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Properties

Label 18.0.240...632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-2.408\times 10^{27}$
Root discriminant \(33.20\)
Ramified primes $2,3,11$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_6^2:D_6$ (as 18T155)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27)
 
Copy content gp:K = bnfinit(y^18 - 21*y^14 + 3*y^12 + 81*y^10 - 18*y^8 + 234*y^6 + 297*y^4 + 135*y^2 + 27, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27)
 

\( x^{18} - 21x^{14} + 3x^{12} + 81x^{10} - 18x^{8} + 234x^{6} + 297x^{4} + 135x^{2} + 27 \) 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:  $[0, 9]$
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:   \(-2407618658213698882221637632\) \(\medspace = -\,2^{30}\cdot 3^{21}\cdot 11^{8}\) 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:  \(33.20\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{13/6}3^{25/18}11^{1/2}\approx 68.48514966410112$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{9}a^{12}$, $\frac{1}{9}a^{13}$, $\frac{1}{99}a^{14}+\frac{5}{99}a^{12}+\frac{2}{33}a^{10}-\frac{4}{33}a^{8}-\frac{1}{11}a^{4}-\frac{1}{11}a^{2}+\frac{4}{11}$, $\frac{1}{99}a^{15}+\frac{5}{99}a^{13}+\frac{2}{33}a^{11}-\frac{4}{33}a^{9}-\frac{1}{11}a^{5}-\frac{1}{11}a^{3}+\frac{4}{11}a$, $\frac{1}{170577}a^{16}-\frac{58}{15507}a^{14}-\frac{1328}{170577}a^{12}+\frac{2056}{18953}a^{10}+\frac{5344}{56859}a^{8}+\frac{1836}{18953}a^{6}+\frac{3744}{18953}a^{4}-\frac{9176}{18953}a^{2}+\frac{8164}{18953}$, $\frac{1}{170577}a^{17}-\frac{58}{15507}a^{15}-\frac{1328}{170577}a^{13}+\frac{2056}{18953}a^{11}+\frac{5344}{56859}a^{9}+\frac{1836}{18953}a^{7}+\frac{3744}{18953}a^{5}-\frac{9176}{18953}a^{3}+\frac{8164}{18953}a$ 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:  $C_{2}$, which has order $2$ (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:  $8$
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:   $\frac{5245}{170577}a^{16}-\frac{410}{18953}a^{14}-\frac{11980}{18953}a^{12}+\frac{31126}{56859}a^{10}+\frac{121849}{56859}a^{8}-\frac{127583}{56859}a^{6}+\frac{165657}{18953}a^{4}+\frac{62467}{18953}a^{2}+\frac{13968}{18953}$, $\frac{950}{56859}a^{16}-\frac{535}{170577}a^{14}-\frac{59674}{170577}a^{12}+\frac{5923}{56859}a^{10}+\frac{24964}{18953}a^{8}-\frac{14237}{56859}a^{6}+\frac{7036}{1723}a^{4}+\frac{43169}{18953}a^{2}+\frac{24130}{18953}$, $\frac{19}{15507}a^{16}-\frac{266}{18953}a^{14}-\frac{208}{18953}a^{12}+\frac{15815}{56859}a^{10}-\frac{4464}{18953}a^{8}-\frac{1299}{1723}a^{6}+\frac{26099}{18953}a^{4}-\frac{77620}{18953}a^{2}-\frac{6386}{18953}$, $\frac{4867}{170577}a^{16}+\frac{1423}{170577}a^{14}-\frac{103783}{170577}a^{12}-\frac{5342}{56859}a^{10}+\frac{145295}{56859}a^{8}+\frac{7894}{56859}a^{6}+\frac{104703}{18953}a^{4}+\frac{203882}{18953}a^{2}+\frac{77620}{18953}$, $\frac{41}{170577}a^{16}-\frac{313}{170577}a^{14}-\frac{115}{18953}a^{12}+\frac{1330}{56859}a^{10}+\frac{2006}{56859}a^{8}-\frac{536}{18953}a^{6}-\frac{5012}{18953}a^{4}+\frac{1355}{1723}a^{2}+\frac{2185}{18953}$, $\frac{7897}{170577}a^{17}+\frac{10609}{170577}a^{16}-\frac{8849}{170577}a^{15}-\frac{598}{170577}a^{14}-\frac{53759}{56859}a^{13}-\frac{225494}{170577}a^{12}+\frac{70129}{56859}a^{11}+\frac{14002}{56859}a^{10}+\frac{15832}{5169}a^{9}+\frac{302429}{56859}a^{8}-\frac{94918}{18953}a^{7}-\frac{24513}{18953}a^{6}+\frac{258138}{18953}a^{5}+\frac{258227}{18953}a^{4}+\frac{25461}{18953}a^{3}+\frac{333855}{18953}a^{2}-\frac{5022}{1723}a+\frac{98323}{18953}$, $\frac{19457}{56859}a^{17}-\frac{16945}{56859}a^{16}-\frac{46297}{170577}a^{15}-\frac{9637}{170577}a^{14}-\frac{402247}{56859}a^{13}+\frac{1085507}{170577}a^{12}+\frac{11469}{1723}a^{11}+\frac{18490}{56859}a^{10}+\frac{1393165}{56859}a^{9}-\frac{1507901}{56859}a^{8}-\frac{1489558}{56859}a^{7}+\frac{12442}{56859}a^{6}+\frac{1804854}{18953}a^{5}-\frac{1142563}{18953}a^{4}+\frac{521553}{18953}a^{3}-\frac{1833933}{18953}a^{2}-\frac{320082}{18953}a-\frac{569966}{18953}$, $\frac{92855}{170577}a^{17}+\frac{1451}{56859}a^{16}-\frac{2938}{15507}a^{15}-\frac{7379}{56859}a^{14}-\frac{645476}{56859}a^{13}-\frac{98330}{170577}a^{12}+\frac{110029}{18953}a^{11}+\frac{139358}{56859}a^{10}+\frac{2415658}{56859}a^{9}+\frac{24766}{18953}a^{8}-\frac{1517605}{56859}a^{7}-\frac{454093}{56859}a^{6}+\frac{2514990}{18953}a^{5}+\frac{275455}{18953}a^{4}+\frac{2155277}{18953}a^{3}-\frac{29027}{1723}a^{2}+\frac{346233}{18953}a-\frac{274663}{18953}$ 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:  \( 917581.32833 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 917581.32833 \cdot 2}{2\cdot\sqrt{2407618658213698882221637632}}\cr\approx \mathstrut & 0.28541022813 \end{aligned}\] (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 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27) 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 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27, 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 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27); 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 = polynomial_ring(QQ); K, a = number_field(x^18 - 21*x^14 + 3*x^12 + 81*x^10 - 18*x^8 + 234*x^6 + 297*x^4 + 135*x^2 + 27); 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

$C_6^2:D_6$ (as 18T155):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_6^2:D_6$
Character table for $C_6^2:D_6$

Intermediate fields

3.1.44.1, 6.0.3345408.1, 9.3.18443443392.2

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(L)]
 
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 36 siblings: data not computed
Minimal sibling: 18.0.980881675568543989053259776.1

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.3.6.30a1.29$x^{18} + 6 x^{16} + 8 x^{15} + 15 x^{14} + 40 x^{13} + 45 x^{12} + 80 x^{11} + 115 x^{10} + 124 x^{9} + 156 x^{8} + 172 x^{7} + 148 x^{6} + 140 x^{5} + 119 x^{4} + 76 x^{3} + 47 x^{2} + 32 x + 13$$6$$3$$30$18T30$$[\frac{8}{3}, \frac{8}{3}]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.3.6.21a11.4$x^{18} + 12 x^{16} + 6 x^{15} + 60 x^{14} + 60 x^{13} + 175 x^{12} + 240 x^{11} + 363 x^{10} + 500 x^{9} + 576 x^{8} + 609 x^{7} + 625 x^{6} + 480 x^{5} + 381 x^{4} + 238 x^{3} + 108 x^{2} + 39 x + 10$$6$$3$$21$18T20not computed
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.1.2.1a1.1$x^{2} + 11$$2$$1$$1$$C_2$$$[\ ]_{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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