↑

Properties

Label 18.0.15987630777...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 5^{6}\cdot 13^{15}$
Root discriminant $150.66$
Ramified primes $2, 3, 5, 13$
Class number $3128736$ (GRH)
Class group $[2, 2, 2, 391092]$ (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![4077568, -7825920, -3165888, 8830864, 2376000, -2969220, 1476580, -753984, 2463540, -2510485, 1597149, -616548, 162220, -25134, 2478, -156, 48, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 48*x^16 - 156*x^15 + 2478*x^14 - 25134*x^13 + 162220*x^12 - 616548*x^11 + 1597149*x^10 - 2510485*x^9 + 2463540*x^8 - 753984*x^7 + 1476580*x^6 - 2969220*x^5 + 2376000*x^4 + 8830864*x^3 - 3165888*x^2 - 7825920*x + 4077568)
 
gp: K = bnfinit(x^18 - 9*x^17 + 48*x^16 - 156*x^15 + 2478*x^14 - 25134*x^13 + 162220*x^12 - 616548*x^11 + 1597149*x^10 - 2510485*x^9 + 2463540*x^8 - 753984*x^7 + 1476580*x^6 - 2969220*x^5 + 2376000*x^4 + 8830864*x^3 - 3165888*x^2 - 7825920*x + 4077568, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 48 x^{16} - 156 x^{15} + 2478 x^{14} - 25134 x^{13} + 162220 x^{12} - 616548 x^{11} + 1597149 x^{10} - 2510485 x^{9} + 2463540 x^{8} - 753984 x^{7} + 1476580 x^{6} - 2969220 x^{5} + 2376000 x^{4} + 8830864 x^{3} - 3165888 x^{2} - 7825920 x + 4077568 \)

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:  \(-1598763077775622674562101105291264000000=-\,2^{18}\cdot 3^{27}\cdot 5^{6}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.66$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} - \frac{3}{32} a^{5} - \frac{7}{16} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} - \frac{1}{16} a^{8} - \frac{3}{32} a^{7} + \frac{5}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{544} a^{12} - \frac{3}{272} a^{11} + \frac{3}{272} a^{10} + \frac{7}{544} a^{9} - \frac{1}{544} a^{8} - \frac{1}{68} a^{7} + \frac{23}{272} a^{6} + \frac{89}{544} a^{5} - \frac{33}{136} a^{4} + \frac{107}{272} a^{3} + \frac{7}{17} a^{2} - \frac{27}{68} a + \frac{4}{17}$, $\frac{1}{1088} a^{13} - \frac{13}{1088} a^{11} - \frac{1}{136} a^{10} + \frac{7}{1088} a^{9} + \frac{5}{272} a^{8} - \frac{19}{1088} a^{7} + \frac{19}{272} a^{6} - \frac{27}{272} a^{5} + \frac{1}{8} a^{4} - \frac{99}{272} a^{3} + \frac{7}{17} a^{2} - \frac{5}{68} a - \frac{5}{17}$, $\frac{1}{1088} a^{14} - \frac{1}{1088} a^{12} - \frac{3}{272} a^{11} + \frac{11}{1088} a^{10} - \frac{1}{34} a^{9} - \frac{31}{1088} a^{8} - \frac{11}{136} a^{7} - \frac{1}{34} a^{6} + \frac{63}{272} a^{5} - \frac{19}{272} a^{4} - \frac{31}{136} a^{3} + \frac{5}{34} a^{2} + \frac{11}{34} a + \frac{7}{17}$, $\frac{1}{11968} a^{15} - \frac{1}{5984} a^{14} + \frac{3}{11968} a^{13} + \frac{127}{11968} a^{11} + \frac{21}{5984} a^{10} + \frac{199}{11968} a^{9} + \frac{65}{1496} a^{8} - \frac{129}{1496} a^{7} - \frac{257}{2992} a^{6} + \frac{1129}{5984} a^{5} + \frac{207}{2992} a^{4} + \frac{643}{2992} a^{3} - \frac{257}{1496} a^{2} + \frac{21}{187} a + \frac{8}{17}$, $\frac{1}{526592} a^{16} + \frac{7}{263296} a^{15} + \frac{13}{263296} a^{14} - \frac{9}{263296} a^{13} + \frac{95}{131648} a^{12} - \frac{1471}{263296} a^{11} + \frac{2487}{263296} a^{10} + \frac{349}{15488} a^{9} - \frac{14745}{526592} a^{8} + \frac{29}{4114} a^{7} - \frac{345}{5984} a^{6} + \frac{5153}{65824} a^{5} - \frac{1457}{131648} a^{4} + \frac{271}{968} a^{3} + \frac{9831}{32912} a^{2} - \frac{2755}{8228} a - \frac{27}{187}$, $\frac{1}{3543856194403283924621598778198270173542912} a^{17} - \frac{1495442561160229114191630876168891261}{1771928097201641962310799389099135086771456} a^{16} + \frac{42558903755453454271741617775872562537}{1771928097201641962310799389099135086771456} a^{15} - \frac{398297744509610120914236755651173967307}{1771928097201641962310799389099135086771456} a^{14} + \frac{172018007646655284785816500457085029905}{885964048600820981155399694549567543385728} a^{13} - \frac{117439816633884807848659447576223199345}{1771928097201641962310799389099135086771456} a^{12} - \frac{23606969092062876099191504817809145743521}{1771928097201641962310799389099135086771456} a^{11} - \frac{23861175910156038182223367916349587861329}{1771928097201641962310799389099135086771456} a^{10} - \frac{691206365069140197920171278154760989475}{25495368305059596580011502001426404126208} a^{9} + \frac{30395689094405001739328538228036921562235}{885964048600820981155399694549567543385728} a^{8} + \frac{623177687899078420268826845574551176877}{13028883067659132075814701390434816814496} a^{7} + \frac{135391001865502477335746619731415655119}{26057766135318264151629402780869633628992} a^{6} + \frac{209176021389351387064185481499608110240075}{885964048600820981155399694549567543385728} a^{5} - \frac{1797625443852307093532204734864599561417}{221491012150205245288849923637391885846432} a^{4} - \frac{61180419787911608998823814036678534817125}{221491012150205245288849923637391885846432} a^{3} + \frac{26311085791973467760597113740242148410099}{55372753037551311322212480909347971461608} a^{2} - \frac{8861053442693424001137294729260826364285}{27686376518775655661106240454673985730804} a - \frac{33216394464501714493257843817236260103}{629235829972173992297869101242590584791}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{391092}$, which has order $3128736$ (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:  \( 1768168461.4286366 \) (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, 3.3.547560.1, 6.0.7308160119.1, 6.0.11693056190400.3, 9.9.164170508913216000.2

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$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$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed