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Properties

Label 18.0.753...672.1
Degree $18$
Signature $[0, 9]$
Discriminant $-7.530\times 10^{21}$
Root discriminant \(16.42\)
Ramified primes $2,107$
Class number $1$
Class group trivial
Galois group $D_9$ (as 18T5)

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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 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*x + 1)
 
Copy content gp:K = bnfinit(y^18 - 9*y^17 + 33*y^16 - 60*y^15 + 50*y^14 - 14*y^13 + 62*y^12 - 268*y^11 + 523*y^10 - 635*y^9 + 523*y^8 - 268*y^7 + 62*y^6 - 14*y^5 + 50*y^4 - 60*y^3 + 33*y^2 - 9*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*x + 1)
 

\( x^{18} - 9 x^{17} + 33 x^{16} - 60 x^{15} + 50 x^{14} - 14 x^{13} + 62 x^{12} - 268 x^{11} + 523 x^{10} + \cdots + 1 \) 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:  $[0, 9]$
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:   \(-7530328934072952860672\) \(\medspace = -\,2^{12}\cdot 107^{9}\) 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:  \(16.42\)
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:  $2^{2/3}107^{1/2}\approx 16.420204160652293$
Ramified primes:   \(2\), \(107\) 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:  \(\Q(\sqrt{-107}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_9$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-107}) \)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{8}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{9}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{3}{8}$, $\frac{1}{40}a^{15}+\frac{1}{20}a^{13}+\frac{1}{20}a^{12}+\frac{1}{40}a^{11}-\frac{1}{10}a^{10}-\frac{7}{40}a^{9}+\frac{1}{20}a^{8}+\frac{7}{40}a^{7}+\frac{1}{5}a^{6}-\frac{9}{40}a^{5}+\frac{2}{5}a^{4}-\frac{9}{20}a^{3}+\frac{1}{20}a^{2}+\frac{3}{8}a-\frac{7}{20}$, $\frac{1}{40}a^{16}+\frac{1}{20}a^{14}+\frac{1}{20}a^{13}+\frac{1}{40}a^{12}-\frac{1}{10}a^{11}+\frac{3}{40}a^{10}-\frac{1}{5}a^{9}-\frac{3}{40}a^{8}+\frac{1}{5}a^{7}+\frac{1}{40}a^{6}+\frac{3}{20}a^{5}-\frac{1}{5}a^{4}+\frac{1}{20}a^{3}+\frac{1}{8}a^{2}+\frac{2}{5}a+\frac{1}{4}$, $\frac{1}{40}a^{17}+\frac{1}{20}a^{14}+\frac{1}{20}a^{13}+\frac{1}{20}a^{12}-\frac{1}{10}a^{11}-\frac{9}{40}a^{9}-\frac{3}{20}a^{8}-\frac{1}{5}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{10}a^{3}+\frac{1}{20}a^{2}-\frac{3}{8}a-\frac{3}{10}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$
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 $1$
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:   $a^{17}-8a^{16}+25a^{15}-35a^{14}+15a^{13}+a^{12}+63a^{11}-205a^{10}+318a^{9}-317a^{8}+206a^{7}-62a^{6}-14a^{4}+36a^{3}-24a^{2}+9a-1$, $\frac{433}{40}a^{17}-\frac{1857}{20}a^{16}+\frac{2543}{8}a^{15}-\frac{20577}{40}a^{14}+\frac{1611}{5}a^{13}-\frac{147}{10}a^{12}+\frac{26669}{40}a^{11}-\frac{104757}{40}a^{10}+\frac{9093}{2}a^{9}-\frac{197611}{40}a^{8}+\frac{142609}{40}a^{7}-\frac{55639}{40}a^{6}+\frac{3361}{40}a^{5}-\frac{4753}{40}a^{4}+489a^{3}-\frac{4411}{10}a^{2}+\frac{3463}{20}a-\frac{1021}{40}$, $\frac{81}{20}a^{17}-\frac{1439}{40}a^{16}+\frac{1283}{10}a^{15}-\frac{8759}{40}a^{14}+\frac{1195}{8}a^{13}-\frac{131}{40}a^{12}+\frac{1929}{8}a^{11}-\frac{2127}{2}a^{10}+\frac{7681}{4}a^{9}-\frac{42373}{20}a^{8}+\frac{61701}{40}a^{7}-\frac{12149}{20}a^{6}+\frac{239}{20}a^{5}-\frac{1191}{40}a^{4}+\frac{8503}{40}a^{3}-\frac{7857}{40}a^{2}+\frac{2841}{40}a-\frac{317}{40}$, $\frac{433}{40}a^{17}-\frac{3647}{40}a^{16}+\frac{12179}{40}a^{15}-\frac{4727}{10}a^{14}+\frac{2117}{8}a^{13}+\frac{151}{20}a^{12}+672a^{11}-\frac{50301}{20}a^{10}+\frac{21017}{5}a^{9}-\frac{176589}{40}a^{8}+\frac{30577}{10}a^{7}-\frac{2165}{2}a^{6}+\frac{577}{40}a^{5}-\frac{549}{4}a^{4}+\frac{18767}{40}a^{3}-\frac{7573}{20}a^{2}+\frac{5253}{40}a-\frac{717}{40}$, $\frac{81}{20}a^{17}-\frac{263}{8}a^{16}+\frac{207}{2}a^{15}-\frac{5701}{40}a^{14}+\frac{1929}{40}a^{13}+\frac{989}{40}a^{12}+\frac{10387}{40}a^{11}-\frac{1723}{2}a^{10}+\frac{6424}{5}a^{9}-\frac{24041}{20}a^{8}+\frac{28239}{40}a^{7}-\frac{519}{4}a^{6}-\frac{127}{2}a^{5}-\frac{567}{8}a^{4}+\frac{6117}{40}a^{3}-\frac{3531}{40}a^{2}+\frac{117}{8}a+\frac{31}{40}$, $a-1$, $\frac{16}{5}a^{17}-\frac{973}{40}a^{16}+\frac{349}{5}a^{15}-\frac{647}{8}a^{14}+\frac{167}{20}a^{13}+\frac{54}{5}a^{12}+\frac{4211}{20}a^{11}-\frac{22797}{40}a^{10}+\frac{3796}{5}a^{9}-656a^{8}+\frac{1717}{5}a^{7}-\frac{1067}{40}a^{6}-\frac{159}{10}a^{5}-\frac{2529}{40}a^{4}+\frac{1753}{20}a^{3}-40a^{2}+\frac{63}{10}a+\frac{2}{5}$, $\frac{11}{4}a^{16}-22a^{15}+\frac{271}{4}a^{14}-\frac{357}{4}a^{13}+\frac{189}{8}a^{12}+\frac{35}{2}a^{11}+\frac{1437}{8}a^{10}-564a^{9}+809a^{8}-732a^{7}+\frac{3299}{8}a^{6}-\frac{241}{4}a^{5}-\frac{147}{4}a^{4}-\frac{223}{4}a^{3}+\frac{759}{8}a^{2}-\frac{95}{2}a+\frac{69}{8}$ Copy content Toggle raw display
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:  \( 5331.83618131 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 5331.83618131 \cdot 1}{2\cdot\sqrt{7530328934072952860672}}\cr\approx \mathstrut & 0.468876690957 \end{aligned}\]

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 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*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 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*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 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*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 = polynomial_ring(QQ); K, a = number_field(x^18 - 9*x^17 + 33*x^16 - 60*x^15 + 50*x^14 - 14*x^13 + 62*x^12 - 268*x^11 + 523*x^10 - 635*x^9 + 523*x^8 - 268*x^7 + 62*x^6 - 14*x^5 + 50*x^4 - 60*x^3 + 33*x^2 - 9*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

$D_9$ (as 18T5):

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 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-107}) \), 3.1.107.1 x3, 6.0.1225043.1, 9.1.8389094464.2 x9

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 9 sibling: 9.1.8389094464.2
Minimal sibling: 9.1.8389094464.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.9.0.1}{9} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.2.0.1}{2} }^{9}$ ${\href{/padicField/11.9.0.1}{9} }^{2}$ ${\href{/padicField/13.9.0.1}{9} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.9.0.1}{9} }^{2}$ ${\href{/padicField/23.9.0.1}{9} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.2.0.1}{2} }^{9}$ ${\href{/padicField/37.9.0.1}{9} }^{2}$ ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.9.0.1}{9} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(107\) Copy content Toggle raw display 107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$C_2$$$[\ ]_{2}$$
107.1.2.1a1.1$x^{2} + 107$$2$$1$$1$$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)