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Properties

Label 18.0.351...832.3
Degree $18$
Signature $[0, 9]$
Discriminant $-3.513\times 10^{26}$
Root discriminant \(29.84\)
Ramified primes $2,3,107$
Class number $1$
Class group trivial
Galois group $C_2^4.S_4^2$ (as 18T546)

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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 + 15*x^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107)
 
Copy content gp:K = bnfinit(y^18 + 15*y^16 + 87*y^14 + 277*y^12 + 576*y^10 + 831*y^8 + 851*y^6 + 633*y^4 + 321*y^2 + 107, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 15*x^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 15*x^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107)
 

\( x^{18} + 15x^{16} + 87x^{14} + 277x^{12} + 576x^{10} + 831x^{8} + 851x^{6} + 633x^{4} + 321x^{2} + 107 \) 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:   \(-351335026748107688667512832\) \(\medspace = -\,2^{18}\cdot 3^{6}\cdot 107^{9}\) 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:  \(29.84\)
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:  $2^{63/32}3^{1/2}107^{3/4}\approx 225.5543797844839$
Ramified primes:   \(2\), \(3\), \(107\) 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:  \(\Q(\sqrt{-107}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{10}-\frac{1}{9}a^{8}-\frac{1}{9}a^{6}-\frac{4}{9}a^{4}+\frac{2}{9}a^{2}+\frac{4}{9}$, $\frac{1}{9}a^{13}-\frac{1}{9}a^{11}-\frac{1}{9}a^{9}-\frac{1}{9}a^{7}-\frac{4}{9}a^{5}+\frac{2}{9}a^{3}+\frac{4}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{1}{9}a^{6}+\frac{4}{9}a^{4}-\frac{1}{3}a^{2}+\frac{4}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{1}{9}a^{7}+\frac{4}{9}a^{5}-\frac{1}{3}a^{3}+\frac{4}{9}a$, $\frac{1}{3681}a^{16}+\frac{88}{3681}a^{14}-\frac{11}{1227}a^{12}-\frac{29}{1227}a^{10}+\frac{40}{409}a^{8}+\frac{13}{409}a^{6}-\frac{1651}{3681}a^{4}+\frac{1583}{3681}a^{2}-\frac{167}{409}$, $\frac{1}{3681}a^{17}+\frac{88}{3681}a^{15}-\frac{11}{1227}a^{13}-\frac{29}{1227}a^{11}+\frac{40}{409}a^{9}+\frac{13}{409}a^{7}-\frac{1651}{3681}a^{5}+\frac{1583}{3681}a^{3}-\frac{167}{409}a$ 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:  Trivial group, which has order $1$
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:  Trivial group, which has order $1$
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:   $\frac{43}{3681}a^{16}+\frac{512}{3681}a^{14}+\frac{1853}{3681}a^{12}+\frac{1985}{3681}a^{10}-\frac{2107}{3681}a^{8}-\frac{7648}{3681}a^{6}-\frac{5144}{3681}a^{4}+\frac{1558}{1227}a^{2}+\frac{590}{409}$, $\frac{146}{3681}a^{16}+\frac{1805}{3681}a^{14}+\frac{828}{409}a^{12}+\frac{1588}{409}a^{10}+\frac{1750}{409}a^{8}+\frac{671}{409}a^{6}-\frac{6689}{3681}a^{4}-\frac{9374}{3681}a^{2}-\frac{1069}{409}$, $\frac{89}{3681}a^{16}+\frac{1288}{3681}a^{14}+\frac{2293}{1227}a^{12}+\frac{18842}{3681}a^{10}+\frac{32858}{3681}a^{8}+\frac{40679}{3681}a^{6}+\frac{3669}{409}a^{4}+\frac{20641}{3681}a^{2}+\frac{3248}{3681}$, $\frac{110}{3681}a^{16}+\frac{500}{1227}a^{14}+\frac{2471}{1227}a^{12}+\frac{19060}{3681}a^{10}+\frac{32647}{3681}a^{8}+\frac{40273}{3681}a^{6}+\frac{35978}{3681}a^{4}+\frac{23209}{3681}a^{2}+\frac{6859}{3681}$, $\frac{146}{3681}a^{16}+\frac{1805}{3681}a^{14}+\frac{828}{409}a^{12}+\frac{1588}{409}a^{10}+\frac{1750}{409}a^{8}+\frac{671}{409}a^{6}-\frac{6689}{3681}a^{4}-\frac{9374}{3681}a^{2}-\frac{660}{409}$, $\frac{10}{409}a^{17}-\frac{236}{3681}a^{16}+\frac{967}{3681}a^{15}-\frac{3590}{3681}a^{14}+\frac{2347}{3681}a^{13}-\frac{20842}{3681}a^{12}-\frac{461}{409}a^{11}-\frac{63722}{3681}a^{10}-\frac{9241}{1227}a^{9}-\frac{120134}{3681}a^{8}-\frac{6601}{409}a^{7}-\frac{149903}{3681}a^{6}-\frac{24172}{1227}a^{5}-\frac{41356}{1227}a^{4}-\frac{49351}{3681}a^{3}-\frac{7381}{409}a^{2}-\frac{20341}{3681}a-\frac{17482}{3681}$, $\frac{2131}{1227}a^{17}-\frac{10861}{3681}a^{16}+\frac{34562}{1227}a^{15}-\frac{152083}{3681}a^{14}+\frac{215977}{1227}a^{13}-\frac{771245}{3681}a^{12}+\frac{232681}{409}a^{11}-\frac{1945087}{3681}a^{10}+\frac{1337306}{1227}a^{9}-\frac{2934904}{3681}a^{8}+\frac{1675919}{1227}a^{7}-\frac{2991400}{3681}a^{6}+\frac{478100}{409}a^{5}-\frac{241068}{409}a^{4}+\frac{813851}{1227}a^{3}-\frac{399400}{1227}a^{2}+\frac{282605}{1227}a-\frac{377432}{3681}$, $\frac{19829}{409}a^{17}+\frac{35311}{3681}a^{16}+\frac{2484052}{3681}a^{15}+\frac{306127}{3681}a^{14}+\frac{12666299}{3681}a^{13}+\frac{5855}{1227}a^{12}+\frac{33396428}{3681}a^{11}-\frac{1772080}{1227}a^{10}+\frac{55481090}{3681}a^{9}-\frac{5927557}{1227}a^{8}+\frac{61361939}{3681}a^{7}-\frac{10362398}{1227}a^{6}+\frac{45027209}{3681}a^{5}-\frac{33018580}{3681}a^{4}+\frac{24953191}{3681}a^{3}-\frac{20618407}{3681}a^{2}+\frac{2118695}{1227}a-\frac{3907511}{1227}$ Copy content Toggle raw display
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:  \( 776442.124776 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 776442.124776 \cdot 1}{2\cdot\sqrt{351335026748107688667512832}}\cr\approx \mathstrut & 0.316109235346 \end{aligned}\]

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^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107) 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^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107, 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^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107); 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^16 + 87*x^14 + 277*x^12 + 576*x^10 + 831*x^8 + 851*x^6 + 633*x^4 + 321*x^2 + 107); 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

$C_2^4.S_4^2$ (as 18T546):

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 9216
The 60 conjugacy class representatives for $C_2^4.S_4^2$
Character table for $C_2^4.S_4^2$

Intermediate fields

3.1.107.1, 3.3.321.1, 9.3.3539149227.1

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

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.6.37592847862047522687423873024.2

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 ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$

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

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 2.3.2.6a1.2$x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 9$$2$$3$$6$$C_6$$$[2]^{3}$$
2.6.2.12a8.1$x^{12} + 2 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 4 x^{7} + 7 x^{6} + 6 x^{5} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 3$$2$$6$$12$12T134$$[2, 2, 2, 2, 2, 2]^{6}$$
\(3\) Copy content Toggle raw display 3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.3.1.0a1.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
3.6.2.6a1.2$x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(107\) Copy content Toggle raw display $\Q_{107}$$x + 105$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{107}$$x + 105$$1$$1$$0$Trivial$$[\ ]$$
107.1.4.3a1.2$x^{4} + 214$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
107.2.2.2a1.2$x^{4} + 206 x^{3} + 10613 x^{2} + 412 x + 111$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
107.4.2.4a1.2$x^{8} + 26 x^{6} + 158 x^{5} + 173 x^{4} + 2054 x^{3} + 6293 x^{2} + 316 x + 111$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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