↑

Properties

Label 18.10.258...864.1
Degree $18$
Signature $[10, 4]$
Discriminant $2.582\times 10^{26}$
Root discriminant \(29.33\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8:C_9$ (as 18T368)

Related objects

Downloads

Learn more

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 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9)
 
Copy content gp:K = bnfinit(y^18 - 18*y^14 + 9*y^12 + 81*y^10 - 54*y^8 - 90*y^6 + 81*y^4 - 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9)
 

\( x^{18} - 18x^{14} + 9x^{12} + 81x^{10} - 54x^{8} - 90x^{6} + 81x^{4} - 9 \) 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:  $[10, 4]$
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:   \(258151783382020583032356864\) \(\medspace = 2^{18}\cdot 3^{44}\) 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:  \(29.33\)
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:   \(2\), \(3\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{1293}a^{16}+\frac{23}{431}a^{14}+\frac{2}{1293}a^{12}+\frac{49}{431}a^{10}-\frac{40}{431}a^{8}-\frac{192}{431}a^{6}+\frac{83}{431}a^{4}+\frac{151}{431}a^{2}+\frac{75}{431}$, $\frac{1}{1293}a^{17}+\frac{23}{431}a^{15}+\frac{2}{1293}a^{13}+\frac{49}{431}a^{11}-\frac{40}{431}a^{9}-\frac{192}{431}a^{7}+\frac{83}{431}a^{5}+\frac{151}{431}a^{3}+\frac{75}{431}a$ 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:  $13$
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:   $\frac{1759}{1293}a^{17}-\frac{374}{431}a^{15}+\frac{30962}{1293}a^{13}+\frac{1302}{431}a^{11}-\frac{140185}{1293}a^{9}+\frac{1979}{431}a^{7}+\frac{53987}{431}a^{5}-\frac{13474}{431}a^{3}-\frac{7797}{431}a$, $\frac{637}{1293}a^{17}+\frac{422}{1293}a^{15}-\frac{11225}{1293}a^{13}-\frac{1612}{1293}a^{11}+\frac{51136}{1293}a^{9}-\frac{762}{431}a^{7}-\frac{19968}{431}a^{5}+\frac{5677}{431}a^{3}+\frac{2951}{431}a$, $\frac{374}{431}a^{16}-\frac{700}{1293}a^{14}+\frac{6579}{431}a^{12}+\frac{2294}{1293}a^{10}-\frac{29683}{431}a^{8}+\frac{1217}{431}a^{6}+\frac{34019}{431}a^{4}-\frac{7797}{431}a^{2}-\frac{4846}{431}$, $\frac{2260}{1293}a^{16}+\frac{1642}{1293}a^{14}-\frac{39442}{1293}a^{12}-\frac{8270}{1293}a^{10}+\frac{58726}{431}a^{8}+\frac{1821}{431}a^{6}-\frac{64986}{431}a^{4}+\frac{14131}{431}a^{2}+\frac{8737}{431}$, $\frac{736}{431}a^{17}-\frac{1502}{1293}a^{15}+\frac{38684}{1293}a^{13}+\frac{2144}{431}a^{11}-\frac{173798}{1293}a^{9}+\frac{263}{431}a^{7}+\frac{65423}{431}a^{5}-\frac{14468}{431}a^{3}-\frac{9147}{431}a$, $\frac{285}{431}a^{16}-\frac{270}{431}a^{14}+\frac{14668}{1293}a^{12}+\frac{2067}{431}a^{10}-\frac{21399}{431}a^{8}-\frac{4792}{431}a^{6}+\frac{22131}{431}a^{4}-\frac{2391}{431}a^{2}-\frac{2492}{431}$, $\frac{673}{1293}a^{17}-\frac{320}{1293}a^{15}+\frac{4005}{431}a^{13}-\frac{232}{1293}a^{11}-\frac{55436}{1293}a^{9}+\frac{2933}{431}a^{7}+\frac{22583}{431}a^{5}-\frac{6372}{431}a^{3}-\frac{3496}{431}a$, $\frac{892}{431}a^{17}-\frac{1900}{1293}a^{15}+\frac{46799}{1293}a^{13}+\frac{9182}{1293}a^{11}-\frac{209872}{1293}a^{9}-\frac{1684}{431}a^{7}+\frac{78730}{431}a^{5}-\frac{16176}{431}a^{3}-\frac{11491}{431}a$, $a+1$, $\frac{3710}{1293}a^{17}-\frac{1254}{431}a^{16}-\frac{854}{431}a^{15}-\frac{2702}{1293}a^{14}+\frac{64988}{1293}a^{13}+\frac{65746}{1293}a^{12}+\frac{11482}{1293}a^{11}+\frac{4440}{431}a^{10}-\frac{292241}{1293}a^{9}-\frac{98207}{431}a^{8}-\frac{554}{431}a^{7}-\frac{2638}{431}a^{6}+\frac{110571}{431}a^{5}+\frac{110134}{431}a^{4}-\frac{23615}{431}a^{3}-\frac{22847}{431}a^{2}-\frac{16202}{431}a-\frac{15792}{431}$, $\frac{1284}{431}a^{17}-\frac{1592}{1293}a^{16}-\frac{2447}{1293}a^{15}-\frac{805}{1293}a^{14}+\frac{67721}{1293}a^{13}+\frac{28279}{1293}a^{12}+\frac{8279}{1293}a^{11}+\frac{3}{431}a^{10}-\frac{101934}{431}a^{9}-\frac{43208}{431}a^{8}+\frac{4729}{431}a^{7}+\frac{6550}{431}a^{6}+\frac{117318}{431}a^{5}+\frac{52332}{431}a^{4}-\frac{29541}{431}a^{3}-\frac{15410}{431}a^{2}-\frac{17801}{431}a-\frac{9064}{431}$, $\frac{944}{1293}a^{17}-\frac{1254}{431}a^{16}+\frac{917}{1293}a^{15}-\frac{2702}{1293}a^{14}-\frac{16214}{1293}a^{13}+\frac{65746}{1293}a^{12}-\frac{2447}{431}a^{11}+\frac{4440}{431}a^{10}+\frac{71188}{1293}a^{9}-\frac{98207}{431}a^{8}+\frac{6237}{431}a^{7}-\frac{2638}{431}a^{6}-\frac{24657}{431}a^{5}+\frac{110134}{431}a^{4}+\frac{745}{431}a^{3}-\frac{22847}{431}a^{2}+\frac{2702}{431}a-\frac{15792}{431}$, $\frac{1259}{431}a^{17}+\frac{5}{431}a^{16}-\frac{2875}{1293}a^{15}+\frac{173}{1293}a^{14}+\frac{65716}{1293}a^{13}+\frac{10}{431}a^{12}+\frac{15856}{1293}a^{11}-\frac{2536}{1293}a^{10}-\frac{97607}{431}a^{9}-\frac{600}{431}a^{8}-\frac{5792}{431}a^{7}+\frac{3154}{431}a^{6}+\frac{108027}{431}a^{5}+\frac{2107}{431}a^{4}-\frac{20802}{431}a^{3}-\frac{2476}{431}a^{2}-\frac{15624}{431}a-\frac{1030}{431}$ 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:  \( 6765640.006566327 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{4}\cdot 6765640.006566327 \cdot 1}{2\cdot\sqrt{258151783382020583032356864}}\cr\approx \mathstrut & 0.336016617164271 \end{aligned}\] (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 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9) 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 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9, 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 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9); 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 - 18*x^14 + 9*x^12 + 81*x^10 - 54*x^8 - 90*x^6 + 81*x^4 - 9); 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

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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 R ${\href{/padicField/5.9.0.1}{9} }^{2}$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.9.0.1}{9} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }^{2}$ ${\href{/padicField/47.9.0.1}{9} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{6}$ ${\href{/padicField/59.9.0.1}{9} }^{2}$

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 2.9.2.18a33.1$x^{18} + 2 x^{17} + 2 x^{14} + 4 x^{13} + 4 x^{12} + 6 x^{9} + 5 x^{8} + 2 x^{7} + 2 x^{5} + 6 x^{4} + 2 x^{3} + 5$$2$$9$$18$18T368$$[2, 2, 2, 2, 2, 2, 2, 2]^{9}$$
\(3\) Copy content Toggle raw display 3.1.9.22a3.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$$9$$1$$22$$C_9$$$[2, 3]$$
3.1.9.22a3.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$$9$$1$$22$$C_9$$$[2, 3]$$

Spectrum of ring of integers

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