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Properties

Label 18.14.881...049.1
Degree $18$
Signature $[14, 2]$
Discriminant $8.813\times 10^{25}$
Root discriminant \(27.63\)
Ramified primes $7,53,287281$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5.(A_4\times S_4)$ (as 18T544)

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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 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 1)
 
Copy content gp:K = bnfinit(y^18 - 9*y^17 + 24*y^16 + 12*y^15 - 158*y^14 + 182*y^13 + 237*y^12 - 590*y^11 + 78*y^10 + 633*y^9 - 417*y^8 - 222*y^7 + 311*y^6 - 26*y^5 - 86*y^4 + 24*y^3 + 10*y^2 - 4*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 1)
 

\( x^{18} - 9 x^{17} + 24 x^{16} + 12 x^{15} - 158 x^{14} + 182 x^{13} + 237 x^{12} - 590 x^{11} + 78 x^{10} + \cdots - 1 \) 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:   \(88133009763711378710013049\) \(\medspace = 7^{12}\cdot 53^{6}\cdot 287281\) 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:  \(27.63\)
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:  $7^{2/3}53^{1/2}287281^{1/2}\approx 14278.746392340992$
Ramified primes:   \(7\), \(53\), \(287281\) 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:  \(\Q(\sqrt{287281}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$ 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:  Yes
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:  $C_{2}$, which has order $2$ (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:   $3a^{16}-24a^{15}+50a^{14}+70a^{13}-371a^{12}+224a^{11}+685a^{10}-950a^{9}-227a^{8}+1064a^{7}-439a^{6}-398a^{5}+378a^{4}-85a^{2}+20a+10$, $a^{16}-8a^{15}+16a^{14}+28a^{13}-129a^{12}+46a^{11}+293a^{10}-266a^{9}-276a^{8}+346a^{7}+124a^{6}-185a^{5}-41a^{4}+51a^{3}+15a^{2}-15a-6$, $a-1$, $a^{16}-8a^{15}+17a^{14}+21a^{13}-121a^{12}+89a^{11}+195a^{10}-337a^{9}+24a^{8}+331a^{7}-257a^{6}-61a^{5}+160a^{4}-45a^{3}-21a^{2}+12a+2$, $a^{17}-8a^{16}+16a^{15}+28a^{14}-130a^{13}+52a^{12}+289a^{11}-301a^{10}-223a^{9}+410a^{8}-7a^{7}-229a^{6}+82a^{5}+56a^{4}-30a^{3}-6a^{2}+3a$, $2a^{14}-14a^{13}+18a^{12}+74a^{11}-186a^{10}-82a^{9}+495a^{8}-102a^{7}-548a^{6}+242a^{5}+251a^{4}-150a^{3}-37a^{2}+37a+9$, $3a^{16}-24a^{15}+50a^{14}+70a^{13}-370a^{12}+218a^{11}+690a^{10}-920a^{9}-280a^{8}+1030a^{7}-331a^{6}-411a^{5}+306a^{4}+33a^{3}-73a^{2}+9a+8$, $a^{16}-8a^{15}+15a^{14}+35a^{13}-139a^{12}+15a^{11}+382a^{10}-260a^{9}-470a^{8}+459a^{7}+266a^{6}-340a^{5}-47a^{4}+119a^{3}-5a^{2}-23a-3$, $4a^{16}-32a^{15}+65a^{14}+105a^{13}-510a^{12}+239a^{11}+1066a^{10}-1205a^{9}-698a^{8}+1497a^{7}-148a^{6}-701a^{5}+287a^{4}+105a^{3}-69a^{2}-5a+3$, $a^{17}-12a^{16}+48a^{15}-38a^{14}-228a^{13}+553a^{12}+13a^{11}-1275a^{10}+1028a^{9}+860a^{8}-1481a^{7}+217a^{6}+709a^{5}-404a^{4}-86a^{3}+109a^{2}-11a-13$, $2a^{15}-14a^{14}+18a^{13}+74a^{12}-186a^{11}-82a^{10}+495a^{9}-102a^{8}-548a^{7}+242a^{6}+251a^{5}-150a^{4}-37a^{3}+38a^{2}+8a-1$, $4a^{16}-32a^{15}+66a^{14}+98a^{13}-500a^{12}+270a^{11}+978a^{10}-1216a^{9}-503a^{8}+1410a^{7}-315a^{6}-583a^{5}+337a^{4}+51a^{3}-70a^{2}+6a+4$, $a^{17}-8a^{16}+16a^{15}+28a^{14}-130a^{13}+52a^{12}+289a^{11}-301a^{10}-223a^{9}+410a^{8}-7a^{7}-229a^{6}+82a^{5}+56a^{4}-30a^{3}-6a^{2}+3a+1$, $a^{17}-8a^{16}+17a^{15}+21a^{14}-121a^{13}+90a^{12}+190a^{11}-338a^{10}+60a^{9}+303a^{8}-342a^{7}+35a^{6}+228a^{5}-147a^{4}-23a^{3}+46a^{2}-6a-5$, $3a^{17}-28a^{16}+82a^{15}+3a^{14}-462a^{13}+715a^{12}+378a^{11}-1835a^{10}+1030a^{9}+1320a^{8}-1800a^{7}+190a^{6}+850a^{5}-466a^{4}-73a^{3}+114a^{2}-11a-11$ 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:  \( 5605712.54201 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 5605712.54201 \cdot 1}{2\cdot\sqrt{88133009763711378710013049}}\cr\approx \mathstrut & 0.193112863263 \end{aligned}\] (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 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 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 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 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 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 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 - 9*x^17 + 24*x^16 + 12*x^15 - 158*x^14 + 182*x^13 + 237*x^12 - 590*x^11 + 78*x^10 + 633*x^9 - 417*x^8 - 222*x^7 + 311*x^6 - 26*x^5 - 86*x^4 + 24*x^3 + 10*x^2 - 4*x - 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

$C_2^5.(A_4\times S_4)$ (as 18T544):

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 96 conjugacy class representatives for $C_2^5.(A_4\times S_4)$
Character table for $C_2^5.(A_4\times S_4)$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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: 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 ${\href{/padicField/2.6.0.1}{6} }^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ R ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(7\) Copy content Toggle raw display 7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
\(53\) Copy content Toggle raw display 53.6.1.0a1.1$x^{6} + x^{4} + 7 x^{3} + 4 x^{2} + 45 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
53.6.2.6a1.2$x^{12} + 2 x^{10} + 14 x^{9} + 9 x^{8} + 104 x^{7} + 61 x^{6} + 146 x^{5} + 650 x^{4} + 388 x^{3} + 2041 x^{2} + 180 x + 57$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(287281\) Copy content Toggle raw display $\Q_{287281}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{287281}$$x$$1$$1$$0$Trivial$$[\ ]$$
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$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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