15 + 24231*x^14 - 118856*x^13 - 825357*x^12 + 4506790*x^11 + 14933513*x^10 - 100422380*x^9 + 48529363*x^8 + 706102334*x^7 + 3550027525*x^6 - 19415573440*x^5 + 48699532177*x^4 + 4438727658*x^3 + 333518260810*x^2 - 1171630181320*x + 3405745481641, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 256 x^{16} + 1138 x^{15} + 24231 x^{14} - 118856 x^{13} - 825357 x^{12} + 4506790 x^{11} + 14933513 x^{10} - 100422380 x^{9} + 48529363 x^{8} + 706102334 x^{7} + 3550027525 x^{6} - 19415573440 x^{5} + 48699532177 x^{4} + 4438727658 x^{3} + 333518260810 x^{2} - 1171630181320 x + 3405745481641 \)

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:  \(-65814407127188227791161915140602399158516689403904=-\,2^{27}\cdot 3^{9}\cdot 7^{14}\cdot 67^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $585.71$
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:  $2, 3, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{3}{16} a^{2} - \frac{3}{8} a - \frac{1}{16}$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{3}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} + \frac{7}{16} a - \frac{3}{8}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{7} + \frac{5}{64} a^{6} - \frac{1}{8} a^{5} - \frac{29}{64} a^{4} + \frac{3}{32} a^{3} - \frac{11}{32} a^{2} - \frac{1}{16} a + \frac{29}{64}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{8} + \frac{5}{64} a^{7} - \frac{1}{8} a^{6} - \frac{29}{64} a^{5} + \frac{3}{32} a^{4} - \frac{11}{32} a^{3} - \frac{1}{16} a^{2} + \frac{29}{64} a$, $\frac{1}{512} a^{12} + \frac{1}{256} a^{11} + \frac{3}{512} a^{10} + \frac{7}{256} a^{9} + \frac{1}{512} a^{8} + \frac{3}{128} a^{7} + \frac{17}{256} a^{6} + \frac{37}{128} a^{5} + \frac{191}{512} a^{4} - \frac{71}{256} a^{3} - \frac{237}{512} a^{2} - \frac{81}{256} a + \frac{183}{512}$, $\frac{1}{1024} a^{13} - \frac{1}{1024} a^{12} + \frac{5}{1024} a^{11} - \frac{3}{1024} a^{10} + \frac{23}{1024} a^{9} - \frac{7}{1024} a^{8} + \frac{59}{512} a^{7} - \frac{29}{512} a^{6} + \frac{219}{1024} a^{5} - \frac{307}{1024} a^{4} + \frac{29}{1024} a^{3} - \frac{75}{1024} a^{2} - \frac{155}{1024} a + \frac{371}{1024}$, $\frac{1}{499712} a^{14} + \frac{3}{7808} a^{13} - \frac{233}{249856} a^{12} + \frac{1283}{249856} a^{11} - \frac{3}{4096} a^{10} + \frac{87}{124928} a^{9} - \frac{3975}{499712} a^{8} - \frac{12951}{124928} a^{7} + \frac{57077}{499712} a^{6} - \frac{737}{3904} a^{5} + \frac{1151}{15616} a^{4} - \frac{15501}{249856} a^{3} - \frac{23881}{62464} a^{2} + \frac{27613}{124928} a - \frac{60103}{499712}$, $\frac{1}{17489920} a^{15} + \frac{3}{4372480} a^{14} - \frac{271}{1249280} a^{13} + \frac{4183}{8744960} a^{12} - \frac{787}{8744960} a^{11} - \frac{981}{624640} a^{10} + \frac{120777}{17489920} a^{9} + \frac{489}{1093120} a^{8} + \frac{1284901}{17489920} a^{7} + \frac{44547}{874496} a^{6} - \frac{3051}{78080} a^{5} - \frac{3728589}{8744960} a^{4} - \frac{5483}{68320} a^{3} - \frac{147117}{624640} a^{2} + \frac{1494485}{3497984} a + \frac{2126843}{4372480}$, $\frac{1}{2571158998876160} a^{16} + \frac{28292823}{1285579499438080} a^{15} + \frac{277001587}{367308428410880} a^{14} - \frac{22228335103}{257115899887616} a^{13} - \frac{14212599139}{64278974971904} a^{12} + \frac{1266637480519}{183654214205440} a^{11} - \frac{1733749779821}{514231799775232} a^{10} - \frac{25719675030029}{1285579499438080} a^{9} + \frac{871532870433}{58435431792640} a^{8} + \frac{4157140036417}{1285579499438080} a^{7} - \frac{31592208392347}{367308428410880} a^{6} - \frac{256117884808237}{1285579499438080} a^{5} - \frac{4789515913041}{128557949943808} a^{4} + \frac{52968240920273}{183654214205440} a^{3} + \frac{487473388428441}{2571158998876160} a^{2} - \frac{208842185375499}{1285579499438080} a + \frac{1649365283529}{367308428410880}$, $\frac{1}{91222907066041414689550408229772228008470006876582734375157760} a^{17} - \frac{2183303538992265587153657040143687486223074029}{18244581413208282937910081645954445601694001375316546875031552} a^{16} + \frac{1691145213498609146219249404409643280519175024473206387}{91222907066041414689550408229772228008470006876582734375157760} a^{15} - \frac{1116705627047291760068896936057617585302480188396959171}{8292991551458310426322764384524748000770000625143884943196160} a^{14} + \frac{1243311598359199689013943046707741100959313150053206536267}{4146495775729155213161382192262374000385000312571942471598080} a^{13} - \frac{33901128915213953043789914526821532919611236358940208213}{480120563505481129945002148577748568465631615139909128290304} a^{12} + \frac{403557314210744874171310454885098636949755403001747496349473}{91222907066041414689550408229772228008470006876582734375157760} a^{11} - \frac{10793108008026447522612368249486959548977525003845292266989}{1658598310291662085264552876904949600154000125028776988639232} a^{10} - \frac{53684126900740666774629344940240941478098013234678549173789}{2400602817527405649725010742888742842328158075699545641451520} a^{9} - \frac{1048761668159290153963721342178728047043621360287772437846601}{45611453533020707344775204114886114004235003438291367187578880} a^{8} + \frac{6023963071477409763192103036927702393840988124850585779101861}{91222907066041414689550408229772228008470006876582734375157760} a^{7} + \frac{90416500667173414396493466233199695764424131469660678476839}{960241127010962259890004297155497136931263230279818256580608} a^{6} - \frac{282540050222099647832850927902252071204482658482967167824821}{4146495775729155213161382192262374000385000312571942471598080} a^{5} - \frac{875263270529922723085097900784346230129362070671519153311735}{9122290706604141468955040822977222800847000687658273437515776} a^{4} - \frac{27410907172038699074579615352114398901097247694401239239352153}{91222907066041414689550408229772228008470006876582734375157760} a^{3} - \frac{1883794023193041590250433876529211613134824077204145183113739}{13031843866577344955650058318538889715495715268083247767879680} a^{2} + \frac{3656626411200275417464173659160729508718049866338742463715401}{91222907066041414689550408229772228008470006876582734375157760} a + \frac{98909241147805613766697437188727106139465950400149343411199}{299091498577184966195247240097613862322852481562566342213632}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{42}\times C_{27006}$, which has order $1959987456$ (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:  \( 4641319478.753615 \) (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

\(\Q(\sqrt{-6}) \), 3.3.219961.2, 3.3.469.1, 6.0.3040740864.3, 6.0.668844401186304.2, 9.9.4991256617582519389.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$67$67.6.4.2$x^{6} - 67 x^{3} + 53868$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
67.12.10.3$x^{12} - 16147 x^{6} + 93083904$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$