↑

Properties

Label 18.0.83965375312...1799.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 13^{9}\cdot 31^{14}$
Root discriminant $187.74$
Ramified primes $3, 13, 31$
Class number $29828800$ (GRH)
Class group $[20, 1491440]$ (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![54730227712, -25618016256, 25896306432, -9029732096, 4889287872, -1307684628, 497306862, -106851099, 33272703, -6349626, 1767420, -309315, 71441, -10053, 1929, -262, 60, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 60*x^16 - 262*x^15 + 1929*x^14 - 10053*x^13 + 71441*x^12 - 309315*x^11 + 1767420*x^10 - 6349626*x^9 + 33272703*x^8 - 106851099*x^7 + 497306862*x^6 - 1307684628*x^5 + 4889287872*x^4 - 9029732096*x^3 + 25896306432*x^2 - 25618016256*x + 54730227712)
 
gp: K = bnfinit(x^18 - 9*x^17 + 60*x^16 - 262*x^15 + 1929*x^14 - 10053*x^13 + 71441*x^12 - 309315*x^11 + 1767420*x^10 - 6349626*x^9 + 33272703*x^8 - 106851099*x^7 + 497306862*x^6 - 1307684628*x^5 + 4889287872*x^4 - 9029732096*x^3 + 25896306432*x^2 - 25618016256*x + 54730227712, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 60 x^{16} - 262 x^{15} + 1929 x^{14} - 10053 x^{13} + 71441 x^{12} - 309315 x^{11} + 1767420 x^{10} - 6349626 x^{9} + 33272703 x^{8} - 106851099 x^{7} + 497306862 x^{6} - 1307684628 x^{5} + 4889287872 x^{4} - 9029732096 x^{3} + 25896306432 x^{2} - 25618016256 x + 54730227712 \)

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:  \(-83965375312753658928066380894847299471799=-\,3^{21}\cdot 13^{9}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $187.74$
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:  $3, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{6} + \frac{5}{12} a^{3} + \frac{1}{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{24} a^{4} - \frac{11}{24} a^{3} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{48} a^{11} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{1}{48} a^{5} - \frac{1}{48} a^{3} - \frac{7}{24} a^{2} - \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{96} a^{12} - \frac{1}{96} a^{10} - \frac{1}{24} a^{9} + \frac{1}{12} a^{7} - \frac{3}{32} a^{6} - \frac{1}{4} a^{5} - \frac{23}{96} a^{4} + \frac{1}{8} a^{3} + \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} + \frac{1}{192} a^{11} + \frac{1}{192} a^{10} + \frac{1}{96} a^{9} + \frac{1}{48} a^{8} - \frac{25}{192} a^{7} + \frac{1}{192} a^{6} + \frac{47}{192} a^{5} + \frac{31}{192} a^{4} + \frac{43}{96} a^{3} + \frac{7}{48} a^{2} - \frac{1}{4} a$, $\frac{1}{384} a^{14} - \frac{1}{384} a^{13} + \frac{1}{384} a^{12} + \frac{1}{384} a^{11} - \frac{1}{64} a^{10} - \frac{1}{96} a^{9} + \frac{23}{384} a^{8} - \frac{31}{384} a^{7} + \frac{5}{128} a^{6} + \frac{31}{384} a^{5} + \frac{47}{192} a^{4} + \frac{11}{32} a^{3} + \frac{1}{4} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{2304} a^{15} + \frac{1}{1152} a^{14} + \frac{1}{1152} a^{13} - \frac{1}{288} a^{12} - \frac{7}{2304} a^{11} + \frac{1}{384} a^{10} - \frac{23}{768} a^{9} + \frac{83}{1152} a^{8} - \frac{233}{1152} a^{7} - \frac{5}{288} a^{6} - \frac{385}{2304} a^{5} - \frac{223}{1152} a^{4} - \frac{265}{576} a^{3} + \frac{1}{12} a^{2} + \frac{13}{36} a + \frac{4}{9}$, $\frac{1}{17747712} a^{16} + \frac{79}{5915904} a^{15} + \frac{3317}{8873856} a^{14} + \frac{123}{123248} a^{13} - \frac{13979}{5915904} a^{12} - \frac{154705}{17747712} a^{11} - \frac{106681}{5915904} a^{10} - \frac{597377}{17747712} a^{9} - \frac{29769}{985984} a^{8} - \frac{195251}{1109232} a^{7} - \frac{4152359}{17747712} a^{6} - \frac{952469}{5915904} a^{5} - \frac{414313}{2957952} a^{4} - \frac{372583}{4436928} a^{3} + \frac{505153}{1109232} a^{2} + \frac{57601}{138654} a + \frac{22522}{69327}$, $\frac{1}{195862774641797244759491790204204512556944228598635685888} a^{17} + \frac{347270594701931855042497788534210764456335352927}{195862774641797244759491790204204512556944228598635685888} a^{16} - \frac{2365684012216122734246705404773746898635456573774345}{48965693660449311189872947551051128139236057149658921472} a^{15} - \frac{62136395424485639586513666601394367560122296309116435}{97931387320898622379745895102102256278472114299317842944} a^{14} - \frac{152770758107336112898065528892548986905039824093518701}{65287591547265748253163930068068170852314742866211895296} a^{13} - \frac{8883023330236895801882448088395474076510302632288483}{4167293077485047760840250855408606650147749544651823104} a^{12} - \frac{1703164032480228925258955142487072025618574958204751999}{195862774641797244759491790204204512556944228598635685888} a^{11} + \frac{1743324068919654485345284580337005447618292577824845941}{195862774641797244759491790204204512556944228598635685888} a^{10} + \frac{29347529682021522632192461279542427593083759373000647}{48965693660449311189872947551051128139236057149658921472} a^{9} - \frac{7000421728562168529101815776969908855927550088538596429}{97931387320898622379745895102102256278472114299317842944} a^{8} - \frac{10068152363785261630441501308465991220733772926788624827}{65287591547265748253163930068068170852314742866211895296} a^{7} - \frac{44031052777519331023591778883731215131840394644695154163}{195862774641797244759491790204204512556944228598635685888} a^{6} - \frac{1011091249368301498153218985166652059377895790286073753}{10881265257877624708860655011344695142052457144368649216} a^{5} - \frac{7607632191192497634125707661793393910402516988057715421}{48965693660449311189872947551051128139236057149658921472} a^{4} + \frac{56117319590584968744631329209571794710156232812205759}{3060355853778081949367059221940695508702253571853682592} a^{3} + \frac{866249083690658422050930502207698623043620206328689}{191272240861130121835441201371293469293890848240855162} a^{2} + \frac{47301330329607732834120535891296510948260707448753027}{765088963444520487341764805485173877175563392963420648} a - \frac{8206054054562032604330121494908582933132785159266459}{95636120430565060917720600685646734646945424120427581}$

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

Class group and class number

$C_{20}\times C_{1491440}$, which has order $29828800$ (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:  \( 67295442.84873295 \) (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

\(\Q(\sqrt{-39}) \), 3.3.961.1, 3.3.837.1, 6.0.54782342199.1, 6.0.4617450279.2, 9.9.541530783546813.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$31$31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.12.10.1$x^{12} + 69161 x^{6} + 2869530624$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$