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Properties

Label 18.14.710...728.1
Degree $18$
Signature $[14, 2]$
Discriminant $7.102\times 10^{25}$
Root discriminant \(27.30\)
Ramified primes $2,7,52879$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_4^3.C_6$ (as 18T768)

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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 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7)
 
Copy content gp:K = bnfinit(y^18 - 12*y^16 + 70*y^14 - 261*y^12 + 637*y^10 - 972*y^8 + 874*y^6 - 427*y^4 + 98*y^2 - 7, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7)
 

\( x^{18} - 12x^{16} + 70x^{14} - 261x^{12} + 637x^{10} - 972x^{8} + 874x^{6} - 427x^{4} + 98x^{2} - 7 \) 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:  $[14, 2]$
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:   \(71020039461348097887305728\) \(\medspace = 2^{18}\cdot 7^{13}\cdot 52879^{2}\) 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:  \(27.30\)
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:  not computed
Ramified primes:   \(2\), \(7\), \(52879\) 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:  \(\Q(\sqrt{7}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{239}a^{16}+\frac{62}{239}a^{14}+\frac{117}{239}a^{12}+\frac{32}{239}a^{10}-\frac{102}{239}a^{8}+\frac{84}{239}a^{6}-\frac{80}{239}a^{4}+\frac{106}{239}a^{2}+\frac{55}{239}$, $\frac{1}{239}a^{17}+\frac{62}{239}a^{15}+\frac{117}{239}a^{13}+\frac{32}{239}a^{11}-\frac{102}{239}a^{9}+\frac{84}{239}a^{7}-\frac{80}{239}a^{5}+\frac{106}{239}a^{3}+\frac{55}{239}a$ 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 $1$ (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:  Trivial group, which has order $1$ (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:  $15$
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:   $\frac{3767}{239}a^{16}-\frac{42969}{239}a^{14}+\frac{238306}{239}a^{12}-\frac{842865}{239}a^{10}+\frac{1904669}{239}a^{8}-\frac{2547509}{239}a^{6}+\frac{1810205}{239}a^{4}-\frac{561717}{239}a^{2}+\frac{46338}{239}$, $\frac{1991}{239}a^{16}-\frac{22826}{239}a^{14}+\frac{127070}{239}a^{12}-\frac{450855}{239}a^{10}+\frac{1023227}{239}a^{8}-\frac{1376218}{239}a^{6}+\frac{983379}{239}a^{4}-\frac{306389}{239}a^{2}+\frac{25377}{239}$, $\frac{4756}{239}a^{16}-\frac{54307}{239}a^{14}+\frac{301439}{239}a^{12}-\frac{1066947}{239}a^{10}+\frac{2413480}{239}a^{8}-\frac{3232579}{239}a^{6}+\frac{2300383}{239}a^{4}-\frac{713808}{239}a^{2}+\frac{58430}{239}$, $\frac{1532}{239}a^{16}-\frac{17585}{239}a^{14}+\frac{97984}{239}a^{12}-\frac{347955}{239}a^{10}+\frac{790654}{239}a^{8}-\frac{1065356}{239}a^{6}+\frac{763413}{239}a^{4}-\frac{238650}{239}a^{2}+\frac{19730}{239}$, $\frac{543}{239}a^{16}-\frac{6247}{239}a^{14}+\frac{34851}{239}a^{12}-\frac{123873}{239}a^{10}+\frac{281843}{239}a^{8}-\frac{380286}{239}a^{6}+\frac{273235}{239}a^{4}-\frac{86798}{239}a^{2}+\frac{7638}{239}$, $\frac{2622}{239}a^{16}-\frac{29831}{239}a^{14}+\frac{165047}{239}a^{12}-\frac{582428}{239}a^{10}+\frac{1311868}{239}a^{8}-\frac{1745527}{239}a^{6}+\frac{1231171}{239}a^{4}-\frac{378601}{239}a^{2}+\frac{30685}{239}$, $\frac{989}{239}a^{16}-\frac{11338}{239}a^{14}+\frac{63133}{239}a^{12}-\frac{224082}{239}a^{10}+\frac{508811}{239}a^{8}-\frac{685070}{239}a^{6}+\frac{490178}{239}a^{4}-\frac{152091}{239}a^{2}+\frac{12331}{239}$, $\frac{1688}{239}a^{17}-\frac{19385}{239}a^{15}+\frac{108110}{239}a^{13}-\frac{384310}{239}a^{11}+\frac{874644}{239}a^{9}-\frac{1182268}{239}a^{7}+\frac{852269}{239}a^{5}-\frac{269914}{239}a^{3}+\frac{23530}{239}a-1$, $a^{17}+\frac{3469}{239}a^{16}-11a^{15}-\frac{39935}{239}a^{14}+59a^{13}+\frac{223038}{239}a^{12}-202a^{11}-\frac{793607}{239}a^{10}+435a^{9}+\frac{1808156}{239}a^{8}-537a^{7}-\frac{2445154}{239}a^{6}+337a^{5}+\frac{1758043}{239}a^{4}-90a^{3}-\frac{550046}{239}a^{2}+8a+\frac{45244}{239}$, $\frac{3767}{239}a^{17}-\frac{846}{239}a^{16}-\frac{42969}{239}a^{15}+\frac{9688}{239}a^{14}+\frac{238306}{239}a^{13}-\frac{53811}{239}a^{12}-\frac{842865}{239}a^{11}+\frac{190418}{239}a^{10}+\frac{1904669}{239}a^{9}-\frac{430426}{239}a^{8}-\frac{2547509}{239}a^{7}+\frac{574236}{239}a^{6}+\frac{1810205}{239}a^{5}-\frac{404345}{239}a^{4}-\frac{561717}{239}a^{3}+\frac{123273}{239}a^{2}+\frac{46338}{239}a-\frac{9963}{239}$, $\frac{1991}{239}a^{17}-\frac{489}{239}a^{16}-\frac{22826}{239}a^{15}+\frac{5771}{239}a^{14}+\frac{127070}{239}a^{13}-\frac{32835}{239}a^{12}-\frac{450855}{239}a^{11}+\frac{118670}{239}a^{10}+\frac{1023227}{239}a^{9}-\frac{276118}{239}a^{8}-\frac{1376218}{239}a^{7}+\frac{383866}{239}a^{6}+\frac{983379}{239}a^{5}-\frac{284486}{239}a^{4}-\frac{306389}{239}a^{3}+\frac{91566}{239}a^{2}+\frac{25377}{239}a-\frac{7775}{239}$, $\frac{543}{239}a^{17}-\frac{1991}{239}a^{16}-\frac{6247}{239}a^{15}+\frac{22826}{239}a^{14}+\frac{34851}{239}a^{13}-\frac{127070}{239}a^{12}-\frac{123873}{239}a^{11}+\frac{450855}{239}a^{10}+\frac{281843}{239}a^{9}-\frac{1023227}{239}a^{8}-\frac{380286}{239}a^{7}+\frac{1376218}{239}a^{6}+\frac{273235}{239}a^{5}-\frac{983379}{239}a^{4}-\frac{86798}{239}a^{3}+\frac{306389}{239}a^{2}+\frac{7877}{239}a-\frac{25377}{239}$, $\frac{2710}{239}a^{17}+\frac{1991}{239}a^{16}-\frac{30828}{239}a^{15}-\frac{22826}{239}a^{14}+\frac{170563}{239}a^{13}+\frac{127070}{239}a^{12}-\frac{601839}{239}a^{11}-\frac{450855}{239}a^{10}+\frac{1355233}{239}a^{9}+\frac{1023227}{239}a^{8}-\frac{1801948}{239}a^{7}-\frac{1376218}{239}a^{6}+\frac{1266673}{239}a^{5}+\frac{983379}{239}a^{4}-\frac{383613}{239}a^{3}-\frac{306389}{239}a^{2}+\frac{29311}{239}a+\frac{25616}{239}$, $\frac{8313}{239}a^{17}-\frac{95000}{239}a^{15}+\frac{527603}{239}a^{13}-\frac{1868254}{239}a^{11}+\frac{4228434}{239}a^{9}-\frac{5666756}{239}a^{7}+\frac{4032983}{239}a^{5}-\frac{1249268}{239}a^{3}+\frac{100866}{239}a-1$, $\frac{2534}{239}a^{17}-\frac{1830}{239}a^{16}-\frac{29073}{239}a^{15}+\frac{20619}{239}a^{14}+\frac{161921}{239}a^{13}-\frac{113252}{239}a^{12}-\frac{574728}{239}a^{11}+\frac{397213}{239}a^{10}+\frac{1305070}{239}a^{9}-\frac{887167}{239}a^{8}-\frac{1756504}{239}a^{7}+\frac{1167711}{239}a^{6}+\frac{1256614}{239}a^{5}-\frac{815575}{239}a^{4}-\frac{393187}{239}a^{3}+\frac{250321}{239}a^{2}+\frac{33254}{239}a-\frac{20824}{239}$ 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:  \( 4750220.74631 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 4750220.74631 \cdot 1}{2\cdot\sqrt{71020039461348097887305728}}\cr\approx \mathstrut & 0.182294269987 \end{aligned}\] (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 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7) 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 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7, 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 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7); 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 - 12*x^16 + 70*x^14 - 261*x^12 + 637*x^10 - 972*x^8 + 874*x^6 - 427*x^4 + 98*x^2 - 7); 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

$S_4^3.C_6$ (as 18T768):

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 82944
The 110 conjugacy class representatives for $S_4^3.C_6$
Character table for $S_4^3.C_6$

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.6221161471.1

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: 18.12.3755468666676626068182839590912.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ $18$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ $18$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }$

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 2.9.2.18a11.1$x^{18} + 2 x^{15} + 2 x^{13} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + x^{8} + 2 x^{7} + 4 x^{6} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 3$$2$$9$$18$18T264$$[2, 2, 2, 2, 2, 2, 2]^{9}$$
\(7\) Copy content Toggle raw display 7.1.6.5a1.1$x^{6} + 7$$6$$1$$5$$C_6$$$[\ ]_{6}$$
7.4.3.8a1.3$x^{12} + 15 x^{10} + 12 x^{9} + 84 x^{8} + 120 x^{7} + 263 x^{6} + 372 x^{5} + 492 x^{4} + 424 x^{3} + 279 x^{2} + 108 x + 34$$3$$4$$8$$C_{12}$$$[\ ]_{3}^{4}$$
\(52879\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$2$$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)