↑

Properties

Label 18.0.61497400484...1947.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 73^{8}$
Root discriminant $34.98$
Ramified primes $3, 73$
Class number $21$ (GRH)
Class group $[21]$ (GRH)
Galois group $C_2\times He_3$ (as 18T15)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16129, 43053, 107682, 114065, 147603, 89958, 115629, 49845, 51657, 10200, 13374, 1020, 2645, -123, 321, -26, 27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 27*x^16 - 26*x^15 + 321*x^14 - 123*x^13 + 2645*x^12 + 1020*x^11 + 13374*x^10 + 10200*x^9 + 51657*x^8 + 49845*x^7 + 115629*x^6 + 89958*x^5 + 147603*x^4 + 114065*x^3 + 107682*x^2 + 43053*x + 16129)
 
gp: K = bnfinit(x^18 - 3*x^17 + 27*x^16 - 26*x^15 + 321*x^14 - 123*x^13 + 2645*x^12 + 1020*x^11 + 13374*x^10 + 10200*x^9 + 51657*x^8 + 49845*x^7 + 115629*x^6 + 89958*x^5 + 147603*x^4 + 114065*x^3 + 107682*x^2 + 43053*x + 16129, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 27 x^{16} - 26 x^{15} + 321 x^{14} - 123 x^{13} + 2645 x^{12} + 1020 x^{11} + 13374 x^{10} + 10200 x^{9} + 51657 x^{8} + 49845 x^{7} + 115629 x^{6} + 89958 x^{5} + 147603 x^{4} + 114065 x^{3} + 107682 x^{2} + 43053 x + 16129 \)

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:  \(-6149740048489888978791661947=-\,3^{27}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.98$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{5} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{14} - \frac{1}{18} a^{13} - \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{8} - \frac{2}{9} a^{7} + \frac{7}{18} a^{6} - \frac{7}{18} a^{5} + \frac{5}{18} a^{4} - \frac{1}{18} a^{3} - \frac{1}{6} a^{2} + \frac{7}{18} a + \frac{5}{18}$, $\frac{1}{9756} a^{16} - \frac{121}{4878} a^{15} - \frac{85}{2439} a^{14} + \frac{295}{4878} a^{13} - \frac{115}{9756} a^{12} + \frac{12}{271} a^{11} + \frac{7}{813} a^{10} - \frac{181}{2439} a^{9} - \frac{511}{2439} a^{8} - \frac{272}{2439} a^{7} - \frac{109}{9756} a^{6} - \frac{383}{4878} a^{5} - \frac{665}{4878} a^{4} - \frac{377}{813} a^{3} + \frac{3793}{9756} a^{2} + \frac{835}{4878} a + \frac{341}{1084}$, $\frac{1}{250276017119298347544860142068652} a^{17} + \frac{1340230125195374753498582770}{62569004279824586886215035517163} a^{16} + \frac{117846303510966442414242729553}{125138008559649173772430071034326} a^{15} + \frac{4410268456454196529016298184915}{125138008559649173772430071034326} a^{14} - \frac{16988242377416074260747967852471}{250276017119298347544860142068652} a^{13} - \frac{1488666364492858348360556047384}{20856334759941528962071678505721} a^{12} + \frac{309593626210754699318975908205}{41712669519883057924143357011442} a^{11} - \frac{671735191306262442204248750290}{62569004279824586886215035517163} a^{10} + \frac{861632603689816227391544375555}{62569004279824586886215035517163} a^{9} + \frac{4060361019468340310229229472561}{125138008559649173772430071034326} a^{8} - \frac{79843131414314007761889214065505}{250276017119298347544860142068652} a^{7} - \frac{31093098695142568399744055147794}{62569004279824586886215035517163} a^{6} - \frac{57842859851013502844513612999597}{125138008559649173772430071034326} a^{5} + \frac{18000760470396080946651395062847}{41712669519883057924143357011442} a^{4} - \frac{34535763032593469083640682835997}{250276017119298347544860142068652} a^{3} - \frac{10476974939716064728606850097079}{62569004279824586886215035517163} a^{2} + \frac{10856767315980764349924504518237}{83425339039766115848286714022884} a + \frac{4213528873990388765392440099}{54741036115332097013311492141}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  \( \frac{758768054739776381090673359}{6952111586647176320690559501907} a^{17} - \frac{125611447366805591397563008159}{250276017119298347544860142068652} a^{16} + \frac{462857105424897622237423891981}{125138008559649173772430071034326} a^{15} - \frac{1045530721312453668982515761077}{125138008559649173772430071034326} a^{14} + \frac{5718175934493584984915838914183}{125138008559649173772430071034326} a^{13} - \frac{19082291857887032483074071576377}{250276017119298347544860142068652} a^{12} + \frac{7741059207925116312918612550391}{20856334759941528962071678505721} a^{11} - \frac{7801678581249395502261862032727}{20856334759941528962071678505721} a^{10} + \frac{107962989190003089972760735535038}{62569004279824586886215035517163} a^{9} - \frac{116307763896491239562218270019335}{125138008559649173772430071034326} a^{8} + \frac{342398746232609220336321979531979}{62569004279824586886215035517163} a^{7} - \frac{219439461694358593390306640054561}{250276017119298347544860142068652} a^{6} + \frac{1201697634955042299250565480787215}{125138008559649173772430071034326} a^{5} - \frac{41126556922402326804045670812458}{62569004279824586886215035517163} a^{4} + \frac{385993330413364849460970183117751}{41712669519883057924143357011442} a^{3} + \frac{614224628392685649641973173847539}{250276017119298347544860142068652} a^{2} + \frac{464646232220153089381446134273675}{125138008559649173772430071034326} a - \frac{484897457648597737466328486097}{656892433383985164159737905692} \) (order $18$)
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:  \( 1105667.84512 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3$ (as 18T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times He_3$
Character table for $C_2\times He_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), 9.9.15091989595281.1

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$73$73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$