Properties

Label 9.3.69534993539.1
Degree $9$
Signature $[3, 3]$
Discriminant $-69534993539$
Root discriminant \(16.02\)
Ramified primes $11,43$
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 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4)
 
gp: K = bnfinit(y^9 - 3*y^7 - 4*y^6 + 3*y^5 + 8*y^4 - 10*y^3 - 4*y^2 + 9*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4)
 

\( x^{9} - 3x^{7} - 4x^{6} + 3x^{5} + 8x^{4} - 10x^{3} - 4x^{2} + 9x + 4 \) 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:   \(-69534993539\) \(\medspace = -\,11\cdot 43^{6}\) 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:  \(16.02\)
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))
 
Ramified primes:   \(11\), \(43\) 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:  \(\Q(\sqrt{-11}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$ 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:   $12a^{8}-7a^{7}-32a^{6}-\frac{59}{2}a^{5}+\frac{107}{2}a^{4}+\frac{131}{2}a^{3}-158a^{2}+\frac{87}{2}a+83$, $\frac{21}{2}a^{8}-6a^{7}-28a^{6}-26a^{5}+46a^{4}+57a^{3}-137a^{2}+\frac{77}{2}a+73$, $\frac{41}{2}a^{8}-12a^{7}-\frac{109}{2}a^{6}-50a^{5}+91a^{4}+\frac{221}{2}a^{3}-\frac{541}{2}a^{2}+76a+141$, $a^{2}-a-1$, $22a^{8}-\frac{25}{2}a^{7}-\frac{117}{2}a^{6}-\frac{109}{2}a^{5}+96a^{4}+\frac{239}{2}a^{3}-288a^{2}+77a+151$ 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:  \( 186.606756006 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 186.606756006 \cdot 1}{2\cdot\sqrt{69534993539}}\cr\approx \mathstrut & 0.702142482130 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4)
 
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 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4, 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 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4);
 
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 - 3*x^7 - 4*x^6 + 3*x^5 + 8*x^4 - 10*x^3 - 4*x^2 + 9*x + 4);
 
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

3.3.1849.1

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 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 ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ R ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} + 22$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
\(43\) Copy content Toggle raw display 43.3.2.1$x^{3} + 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.4.1$x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
1.473.6t1.b.a$1$ $ 11 \cdot 43 $ 6.0.4550424131.1 $C_6$ (as 6T1) $0$ $-1$
* 1.43.3t1.a.a$1$ $ 43 $ 3.3.1849.1 $C_3$ (as 3T1) $0$ $1$
* 1.43.3t1.a.b$1$ $ 43 $ 3.3.1849.1 $C_3$ (as 3T1) $0$ $1$
1.473.6t1.b.b$1$ $ 11 \cdot 43 $ 6.0.4550424131.1 $C_6$ (as 6T1) $0$ $-1$
3.20339.6t6.a.a$3$ $ 11 \cdot 43^{2}$ 6.0.37606811.1 $A_4\times C_2$ (as 6T6) $1$ $-3$
3.223729.4t4.a.a$3$ $ 11^{2} \cdot 43^{2}$ 4.4.223729.1 $A_4$ (as 4T4) $1$ $3$
6.4550424131.18t197.a.a$6$ $ 11^{3} \cdot 43^{4}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
* 6.37606811.9t28.a.a$6$ $ 11 \cdot 43^{4}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
6.550601319851.18t202.a.a$6$ $ 11^{5} \cdot 43^{4}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
6.4550424131.18t197.b.a$6$ $ 11^{3} \cdot 43^{4}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
8.50054665441.12t176.a.a$8$ $ 11^{4} \cdot 43^{4}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
8.925...409.24t1539.a.a$8$ $ 11^{4} \cdot 43^{6}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $0$ $0$
8.925...409.24t1539.a.b$8$ $ 11^{4} \cdot 43^{6}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $0$ $0$
12.171...241.18t206.a.a$12$ $ 11^{4} \cdot 43^{8}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$
12.250...481.36t1101.a.a$12$ $ 11^{8} \cdot 43^{8}$ 9.3.69534993539.1 $S_3 \wr C_3 $ (as 9T28) $1$ $0$

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 *.