Properties

Label 9.3.49633171875.1
Degree $9$
Signature $[3, 3]$
Discriminant $-49633171875$
Root discriminant \(15.43\)
Ramified primes $3,5,7$
Class number $1$
Class group trivial
Galois group $S_3\times C_3$ (as 9T4)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*x^2 - 4*x - 1)
 
gp: K = bnfinit(y^9 - 4*y^7 - 3*y^6 - 18*y^5 + 8*y^4 - 24*y^3 + 18*y^2 - 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*x^2 - 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*x^2 - 4*x - 1)
 

\( x^{9} - 4x^{7} - 3x^{6} - 18x^{5} + 8x^{4} - 24x^{3} + 18x^{2} - 4x - 1 \) 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:   \(-49633171875\) \(\medspace = -\,3^{3}\cdot 5^{6}\cdot 7^{6}\) 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:  \(15.43\)
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:  $3^{1/2}5^{2/3}7^{2/3}\approx 18.532726798013343$
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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{4}{9}a^{4}+\frac{2}{9}a^{3}-\frac{2}{9}a^{2}-\frac{4}{9}$, $\frac{1}{387}a^{8}+\frac{5}{129}a^{7}+\frac{49}{387}a^{6}+\frac{1}{387}a^{5}-\frac{89}{387}a^{4}+\frac{2}{129}a^{3}-\frac{20}{387}a^{2}+\frac{191}{387}a+\frac{152}{387}$ 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:  $5$
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:   $\frac{47}{387}a^{8}-\frac{26}{387}a^{7}-\frac{191}{387}a^{6}+\frac{4}{387}a^{5}-\frac{743}{387}a^{4}+\frac{626}{387}a^{3}-\frac{157}{43}a^{2}+\frac{1108}{387}a-\frac{170}{129}$, $\frac{47}{387}a^{8}-\frac{26}{387}a^{7}-\frac{191}{387}a^{6}+\frac{4}{387}a^{5}-\frac{743}{387}a^{4}+\frac{626}{387}a^{3}-\frac{157}{43}a^{2}+\frac{1108}{387}a-\frac{41}{129}$, $\frac{71}{387}a^{8}+\frac{119}{387}a^{7}-\frac{73}{129}a^{6}-\frac{220}{129}a^{5}-\frac{210}{43}a^{4}-\frac{1853}{387}a^{3}-\frac{1979}{387}a^{2}-\frac{1661}{387}a+\frac{386}{387}$, $\frac{4}{387}a^{8}+\frac{17}{387}a^{7}-\frac{19}{387}a^{6}-\frac{82}{387}a^{5}-\frac{55}{387}a^{4}-\frac{320}{387}a^{3}-\frac{41}{129}a^{2}-\frac{10}{387}a-\frac{41}{129}$, $\frac{97}{387}a^{8}+\frac{79}{387}a^{7}-\frac{107}{129}a^{6}-\frac{197}{129}a^{5}-\frac{742}{129}a^{4}-\frac{880}{387}a^{3}-\frac{3187}{387}a^{2}-\frac{49}{387}a-\frac{5}{387}$ 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:  \( 53.7202008145 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 53.7202008145 \cdot 1}{2\cdot\sqrt{49633171875}}\cr\approx \mathstrut & 0.239249593597 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 4*x^7 - 3*x^6 - 18*x^5 + 8*x^4 - 24*x^3 + 18*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

$C_3\times S_3$ (as 9T4):

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 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.3675.1

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

Galois closure: data not computed
Degree 6 sibling: 6.0.826875.2
Minimal sibling: 6.0.826875.2

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 ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.1.0.1}{1} }^{9}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(3\) Copy content Toggle raw display 3.3.0.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(5\) Copy content Toggle raw display 5.9.6.1$x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
\(7\) Copy content Toggle raw display 7.9.6.1$x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.21.6t1.a.a$1$ $ 3 \cdot 7 $ 6.0.64827.1 $C_6$ (as 6T1) $0$ $-1$
1.21.6t1.a.b$1$ $ 3 \cdot 7 $ 6.0.64827.1 $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 2.3675.3t2.a.a$2$ $ 3 \cdot 5^{2} \cdot 7^{2}$ 3.1.3675.1 $S_3$ (as 3T2) $1$ $0$
* 2.525.6t5.a.a$2$ $ 3 \cdot 5^{2} \cdot 7 $ 9.3.49633171875.1 $S_3\times C_3$ (as 9T4) $0$ $0$
* 2.525.6t5.a.b$2$ $ 3 \cdot 5^{2} \cdot 7 $ 9.3.49633171875.1 $S_3\times C_3$ (as 9T4) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.