↑

Properties

Label 18.0.24194757339...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{6}\cdot 13^{15}\cdot 73^{6}$
Root discriminant $629.64$
Ramified primes $2, 3, 5, 13, 73$
Class number $1326830232$ (GRH)
Class group $[2, 663415116]$ (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![27667283968, 35231984640, 22818506112, -3995186848, 1777895376, -49951008, -257271072, 78891408, -510672, -6314064, 755604, 222126, -47197, -13386, 4083, 322, -75, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 75*x^16 + 322*x^15 + 4083*x^14 - 13386*x^13 - 47197*x^12 + 222126*x^11 + 755604*x^10 - 6314064*x^9 - 510672*x^8 + 78891408*x^7 - 257271072*x^6 - 49951008*x^5 + 1777895376*x^4 - 3995186848*x^3 + 22818506112*x^2 + 35231984640*x + 27667283968)
 
gp: K = bnfinit(x^18 - 6*x^17 - 75*x^16 + 322*x^15 + 4083*x^14 - 13386*x^13 - 47197*x^12 + 222126*x^11 + 755604*x^10 - 6314064*x^9 - 510672*x^8 + 78891408*x^7 - 257271072*x^6 - 49951008*x^5 + 1777895376*x^4 - 3995186848*x^3 + 22818506112*x^2 + 35231984640*x + 27667283968, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 75 x^{16} + 322 x^{15} + 4083 x^{14} - 13386 x^{13} - 47197 x^{12} + 222126 x^{11} + 755604 x^{10} - 6314064 x^{9} - 510672 x^{8} + 78891408 x^{7} - 257271072 x^{6} - 49951008 x^{5} + 1777895376 x^{4} - 3995186848 x^{3} + 22818506112 x^{2} + 35231984640 x + 27667283968 \)

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:  \(-241947573394594188600060412651445161430839296000000=-\,2^{18}\cdot 3^{27}\cdot 5^{6}\cdot 13^{15}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $629.64$
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:  $2, 3, 5, 13, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{352} a^{12} - \frac{5}{176} a^{11} + \frac{5}{352} a^{10} - \frac{9}{176} a^{9} - \frac{21}{352} a^{8} - \frac{19}{176} a^{7} + \frac{39}{352} a^{6} - \frac{39}{176} a^{5} - \frac{7}{88} a^{4} + \frac{3}{44} a^{3} + \frac{3}{8} a^{2} - \frac{9}{44} a + \frac{5}{11}$, $\frac{1}{1408} a^{13} - \frac{1}{704} a^{12} - \frac{31}{1408} a^{11} - \frac{3}{64} a^{10} + \frac{1}{128} a^{9} - \frac{15}{704} a^{8} + \frac{131}{1408} a^{7} + \frac{29}{704} a^{6} + \frac{79}{352} a^{5} - \frac{9}{44} a^{4} - \frac{139}{352} a^{3} - \frac{75}{176} a^{2} + \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{1408} a^{14} + \frac{1}{1408} a^{12} + \frac{5}{176} a^{11} + \frac{59}{1408} a^{10} + \frac{3}{88} a^{9} + \frac{19}{1408} a^{8} - \frac{21}{176} a^{7} + \frac{19}{352} a^{6} + \frac{81}{352} a^{4} - \frac{31}{88} a^{3} - \frac{1}{44} a^{2} + \frac{21}{44} a - \frac{1}{11}$, $\frac{1}{5632} a^{15} - \frac{1}{2816} a^{14} - \frac{1}{5632} a^{13} - \frac{3}{2816} a^{12} + \frac{169}{5632} a^{11} - \frac{89}{2816} a^{10} - \frac{291}{5632} a^{9} - \frac{97}{2816} a^{8} + \frac{283}{2816} a^{7} - \frac{95}{1408} a^{6} - \frac{21}{1408} a^{5} - \frac{79}{704} a^{4} + \frac{115}{704} a^{3} + \frac{15}{32} a^{2} - \frac{1}{8} a - \frac{3}{11}$, $\frac{1}{45056} a^{16} + \frac{1}{22528} a^{15} - \frac{1}{45056} a^{14} + \frac{7}{22528} a^{13} - \frac{23}{45056} a^{12} - \frac{27}{22528} a^{11} + \frac{765}{45056} a^{10} + \frac{93}{22528} a^{9} + \frac{955}{22528} a^{8} - \frac{1383}{11264} a^{7} + \frac{1139}{11264} a^{6} + \frac{1371}{5632} a^{5} - \frac{1173}{5632} a^{4} + \frac{881}{2816} a^{3} - \frac{163}{704} a^{2} + \frac{39}{88} a - \frac{2}{11}$, $\frac{1}{5851018152504888383208235899998918583408091245480150779808305016832} a^{17} - \frac{7426735242774968091297671063944475064033737111220065285329227}{731377269063111047901029487499864822926011405685018847476038127104} a^{16} + \frac{169532796375790388752913010457852943449916253372908690808335107}{5851018152504888383208235899998918583408091245480150779808305016832} a^{15} + \frac{19630950398981049134056746473692952865569421393192046500498217}{66488842642101004354639044318169529356910127789547167952367102464} a^{14} + \frac{1808533011251788752718686973170582882783166847991082941046312133}{5851018152504888383208235899998918583408091245480150779808305016832} a^{13} - \frac{512424843088902808093099763046225257465275142525906898553446573}{365688634531555523950514743749932411463005702842509423738019063552} a^{12} - \frac{11668927884676410979760662707376911895556985152913949990323282447}{5851018152504888383208235899998918583408091245480150779808305016832} a^{11} - \frac{1674205525921566815242780056584964398764787514049985650752536923}{66488842642101004354639044318169529356910127789547167952367102464} a^{10} - \frac{24273360803947345034435067902752083395451026572507871031440187979}{2925509076252444191604117949999459291704045622740075389904152508416} a^{9} - \frac{2501708698241677640211935505171523259264420924390696759774876213}{731377269063111047901029487499864822926011405685018847476038127104} a^{8} + \frac{6943455397972628426768324133517321383563272878955086113749273213}{1462754538126222095802058974999729645852022811370037694952076254208} a^{7} + \frac{745239803968988201128419371931244341045559321811546911241687647}{22855539658222220246907171484370775716437856427656838983626191472} a^{6} + \frac{80663575197698347877996185466803569637786918599466636938121596929}{731377269063111047901029487499864822926011405685018847476038127104} a^{5} + \frac{22352530633253718900420903124043917708255876988498044705597924483}{182844317265777761975257371874966205731502851421254711869009531776} a^{4} + \frac{84706025877090857051816970230444623901985340237193758391987296071}{182844317265777761975257371874966205731502851421254711869009531776} a^{3} - \frac{983221380975526637942758019853013599622707973246302565273853937}{2077776332565656386082470134942797792403441493423348998511471952} a^{2} - \frac{4206128562649656270135680583400582717589414464298840125717746203}{11427769829111110123453585742185387858218928213828419491813095736} a - \frac{477316852875427376736661963669328055665646120758479417678787219}{1428471228638888765431698217773173482277366026728552436476636967}$

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

Class group and class number

$C_{2}\times C_{663415116}$, which has order $1326830232$ (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:  \( 915173418366.1372 \) (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

\(\Q(\sqrt{-39}) \), 3.3.13689.1, 3.3.2920.1, 6.0.505777521600.3, 6.0.7308160119.1, 9.9.63865118865892548672000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$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]$
2.2.3.2$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$$[3]$
3Data not computed
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.12.10.2$x^{12} + 39 x^{6} + 676$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$73$73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$