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Properties

Label 18.6.103...616.3
Degree $18$
Signature $[6, 6]$
Discriminant $1.035\times 10^{29}$
Root discriminant \(40.92\)
Ramified primes $2,19,37$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8:C_9$ (as 18T368)

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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 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369)
 
Copy content gp:K = bnfinit(y^18 - 5*y^16 - 48*y^14 + 58*y^12 + 756*y^10 + 853*y^8 - 2268*y^6 - 6053*y^4 - 4958*y^2 - 1369, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369)
 

\( x^{18} - 5x^{16} - 48x^{14} + 58x^{12} + 756x^{10} + 853x^{8} - 2268x^{6} - 6053x^{4} - 4958x^{2} - 1369 \) 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:  $[6, 6]$
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:   \(103514451522112291747997679616\) \(\medspace = 2^{18}\cdot 19^{16}\cdot 37^{2}\) 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:  \(40.92\)
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:   \(2\), \(19\), \(37\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{8405444771}a^{16}-\frac{730992133}{8405444771}a^{14}-\frac{2798027183}{8405444771}a^{12}+\frac{933627126}{8405444771}a^{10}-\frac{1304929637}{8405444771}a^{8}-\frac{2899868597}{8405444771}a^{6}-\frac{3132220840}{8405444771}a^{4}+\frac{3293144648}{8405444771}a^{2}-\frac{7628236}{227174183}$, $\frac{1}{8405444771}a^{17}-\frac{730992133}{8405444771}a^{15}-\frac{2798027183}{8405444771}a^{13}+\frac{933627126}{8405444771}a^{11}-\frac{1304929637}{8405444771}a^{9}-\frac{2899868597}{8405444771}a^{7}-\frac{3132220840}{8405444771}a^{5}+\frac{3293144648}{8405444771}a^{3}-\frac{7628236}{227174183}a$ 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:  $C_{2}\times C_{2}$, 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:  $11$
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:   $\frac{41747615}{8405444771}a^{16}-\frac{511350271}{8405444771}a^{14}+\frac{308409944}{8405444771}a^{12}+\frac{11187430123}{8405444771}a^{10}-\frac{12030859257}{8405444771}a^{8}-\frac{86220142927}{8405444771}a^{6}+\frac{23305455323}{8405444771}a^{4}+\frac{236223712412}{8405444771}a^{2}+\frac{3923129776}{227174183}$, $\frac{1620653394}{8405444771}a^{16}-\frac{10992174726}{8405444771}a^{14}-\frac{58063576335}{8405444771}a^{12}+\frac{196423480777}{8405444771}a^{10}+\frac{871630290255}{8405444771}a^{8}-\frac{150842070634}{8405444771}a^{6}-\frac{3358155470823}{8405444771}a^{4}-\frac{3891313811379}{8405444771}a^{2}-\frac{35423838013}{227174183}$, $\frac{130798098}{8405444771}a^{16}-\frac{890665171}{8405444771}a^{14}-\frac{4450775030}{8405444771}a^{12}+\frac{13975272010}{8405444771}a^{10}+\frac{67496661873}{8405444771}a^{8}+\frac{20604687297}{8405444771}a^{6}-\frac{249379265329}{8405444771}a^{4}-\frac{431597522672}{8405444771}a^{2}-\frac{5431849651}{227174183}$, $\frac{130798098}{8405444771}a^{16}-\frac{890665171}{8405444771}a^{14}-\frac{4450775030}{8405444771}a^{12}+\frac{13975272010}{8405444771}a^{10}+\frac{67496661873}{8405444771}a^{8}+\frac{20604687297}{8405444771}a^{6}-\frac{249379265329}{8405444771}a^{4}-\frac{431597522672}{8405444771}a^{2}-\frac{5659023834}{227174183}$, $\frac{1079672073}{8405444771}a^{16}-\frac{7959991096}{8405444771}a^{14}-\frac{33659536854}{8405444771}a^{12}+\frac{148134075873}{8405444771}a^{10}+\frac{484616823584}{8405444771}a^{8}-\frac{336975584821}{8405444771}a^{6}-\frac{1922262682955}{8405444771}a^{4}-\frac{1622920781656}{8405444771}a^{2}-\frac{10549307176}{227174183}$, $\frac{621644162}{8405444771}a^{16}-\frac{4350615468}{8405444771}a^{14}-\frac{21191061069}{8405444771}a^{12}+\frac{78820106362}{8405444771}a^{10}+\frac{313477653891}{8405444771}a^{8}-\frac{104939733069}{8405444771}a^{6}-\frac{1214490863328}{8405444771}a^{4}-\frac{1295071057408}{8405444771}a^{2}-\frac{11299377670}{227174183}$, $\frac{420799300}{8405444771}a^{16}-\frac{2865128347}{8405444771}a^{14}-\frac{15050845703}{8405444771}a^{12}+\frac{51867862297}{8405444771}a^{10}+\frac{225619587414}{8405444771}a^{8}-\frac{54069461411}{8405444771}a^{6}-\frac{875643658547}{8405444771}a^{4}-\frac{947583279011}{8405444771}a^{2}-\frac{7880663100}{227174183}$, $\frac{621644162}{8405444771}a^{16}-\frac{4350615468}{8405444771}a^{14}-\frac{21191061069}{8405444771}a^{12}+\frac{78820106362}{8405444771}a^{10}+\frac{313477653891}{8405444771}a^{8}-\frac{104939733069}{8405444771}a^{6}-\frac{1214490863328}{8405444771}a^{4}-\frac{1295071057408}{8405444771}a^{2}-\frac{11072203487}{227174183}$, $\frac{426579502}{8405444771}a^{17}-\frac{428179060}{8405444771}a^{16}-\frac{2455013849}{8405444771}a^{15}+\frac{2158086795}{8405444771}a^{14}-\frac{17328297210}{8405444771}a^{13}+\frac{18943091809}{8405444771}a^{12}+\frac{32326162306}{8405444771}a^{11}-\frac{19286664273}{8405444771}a^{10}+\frac{237479758697}{8405444771}a^{9}-\frac{250939440681}{8405444771}a^{8}+\frac{190385192027}{8405444771}a^{7}-\frac{353150459816}{8405444771}a^{6}-\frac{391554399914}{8405444771}a^{5}+\frac{133666908931}{8405444771}a^{4}-\frac{615962238814}{8405444771}a^{3}+\frac{505386350113}{8405444771}a^{2}-\frac{5889552289}{227174183}a+\frac{6170103047}{227174183}$, $\frac{606019088}{8405444771}a^{17}+\frac{579863592}{8405444771}a^{16}-\frac{4049704144}{8405444771}a^{15}-\frac{3998235868}{8405444771}a^{14}-\frac{22182637934}{8405444771}a^{13}-\frac{20242793118}{8405444771}a^{12}+\frac{71505810378}{8405444771}a^{11}+\frac{72284045540}{8405444771}a^{10}+\frac{335507426298}{8405444771}a^{9}+\frac{301992917734}{8405444771}a^{8}-\frac{27429114224}{8405444771}a^{7}-\frac{82295135674}{8405444771}a^{6}-\frac{1289020023554}{8405444771}a^{5}-\frac{1175482210934}{8405444771}a^{4}-\frac{1581608363332}{8405444771}a^{3}-\frac{1290421307920}{8405444771}a^{2}-\frac{15115118110}{227174183}a-\frac{11226675353}{227174183}$, $\frac{25049442189}{8405444771}a^{17}+\frac{53390057801}{8405444771}a^{16}-\frac{14244197554}{8405444771}a^{15}-\frac{32260632956}{8405444771}a^{14}-\frac{1256616844229}{8405444771}a^{13}-\frac{2673859844648}{8405444771}a^{12}-\frac{4133312050495}{8405444771}a^{11}-\frac{8722001809258}{8405444771}a^{10}+\frac{203785977731}{8405444771}a^{9}+\frac{594510832084}{8405444771}a^{8}+\frac{21410566114188}{8405444771}a^{7}+\frac{45338262119099}{8405444771}a^{6}+\frac{39335769265753}{8405444771}a^{5}+\frac{82729336408969}{8405444771}a^{4}+\frac{28337776145927}{8405444771}a^{3}+\frac{59351210145287}{8405444771}a^{2}+\frac{196941453154}{227174183}a+\frac{411482496155}{227174183}$ 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:  \( 31418398.4612 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 31418398.4612 \cdot 1}{2\cdot\sqrt{103514451522112291747997679616}}\cr\approx \mathstrut & 0.192270536930 \end{aligned}\] (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 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369) 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 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369, 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 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369); 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 - 5*x^16 - 48*x^14 + 58*x^12 + 756*x^10 + 853*x^8 - 2268*x^6 - 6053*x^4 - 4958*x^2 - 1369); 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

$C_2^8:C_9$ (as 18T368):

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 2304
The 40 conjugacy class representatives for $C_2^8:C_9$
Character table for $C_2^8:C_9$

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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: 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 ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ R ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 2.9.2.18a14.2$x^{18} + 2 x^{15} + 4 x^{13} + 2 x^{12} + 4 x^{10} + 4 x^{9} + 3 x^{8} + 2 x^{7} + 2 x^{6} + 2 x^{5} + 6 x^{4} + 2 x^{3} + 6 x + 5$$2$$9$$18$18T368$$[2, 2, 2, 2, 2, 2, 2, 2]^{9}$$
\(19\) Copy content Toggle raw display 19.1.9.8a1.1$x^{9} + 19$$9$$1$$8$$C_9$$$[\ ]_{9}$$
19.1.9.8a1.1$x^{9} + 19$$9$$1$$8$$C_9$$$[\ ]_{9}$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.1.2.1a1.2$x^{2} + 74$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_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)