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Properties

Label 18.14.756...064.1
Degree $18$
Signature $[14, 2]$
Discriminant $7.561\times 10^{25}$
Root discriminant \(27.40\)
Ramified primes $2,19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^6:C_9$ (as 18T177)

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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 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1)
 
Copy content gp:K = bnfinit(y^18 - 7*y^16 + 7*y^14 + 45*y^12 - 100*y^10 - 13*y^8 + 170*y^6 - 139*y^4 + 36*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1)
 

\( x^{18} - 7x^{16} + 7x^{14} + 45x^{12} - 100x^{10} - 13x^{8} + 170x^{6} - 139x^{4} + 36x^{2} - 1 \) 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:   \(75613185918270483380568064\) \(\medspace = 2^{18}\cdot 19^{16}\) 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:  \(27.40\)
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:  $2^{63/32}19^{8/9}\approx 53.61948908490599$
Ramified primes:   \(2\), \(19\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{113}a^{16}+\frac{37}{113}a^{14}+\frac{53}{113}a^{12}+\frac{4}{113}a^{10}-\frac{37}{113}a^{8}+\frac{54}{113}a^{6}-\frac{53}{113}a^{4}+\frac{15}{113}a^{2}+\frac{18}{113}$, $\frac{1}{113}a^{17}+\frac{37}{113}a^{15}+\frac{53}{113}a^{13}+\frac{4}{113}a^{11}-\frac{37}{113}a^{9}+\frac{54}{113}a^{7}-\frac{53}{113}a^{5}+\frac{15}{113}a^{3}+\frac{18}{113}a$ 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:   $\frac{169}{113}a^{17}-\frac{1092}{113}a^{15}+\frac{595}{113}a^{13}+\frac{7908}{113}a^{11}-\frac{12581}{113}a^{9}-\frac{8841}{113}a^{7}+\frac{23474}{113}a^{5}-\frac{11025}{113}a^{3}+\frac{1008}{113}a$, $\frac{99}{113}a^{16}-\frac{631}{113}a^{14}+\frac{275}{113}a^{12}+\frac{4690}{113}a^{10}-\frac{6714}{113}a^{8}-\frac{6067}{113}a^{6}+\frac{12381}{113}a^{4}-\frac{4843}{113}a^{2}+\frac{200}{113}$, $a$, $\frac{290}{113}a^{17}-\frac{1813}{113}a^{15}+\frac{680}{113}a^{13}+\frac{13477}{113}a^{11}-\frac{18753}{113}a^{9}-\frac{17223}{113}a^{7}+\frac{34915}{113}a^{5}-\frac{14860}{113}a^{3}+\frac{1604}{113}a$, $\frac{675}{113}a^{17}-\frac{4292}{113}a^{15}+\frac{1988}{113}a^{13}+\frac{31515}{113}a^{11}-\frac{47123}{113}a^{9}-\frac{38017}{113}a^{7}+\frac{88412}{113}a^{5}-\frac{38352}{113}a^{3}+\frac{2884}{113}a$, $\frac{95}{113}a^{17}-\frac{553}{113}a^{15}-\frac{50}{113}a^{13}+\frac{4561}{113}a^{11}-\frac{4080}{113}a^{9}-\frac{8656}{113}a^{7}+\frac{8525}{113}a^{5}+\frac{521}{113}a^{3}-\frac{1228}{113}a$, $\frac{420}{113}a^{16}-\frac{2653}{113}a^{14}+\frac{1129}{113}a^{12}+\frac{19647}{113}a^{10}-\frac{28535}{113}a^{8}-\frac{24780}{113}a^{6}+\frac{54128}{113}a^{4}-\frac{21611}{113}a^{2}+\frac{780}{113}$, $\frac{420}{113}a^{16}-\frac{2653}{113}a^{14}+\frac{1129}{113}a^{12}+\frac{19647}{113}a^{10}-\frac{28535}{113}a^{8}-\frac{24780}{113}a^{6}+\frac{54128}{113}a^{4}-\frac{21611}{113}a^{2}+\frac{667}{113}$, $\frac{385}{113}a^{17}-\frac{216}{113}a^{16}-\frac{2479}{113}a^{15}+\frac{1387}{113}a^{14}+\frac{1308}{113}a^{13}-\frac{713}{113}a^{12}+\frac{18038}{113}a^{11}-\frac{10130}{113}a^{10}-\frac{28370}{113}a^{9}+\frac{15789}{113}a^{8}-\frac{20794}{113}a^{7}+\frac{11953}{113}a^{6}+\frac{53497}{113}a^{5}-\frac{30023}{113}a^{4}-\frac{23492}{113}a^{3}+\frac{12467}{113}a^{2}+\frac{1393}{113}a-\frac{385}{113}$, $\frac{94}{113}a^{17}-\frac{26}{113}a^{16}-\frac{477}{113}a^{15}+\frac{168}{113}a^{14}-\frac{442}{113}a^{13}-\frac{22}{113}a^{12}+\frac{4557}{113}a^{11}-\frac{1460}{113}a^{10}-\frac{1218}{113}a^{9}+\frac{1414}{113}a^{8}-\frac{12326}{113}a^{7}+\frac{3229}{113}a^{6}+\frac{5414}{113}a^{5}-\frac{2916}{113}a^{4}+\frac{8642}{113}a^{3}-\frac{1068}{113}a^{2}-\frac{4749}{113}a+\frac{888}{113}$, $\frac{874}{113}a^{17}-\frac{77}{113}a^{16}-\frac{5517}{113}a^{15}+\frac{428}{113}a^{14}+\frac{2365}{113}a^{13}+\frac{100}{113}a^{12}+\frac{40786}{113}a^{11}-\frac{3472}{113}a^{10}-\frac{59571}{113}a^{9}+\frac{2736}{113}a^{8}-\frac{51001}{113}a^{7}+\frac{6238}{113}a^{6}+\frac{113234}{113}a^{5}-\frac{5411}{113}a^{4}-\frac{45650}{113}a^{3}+\frac{314}{113}a^{2}+\frac{1720}{113}a-\frac{30}{113}$, $\frac{273}{113}a^{17}+\frac{385}{113}a^{16}-\frac{1651}{113}a^{15}-\frac{2366}{113}a^{14}+\frac{344}{113}a^{13}+\frac{630}{113}a^{12}+\frac{12618}{113}a^{11}+\frac{18038}{113}a^{10}-\frac{15412}{113}a^{9}-\frac{22833}{113}a^{8}-\frac{18254}{113}a^{7}-\frac{25879}{113}a^{6}+\frac{29714}{113}a^{5}+\frac{43440}{113}a^{4}-\frac{9578}{113}a^{3}-\frac{14339}{113}a^{2}-\frac{58}{113}a+\frac{263}{113}$, $\frac{207}{113}a^{17}-\frac{601}{113}a^{16}-\frac{1268}{113}a^{15}+\frac{3753}{113}a^{14}+\frac{349}{113}a^{13}-\frac{1343}{113}a^{12}+\frac{9642}{113}a^{11}-\frac{28168}{113}a^{10}-\frac{12518}{113}a^{9}+\frac{38622}{113}a^{8}-\frac{13795}{113}a^{7}+\frac{37832}{113}a^{6}+\frac{24624}{113}a^{5}-\frac{73463}{113}a^{4}-\frac{7065}{113}a^{3}+\frac{26806}{113}a^{2}-\frac{681}{113}a-\frac{648}{113}$, $a^{17}+\frac{47}{113}a^{16}-7a^{15}-\frac{295}{113}a^{14}+7a^{13}+\frac{118}{113}a^{12}+45a^{11}+\frac{2222}{113}a^{10}-100a^{9}-\frac{3208}{113}a^{8}-13a^{7}-\frac{3112}{113}a^{6}+170a^{5}+\frac{6549}{113}a^{4}-139a^{3}-\frac{1555}{113}a^{2}+36a-\frac{623}{113}$, $\frac{95}{113}a^{17}+\frac{601}{113}a^{16}-\frac{553}{113}a^{15}-\frac{3753}{113}a^{14}-\frac{50}{113}a^{13}+\frac{1343}{113}a^{12}+\frac{4561}{113}a^{11}+\frac{28168}{113}a^{10}-\frac{4080}{113}a^{9}-\frac{38622}{113}a^{8}-\frac{8656}{113}a^{7}-\frac{37832}{113}a^{6}+\frac{8525}{113}a^{5}+\frac{73463}{113}a^{4}+\frac{521}{113}a^{3}-\frac{26806}{113}a^{2}-\frac{1228}{113}a+\frac{535}{113}$ 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:  \( 4500826.05302 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 4500826.05302 \cdot 1}{2\cdot\sqrt{75613185918270483380568064}}\cr\approx \mathstrut & 0.167395253923 \end{aligned}\] (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 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1) 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 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1, 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 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1); 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 - 7*x^16 + 7*x^14 + 45*x^12 - 100*x^10 - 13*x^8 + 170*x^6 - 139*x^4 + 36*x^2 - 1); 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

$C_2^6:C_9$ (as 18T177):

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 576
The 16 conjugacy class representatives for $C_2^6:C_9$
Character table for $C_2^6:C_9$

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 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 12.4.1607222233084985344.7

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.9.0.1}{9} }^{2}$ R ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 2.9.2.18a59.1$x^{18} + 2 x^{17} + 2 x^{16} + 2 x^{15} + 2 x^{14} + 4 x^{13} + 2 x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 5 x^{8} + 2 x^{7} + 4 x^{6} + 2 x^{5} + 4 x^{4} + 2 x^{2} + 3$$2$$9$$18$18T177$$[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$$$[\ ]_{9}$$

Spectrum of ring of integers

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