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Properties

Label 18.2.535...000.1
Degree $18$
Signature $[2, 8]$
Discriminant $5.356\times 10^{27}$
Root discriminant \(34.71\)
Ramified primes $2,3,5$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216)
 
Copy content gp:K = bnfinit(y^18 - 18*y^15 - 50*y^12 - 360*y^9 - 860*y^6 - 648*y^3 - 216, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216)
 

\( x^{18} - 18x^{15} - 50x^{12} - 360x^{9} - 860x^{6} - 648x^{3} - 216 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(5355700839936000000000000000\) \(\medspace = 2^{24}\cdot 3^{21}\cdot 5^{15}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(34.71\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{4/3}3^{7/6}5^{5/6}\approx 34.71281190206154$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{15}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{6}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}+\frac{1}{3}a^{3}$, $\frac{1}{36}a^{10}+\frac{1}{18}a^{4}$, $\frac{1}{36}a^{11}+\frac{1}{18}a^{5}$, $\frac{1}{36}a^{12}+\frac{1}{18}a^{6}$, $\frac{1}{36}a^{13}+\frac{1}{18}a^{7}$, $\frac{1}{72}a^{14}-\frac{1}{18}a^{8}-\frac{1}{6}a^{2}$, $\frac{1}{293112}a^{15}-\frac{1}{108}a^{13}+\frac{211}{24426}a^{12}-\frac{1}{108}a^{11}+\frac{1343}{73278}a^{9}+\frac{4}{27}a^{7}+\frac{5711}{24426}a^{6}-\frac{1}{54}a^{5}+\frac{1}{3}a^{4}-\frac{7583}{24426}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{422}{1357}$, $\frac{1}{293112}a^{16}+\frac{1}{216}a^{14}+\frac{211}{24426}a^{13}-\frac{1}{108}a^{12}-\frac{1385}{146556}a^{10}-\frac{2}{27}a^{8}+\frac{5711}{24426}a^{7}-\frac{1}{54}a^{6}+\frac{1}{3}a^{5}-\frac{1490}{4071}a^{4}+\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{422}{1357}a$, $\frac{1}{293112}a^{17}-\frac{57}{10856}a^{14}-\frac{7}{36639}a^{11}+\frac{1}{18}a^{9}-\frac{179}{4071}a^{8}-\frac{1}{6}a^{7}-\frac{25463}{73278}a^{5}+\frac{1}{9}a^{3}-\frac{1175}{8142}a^{2}-\frac{1}{3}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{6}$, which has order $6$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{6}$, which has order $12$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{1135}{97704}a^{17}-\frac{6887}{32568}a^{14}-\frac{732}{1357}a^{11}-\frac{16486}{4071}a^{8}-\frac{218953}{24426}a^{5}-\frac{38435}{8142}a^{2}$, $\frac{4}{4071}a^{17}+\frac{151}{73278}a^{16}-\frac{247}{146556}a^{15}-\frac{3139}{146556}a^{14}-\frac{1528}{36639}a^{13}+\frac{1061}{36639}a^{12}+\frac{698}{36639}a^{11}-\frac{1985}{146556}a^{10}+\frac{8273}{73278}a^{9}-\frac{6659}{36639}a^{8}-\frac{23144}{36639}a^{7}+\frac{43331}{73278}a^{6}+\frac{14869}{36639}a^{5}-\frac{8399}{24426}a^{4}+\frac{24767}{12213}a^{3}+\frac{7210}{4071}a^{2}+\frac{2041}{4071}a+\frac{847}{1357}$, $\frac{4}{4071}a^{17}+\frac{1343}{146556}a^{16}+\frac{247}{73278}a^{15}-\frac{3139}{146556}a^{14}-\frac{6148}{36639}a^{13}-\frac{2122}{36639}a^{12}+\frac{698}{36639}a^{11}-\frac{60281}{146556}a^{10}-\frac{8273}{36639}a^{9}-\frac{6659}{36639}a^{8}-\frac{112304}{36639}a^{7}-\frac{43331}{36639}a^{6}+\frac{14869}{36639}a^{5}-\frac{166279}{24426}a^{4}-\frac{49534}{12213}a^{3}+\frac{7210}{4071}a^{2}-\frac{12047}{4071}a-\frac{4408}{1357}$, $\frac{209}{24426}a^{17}-\frac{1343}{146556}a^{16}+\frac{247}{146556}a^{15}-\frac{23321}{146556}a^{14}+\frac{6148}{36639}a^{13}-\frac{1061}{36639}a^{12}-\frac{12128}{36639}a^{11}+\frac{60281}{146556}a^{10}-\frac{8273}{73278}a^{9}-\frac{214247}{73278}a^{8}+\frac{112304}{36639}a^{7}-\frac{43331}{73278}a^{6}-\frac{202585}{36639}a^{5}+\frac{166279}{24426}a^{4}-\frac{24767}{12213}a^{3}-\frac{10604}{4071}a^{2}+\frac{12047}{4071}a-\frac{847}{1357}$, $\frac{85}{97704}a^{16}-\frac{475}{24426}a^{13}+\frac{1691}{48852}a^{10}-\frac{3949}{12213}a^{7}+\frac{6815}{12213}a^{4}-\frac{1764}{1357}a$, $\frac{1441}{293112}a^{17}-\frac{187}{48852}a^{16}-\frac{1369}{146556}a^{15}-\frac{9167}{97704}a^{14}+\frac{11183}{146556}a^{13}+\frac{4435}{24426}a^{12}-\frac{10675}{73278}a^{11}+\frac{1165}{24426}a^{10}+\frac{15263}{73278}a^{9}-\frac{19325}{12213}a^{8}+\frac{88241}{73278}a^{7}+\frac{73271}{24426}a^{6}-\frac{60247}{24426}a^{5}+\frac{10724}{12213}a^{4}+\frac{5135}{1357}a^{3}+\frac{361}{8142}a^{2}-\frac{109}{1357}a+\frac{629}{1357}$, $\frac{5513}{293112}a^{17}+\frac{1009}{146556}a^{16}+\frac{53}{16284}a^{15}-\frac{103675}{293112}a^{14}-\frac{1507}{12213}a^{13}-\frac{2027}{36639}a^{12}-\frac{48685}{73278}a^{11}-\frac{12769}{36639}a^{10}-\frac{5809}{24426}a^{9}-\frac{443495}{73278}a^{8}-\frac{31991}{12213}a^{7}-\frac{76175}{73278}a^{6}-\frac{823639}{73278}a^{5}-\frac{75116}{12213}a^{4}-\frac{35968}{12213}a^{3}-\frac{18455}{8142}a^{2}-\frac{13127}{4071}a-\frac{4987}{1357}$, $\frac{47}{2484}a^{17}-\frac{85}{4071}a^{16}-\frac{53}{293112}a^{15}-\frac{283}{828}a^{14}+\frac{27415}{73278}a^{13}+\frac{703}{48852}a^{12}-\frac{385}{414}a^{11}+\frac{17683}{16284}a^{10}-\frac{14185}{73278}a^{9}-\frac{1373}{207}a^{8}+\frac{542873}{73278}a^{7}-\frac{5464}{12213}a^{6}-\frac{9869}{621}a^{5}+\frac{144719}{8142}a^{4}-\frac{97477}{24426}a^{3}-\frac{223}{23}a^{2}+\frac{48302}{4071}a-\frac{8845}{1357}$, $\frac{2839}{97704}a^{17}-\frac{51247}{97704}a^{14}-\frac{7777}{5428}a^{11}-\frac{125383}{12213}a^{8}-\frac{294824}{12213}a^{5}-\frac{125801}{8142}a^{2}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 6009745.31026225 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 6009745.31026225 \cdot 6}{2\cdot\sqrt{5355700839936000000000000000}}\cr\approx \mathstrut & 2.39369110030110 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 18*x^15 - 50*x^12 - 360*x^9 - 860*x^6 - 648*x^3 - 216); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_6:S_3$ (as 18T12):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{15}) \), 3.1.675.1, 3.1.108.1, 3.1.300.1, 3.1.2700.1, 6.2.437400000.2, 6.2.21600000.1, 6.2.69984000.1, 6.2.1749600000.1, 9.1.59049000000.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 18.0.1785233613312000000000000000.3
Minimal sibling: 18.0.1785233613312000000000000000.3

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 ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.6.8a1.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$$[2]_{3}^{2}$$
2.2.6.16a1.5$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 143 x^{6} + 132 x^{5} + 102 x^{4} + 64 x^{3} + 33 x^{2} + 12 x + 5$$6$$2$$16$$D_6$$$[2]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
\(5\) Copy content Toggle raw display 5.1.6.5a1.2$x^{6} + 10$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$
5.2.6.10a1.2$x^{12} + 24 x^{11} + 252 x^{10} + 1520 x^{9} + 5820 x^{8} + 14784 x^{7} + 25376 x^{6} + 29568 x^{5} + 23280 x^{4} + 12160 x^{3} + 4032 x^{2} + 768 x + 69$$6$$2$$10$$D_6$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)