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Properties

Label 18.0.360...000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-3.602\times 10^{27}$
Root discriminant \(33.96\)
Ramified primes $2,3,5$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000)
 
Copy content gp:K = bnfinit(y^18 - 15*y^15 + 39*y^12 + 475*y^9 + 834*y^6 - 300*y^3 + 1000, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000)
 

\( x^{18} - 15x^{15} + 39x^{12} + 475x^{9} + 834x^{6} - 300x^{3} + 1000 \) 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:  $[0, 9]$
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:   \(-3602271247127978904000000000\) \(\medspace = -\,2^{12}\cdot 3^{37}\cdot 5^{9}\) 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:  \(33.96\)
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:  $2^{2/3}3^{37/18}5^{1/2}\approx 33.956342994275154$
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.
Maximal CM subfield:  \(\Q(\sqrt{-15}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{132}a^{12}+\frac{17}{132}a^{9}+\frac{13}{44}a^{6}-\frac{1}{132}a^{3}+\frac{23}{66}$, $\frac{1}{264}a^{13}-\frac{9}{88}a^{10}+\frac{13}{88}a^{7}+\frac{131}{264}a^{4}+\frac{15}{44}a$, $\frac{1}{3960}a^{14}-\frac{1}{792}a^{13}+\frac{1}{396}a^{12}-\frac{23}{792}a^{11}+\frac{115}{792}a^{10}+\frac{17}{396}a^{9}+\frac{151}{440}a^{8}+\frac{25}{88}a^{7}+\frac{19}{44}a^{6}-\frac{185}{792}a^{5}+\frac{133}{792}a^{4}-\frac{133}{396}a^{3}-\frac{43}{1980}a^{2}+\frac{43}{396}a+\frac{89}{198}$, $\frac{1}{4007520}a^{15}+\frac{2027}{801504}a^{12}-\frac{475151}{4007520}a^{9}-\frac{395699}{801504}a^{6}+\frac{131533}{500940}a^{3}-\frac{20579}{200376}$, $\frac{1}{20037600}a^{16}+\frac{2027}{4007520}a^{13}+\frac{860689}{20037600}a^{10}+\frac{81161}{801504}a^{7}+\frac{632473}{2504700}a^{4}+\frac{22601}{200376}a$, $\frac{1}{40075200}a^{17}+\frac{1}{2671680}a^{14}+\frac{1}{792}a^{13}-\frac{1}{396}a^{12}+\frac{2024489}{40075200}a^{11}-\frac{115}{792}a^{10}-\frac{17}{396}a^{9}-\frac{2344811}{8015040}a^{8}-\frac{25}{88}a^{7}-\frac{19}{44}a^{6}-\frac{117017}{834900}a^{5}-\frac{133}{792}a^{4}+\frac{133}{396}a^{3}+\frac{156521}{2003760}a^{2}-\frac{43}{396}a-\frac{89}{198}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{2}\times C_{2}$, which has order $4$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $8$
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:   $\frac{1093}{1252350}a^{16}-\frac{14461}{1001880}a^{13}+\frac{284683}{5009400}a^{10}+\frac{63353}{200376}a^{7}+\frac{1667873}{5009400}a^{4}-\frac{53093}{100188}a$, $\frac{389}{404800}a^{17}+\frac{27}{89056}a^{15}-\frac{3307}{242880}a^{14}-\frac{809}{267168}a^{12}+\frac{10821}{404800}a^{11}-\frac{965}{89056}a^{9}+\frac{37297}{80960}a^{8}+\frac{18283}{89056}a^{6}+\frac{204241}{151800}a^{5}+\frac{36775}{33396}a^{3}+\frac{6313}{20240}a^{2}+\frac{8883}{22264}$, $\frac{7499}{20037600}a^{16}-\frac{103}{2003760}a^{15}-\frac{25067}{4007520}a^{13}+\frac{703}{400752}a^{12}+\frac{453911}{20037600}a^{10}-\frac{45307}{2003760}a^{9}+\frac{148183}{801504}a^{7}+\frac{35129}{400752}a^{6}-\frac{474521}{5009400}a^{4}+\frac{194047}{500940}a^{3}-\frac{86327}{200376}a-\frac{46549}{100188}$, $\frac{389}{404800}a^{17}-\frac{39}{22264}a^{15}-\frac{3307}{242880}a^{14}+\frac{1787}{66792}a^{12}+\frac{10821}{404800}a^{11}-\frac{5095}{66792}a^{9}+\frac{37297}{80960}a^{8}-\frac{18369}{22264}a^{6}+\frac{204241}{151800}a^{5}-\frac{34675}{33396}a^{3}+\frac{6313}{20240}a^{2}+\frac{4409}{8349}$, $\frac{871}{404800}a^{17}-\frac{44059}{20037600}a^{16}-\frac{1}{125235}a^{15}-\frac{6881}{242880}a^{14}+\frac{132967}{4007520}a^{13}-\frac{259}{50094}a^{12}+\frac{26557}{1214400}a^{11}-\frac{1760251}{20037600}a^{10}+\frac{20527}{250470}a^{9}+\frac{99059}{80960}a^{8}-\frac{836771}{801504}a^{7}-\frac{12383}{50094}a^{6}+\frac{263437}{75900}a^{5}-\frac{4642057}{2504700}a^{4}-\frac{620683}{250470}a^{3}+\frac{10343}{5520}a^{2}+\frac{419149}{200376}a-\frac{80869}{25047}$, $\frac{81809}{20037600}a^{16}+\frac{119}{2003760}a^{15}-\frac{227177}{4007520}a^{13}+\frac{1369}{400752}a^{12}+\frac{1877501}{20037600}a^{10}-\frac{118909}{2003760}a^{9}+\frac{1653229}{801504}a^{7}+\frac{63935}{400752}a^{6}+\frac{29515939}{5009400}a^{4}+\frac{1047319}{500940}a^{3}+\frac{589207}{200376}a+\frac{570401}{100188}$, $\frac{1}{12650}a^{16}+\frac{59}{30360}a^{13}-\frac{7357}{151800}a^{10}+\frac{459}{2024}a^{7}+\frac{194633}{151800}a^{4}+\frac{4247}{3036}a+2$, $\frac{135739}{6679200}a^{17}+\frac{961}{139150}a^{16}-\frac{8}{605}a^{15}-\frac{457031}{1335840}a^{14}-\frac{17057}{111320}a^{13}+\frac{20}{121}a^{12}+\frac{3049257}{2226400}a^{11}+\frac{608391}{556600}a^{10}+\frac{48}{605}a^{9}+\frac{10658479}{1335840}a^{8}+\frac{2765}{22264}a^{7}-\frac{1088}{121}a^{6}-\frac{85003}{834900}a^{5}-\frac{6895979}{556600}a^{4}-\frac{12352}{605}a^{3}-\frac{3556803}{111320}a^{2}-\frac{340149}{11132}a-\frac{3298}{121}$ 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:  \( 5300270.93735502 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 5300270.93735502 \cdot 4}{2\cdot\sqrt{3602271247127978904000000000}}\cr\approx \mathstrut & 2.69562003364532 \end{aligned}\] (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 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000) 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 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000, 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 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000); 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 - 15*x^15 + 39*x^12 + 475*x^9 + 834*x^6 - 300*x^3 + 1000); 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

\(\Q(\sqrt{-15}) \), 3.1.243.1, 3.1.972.2, 3.1.108.1, 3.1.972.1, 6.0.22143375.1, 6.0.4374000.1, 6.0.354294000.5, 6.0.354294000.4, 9.1.24794911296.1

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: 18.2.1200757082375992968000000000.2
Minimal sibling: 18.2.1200757082375992968000000000.2

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 ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.18.37c4.77$x^{18} + 9 x^{8} + 3 x^{6} + 18 x^{4} + 9 x^{2} + 6$$18$$1$$37$not computednot computed
\(5\) Copy content Toggle raw display 5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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