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Properties

Label 9.5.2008387814976000.3
Degree $9$
Signature $[5, 2]$
Discriminant $2.008\times 10^{15}$
Root discriminant \(50.16\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $S_3 \wr C_3 $ (as 9T28)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008)
 
gp: K = bnfinit(y^9 - 18*y^7 - 54*y^6 + 81*y^5 + 522*y^4 + 357*y^3 - 1080*y^2 - 3564*y + 2008, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008)
 

\( x^{9} - 18x^{7} - 54x^{6} + 81x^{5} + 522x^{4} + 357x^{3} - 1080x^{2} - 3564x + 2008 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[5, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2008387814976000\) \(\medspace = 2^{9}\cdot 3^{22}\cdot 5^{3}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(50.16\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{9/4}3^{22/9}5^{1/2}\approx 155.99076676750158$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{10}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{1175825956}a^{8}-\frac{70559971}{587912978}a^{7}+\frac{53650404}{293956489}a^{6}-\frac{79710053}{587912978}a^{5}+\frac{13367339}{1175825956}a^{4}+\frac{26895607}{587912978}a^{3}-\frac{562100743}{1175825956}a^{2}-\frac{240675075}{587912978}a+\frac{30561918}{293956489}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{681615}{293956489}a^{8}+\frac{3823923}{587912978}a^{7}-\frac{8001450}{293956489}a^{6}-\frac{61696917}{293956489}a^{5}-\frac{96560559}{293956489}a^{4}+\frac{292784747}{587912978}a^{3}+\frac{710692875}{293956489}a^{2}+\frac{1932323931}{587912978}a-\frac{235247616}{293956489}$, $\frac{681615}{293956489}a^{8}+\frac{3823923}{587912978}a^{7}-\frac{8001450}{293956489}a^{6}-\frac{61696917}{293956489}a^{5}-\frac{96560559}{293956489}a^{4}+\frac{292784747}{587912978}a^{3}+\frac{710692875}{293956489}a^{2}+\frac{1932323931}{587912978}a-\frac{529204105}{293956489}$, $\frac{2147633}{1175825956}a^{8}-\frac{467231}{587912978}a^{7}-\frac{24016751}{587912978}a^{6}-\frac{27233487}{587912978}a^{5}+\frac{347642847}{1175825956}a^{4}+\frac{162464599}{293956489}a^{3}-\frac{1458657777}{1175825956}a^{2}-\frac{53696473}{293956489}a+\frac{2950218}{293956489}$, $\frac{4635397}{587912978}a^{8}+\frac{4358017}{587912978}a^{7}-\frac{63555387}{587912978}a^{6}-\frac{321090405}{587912978}a^{5}-\frac{165217727}{587912978}a^{4}+\frac{1213293977}{587912978}a^{3}+\frac{3198616537}{587912978}a^{2}+\frac{910161959}{293956489}a-\frac{2301073049}{293956489}$, $\frac{187448861}{1175825956}a^{8}-\frac{342002057}{587912978}a^{7}-\frac{880385679}{587912978}a^{6}-\frac{821013864}{293956489}a^{5}+\frac{36753563779}{1175825956}a^{4}+\frac{4864159821}{587912978}a^{3}+\frac{4917581289}{1175825956}a^{2}-\frac{87583255098}{293956489}a+\frac{43320588238}{293956489}$, $\frac{54763503}{1175825956}a^{8}+\frac{35245548}{293956489}a^{7}-\frac{170759695}{293956489}a^{6}-\frac{1123448062}{293956489}a^{5}-\frac{6941735831}{1175825956}a^{4}+\frac{5149977701}{587912978}a^{3}+\frac{49235853805}{1175825956}a^{2}+\frac{16509159215}{293956489}a-\frac{2972212272}{293956489}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 25182.0355995 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 25182.0355995 \cdot 1}{2\cdot\sqrt{2008387814976000}}\cr\approx \mathstrut & 0.354933313939 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 18*x^7 - 54*x^6 + 81*x^5 + 522*x^4 + 357*x^3 - 1080*x^2 - 3564*x + 2008);
 
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

$S_3\wr C_3$ (as 9T28):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 648
The 17 conjugacy class representatives for $S_3 \wr C_3 $
Character table for $S_3 \wr C_3 $

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 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 R ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.4$x^{6} + 4 x^{5} + 14 x^{4} + 224 x^{3} + 1244 x^{2} + 3184 x + 2376$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
\(3\) Copy content Toggle raw display 3.9.22.11$x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$$9$$1$$22$$C_9:C_3$$[2, 3]^{3}$
\(5\) Copy content Toggle raw display 5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$