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Properties

Label 9.3.3230456496696835875.4
Degree $9$
Signature $[3, 3]$
Discriminant $-3.230\times 10^{18}$
Root discriminant \(113.92\)
Ramified primes $3,5,7$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_3 \wr S_3 $ (as 9T20)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421)
 
gp: K = bnfinit(y^9 - 63*y^6 + 378*y^5 - 1008*y^4 - 7266*y^3 + 29106*y^2 - 28224*y - 1421, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421)
 

\( x^{9} - 63x^{6} + 378x^{5} - 1008x^{4} - 7266x^{3} + 29106x^{2} - 28224x - 1421 \) 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:  $[3, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-3230456496696835875\) \(\medspace = -\,3^{22}\cdot 5^{3}\cdot 7^{7}\) 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:  \(113.92\)
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:  $3^{22/9}5^{1/2}7^{5/6}\approx 165.97006698500886$
Ramified primes:   \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{-35}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{35}a^{6}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{17605}a^{7}-\frac{33}{3521}a^{6}+\frac{384}{2515}a^{5}-\frac{1003}{2515}a^{4}+\frac{1056}{2515}a^{3}+\frac{198}{503}a^{2}-\frac{946}{2515}a-\frac{92}{503}$, $\frac{1}{4560487225}a^{8}+\frac{3617}{651498175}a^{7}-\frac{10446414}{4560487225}a^{6}-\frac{128084932}{651498175}a^{5}+\frac{175521896}{651498175}a^{4}-\frac{8842636}{26059927}a^{3}-\frac{16402078}{651498175}a^{2}+\frac{111027656}{651498175}a-\frac{105406893}{651498175}$ 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

$C_{3}\times C_{3}$, which has order $9$ (assuming GRH)

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:  $5$
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:   $\frac{7668}{8855315}a^{8}+\frac{13242}{8855315}a^{7}+\frac{4336}{1771063}a^{6}-\frac{64116}{1265045}a^{5}+\frac{310476}{1265045}a^{4}-\frac{551396}{1265045}a^{3}-\frac{9281412}{1265045}a^{2}+\frac{15705648}{1265045}a+\frac{791209}{1265045}$, $\frac{682241}{651498175}a^{8}+\frac{3129729}{651498175}a^{7}+\frac{11659121}{651498175}a^{6}+\frac{15268341}{651498175}a^{5}+\frac{222194087}{651498175}a^{4}+\frac{95551856}{130299635}a^{3}-\frac{2997098096}{651498175}a^{2}+\frac{626224397}{651498175}a+\frac{5376922269}{651498175}$, $\frac{1078008}{4560487225}a^{8}+\frac{2398577}{4560487225}a^{7}+\frac{8784444}{651498175}a^{6}-\frac{20431831}{651498175}a^{5}-\frac{2291052}{651498175}a^{4}-\frac{214342047}{130299635}a^{3}+\frac{3770457371}{651498175}a^{2}-\frac{3526570127}{651498175}a-\frac{177595889}{651498175}$, $\frac{11083426}{4560487225}a^{8}+\frac{3772429}{4560487225}a^{7}+\frac{3565993}{651498175}a^{6}-\frac{99918367}{651498175}a^{5}+\frac{505843776}{651498175}a^{4}-\frac{364078467}{130299635}a^{3}-\frac{10442545388}{651498175}a^{2}+\frac{46512814496}{651498175}a-\frac{48339409583}{651498175}$, $\frac{2503203}{4560487225}a^{8}+\frac{3755597}{4560487225}a^{7}-\frac{1361631}{651498175}a^{6}-\frac{13852761}{651498175}a^{5}+\frac{68147018}{651498175}a^{4}-\frac{6387952}{130299635}a^{3}-\frac{3437359584}{651498175}a^{2}+\frac{9104768178}{651498175}a+\frac{445197246}{651498175}$ Copy content Toggle raw display (assuming GRH)
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:  \( 244105.73301 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 244105.73301 \cdot 9}{2\cdot\sqrt{3230456496696835875}}\cr\approx \mathstrut & 1.2127970632 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421)
 
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 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421, 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 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421);
 
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 - 63*x^6 + 378*x^5 - 1008*x^4 - 7266*x^3 + 29106*x^2 - 28224*x - 1421);
 
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

$C_3\wr S_3$ (as 9T20):

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 162
The 22 conjugacy class representatives for $C_3 \wr S_3 $
Character table for $C_3 \wr S_3 $

Intermediate fields

3.1.2835.1

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 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 siblings: data not computed
Minimal sibling: 9.3.3230456496696835875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ R R R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$

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
\(3\) 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 5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} + 3 x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
\(7\) Copy content Toggle raw display 7.3.2.3$x^{3} + 21$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} + 35$$6$$1$$5$$C_6$$[\ ]_{6}$