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Properties

Label 18.0.414...792.2
Degree $18$
Signature $[0, 9]$
Discriminant $-4.146\times 10^{29}$
Root discriminant \(44.20\)
Ramified primes $2,3,7$
Class number $6$ (GRH)
Class group [6] (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 - 58*x^12 + 3996*x^6 + 5832)
 
Copy content gp:K = bnfinit(y^18 - 58*y^12 + 3996*y^6 + 5832, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 58*x^12 + 3996*x^6 + 5832);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 58*x^12 + 3996*x^6 + 5832)
 

\( x^{18} - 58x^{12} + 3996x^{6} + 5832 \) 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:   \(-414563852462258547447973281792\) \(\medspace = -\,2^{33}\cdot 3^{20}\cdot 7^{12}\) 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:  \(44.20\)
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:  $2^{11/6}3^{7/6}7^{2/3}\approx 46.98167351748441$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{-2}) \)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-2}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{2}$, $\frac{1}{12}a^{9}+\frac{1}{6}a^{3}$, $\frac{1}{36}a^{10}+\frac{7}{18}a^{4}$, $\frac{1}{108}a^{11}-\frac{1}{36}a^{9}-\frac{1}{6}a^{7}+\frac{25}{54}a^{5}-\frac{7}{18}a^{3}-\frac{1}{3}a$, $\frac{1}{9720}a^{12}+\frac{8}{243}a^{6}-\frac{13}{90}$, $\frac{1}{29160}a^{13}+\frac{259}{1458}a^{7}-\frac{103}{270}a$, $\frac{1}{29160}a^{14}+\frac{8}{729}a^{8}+\frac{77}{270}a^{2}$, $\frac{1}{87480}a^{15}-\frac{211}{8748}a^{9}+\frac{16}{405}a^{3}$, $\frac{1}{524880}a^{16}+\frac{1}{87480}a^{14}-\frac{1}{29160}a^{12}+\frac{259}{26244}a^{10}-\frac{211}{8748}a^{8}-\frac{259}{1458}a^{6}-\frac{1453}{4860}a^{4}+\frac{16}{405}a^{2}+\frac{103}{270}$, $\frac{1}{524880}a^{17}+\frac{4}{6561}a^{11}+\frac{1}{36}a^{9}+\frac{1}{6}a^{7}+\frac{1157}{4860}a^{5}+\frac{7}{18}a^{3}+\frac{1}{3}a$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:  $C_{6}$, which has order $6$ (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:   $\frac{19}{87480}a^{16}-\frac{121}{8748}a^{10}+\frac{394}{405}a^{4}-1$, $\frac{11}{524880}a^{16}-\frac{5}{17496}a^{14}-\frac{7}{14580}a^{12}-\frac{67}{26244}a^{10}+\frac{43}{2187}a^{8}+\frac{19}{1458}a^{6}+\frac{757}{4860}a^{4}-\frac{187}{162}a^{2}-\frac{224}{135}$, $\frac{1}{9720}a^{12}+\frac{8}{243}a^{6}-\frac{13}{90}$, $\frac{103}{524880}a^{17}+\frac{19}{87480}a^{16}-\frac{13}{29160}a^{14}-\frac{7}{14580}a^{13}-\frac{74}{6561}a^{11}-\frac{121}{8748}a^{10}+\frac{35}{1458}a^{8}+\frac{19}{1458}a^{7}+\frac{3971}{4860}a^{5}+\frac{394}{405}a^{4}-\frac{371}{270}a^{2}-\frac{224}{135}a-1$, $\frac{13}{34992}a^{16}-\frac{175}{8748}a^{10}+\frac{479}{324}a^{4}-2$, $\frac{433}{524880}a^{17}+\frac{7}{7290}a^{16}+\frac{11}{10935}a^{15}+\frac{1}{1080}a^{14}-\frac{2}{3645}a^{13}+\frac{1}{4860}a^{12}-\frac{667}{13122}a^{11}-\frac{157}{2916}a^{10}-\frac{293}{4374}a^{9}-\frac{1}{27}a^{8}-\frac{13}{1458}a^{7}+\frac{16}{243}a^{6}+\frac{16241}{4860}a^{5}+\frac{1061}{270}a^{4}+\frac{1723}{405}a^{3}+\frac{91}{30}a^{2}-\frac{31}{135}a-\frac{148}{45}$, $\frac{49}{87480}a^{17}+\frac{173}{524880}a^{16}+\frac{17}{43740}a^{15}+\frac{5}{8748}a^{14}-\frac{1}{5832}a^{13}-\frac{59}{29160}a^{12}-\frac{295}{8748}a^{11}-\frac{391}{26244}a^{10}-\frac{127}{8748}a^{9}-\frac{86}{2187}a^{8}-\frac{40}{729}a^{7}+\frac{14}{729}a^{6}+\frac{949}{405}a^{5}+\frac{5671}{4860}a^{4}+\frac{503}{810}a^{3}+\frac{25}{81}a^{2}-\frac{77}{54}a-\frac{943}{270}$, $\frac{119}{43740}a^{15}+\frac{47}{1215}a^{12}+\frac{1541}{8748}a^{9}-\frac{59}{486}a^{6}-\frac{619}{810}a^{3}+\frac{76}{45}$ 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:  \( 50024296.556075595 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 50024296.556075595 \cdot 6}{2\cdot\sqrt{414563852462258547447973281792}}\cr\approx \mathstrut & 3.55733974394770 \end{aligned}\] (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 - 58*x^12 + 3996*x^6 + 5832) 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 - 58*x^12 + 3996*x^6 + 5832, 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 - 58*x^12 + 3996*x^6 + 5832); 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 - 58*x^12 + 3996*x^6 + 5832); 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

\(\Q(\sqrt{-2}) \), 3.1.1323.1, 3.1.108.1, 3.1.588.1, 3.1.5292.1, 6.0.896168448.4, 6.0.44255232.2, 6.0.1492992.1, 6.0.3584673792.8, 9.1.444611571264.1

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: 18.2.1243691557386775642343919845376.1
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.2.0.1}{2} }^{9}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.6.11a1.1$x^{6} + 2$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
2.2.6.22a1.1$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 3$$6$$2$$22$$D_6$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.3.3a1.1$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.3.3a1.1$x^{3} + 3 x + 3$$3$$1$$3$$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$$$[\frac{3}{2}]_{2}$$
\(7\) Copy content Toggle raw display 7.6.3.12a1.3$x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$

Spectrum of ring of integers

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