Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64)
gp: K = bnfinit(y^9 - 3*y^8 + 9*y^7 + 21*y^6 - 6*y^5 + 72*y^4 + 33*y^3 + 60*y^2 + 64, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 3*x^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64)
\( x^{9} - 3x^{8} + 9x^{7} + 21x^{6} - 6x^{5} + 72x^{4} + 33x^{3} + 60x^{2} + 64 \)
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: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(32257648686537\)
\(\medspace = 3^{15}\cdot 131^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(31.69\) | 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: | $3^{31/18}131^{1/2}\approx 75.91795052547229$ | ||
| Ramified primes: |
\(3\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{393}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{68}a^{7}+\frac{5}{68}a^{6}+\frac{33}{68}a^{5}+\frac{1}{68}a^{4}+\frac{9}{34}a^{3}-\frac{1}{17}a^{2}-\frac{15}{68}a+\frac{1}{17}$, $\frac{1}{337552}a^{8}+\frac{1905}{337552}a^{7}-\frac{78323}{337552}a^{6}+\frac{95273}{337552}a^{5}-\frac{79825}{168776}a^{4}-\frac{8764}{21097}a^{3}+\frac{131377}{337552}a^{2}-\frac{8360}{21097}a-\frac{1548}{21097}$
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
$C_{6}$, which has order $6$
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: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{2679}{168776}a^{8}-\frac{11907}{168776}a^{7}+\frac{38193}{168776}a^{6}+\frac{14789}{168776}a^{5}-\frac{9511}{21097}a^{4}+\frac{69629}{42194}a^{3}-\frac{236817}{168776}a^{2}+\frac{121955}{84388}a-\frac{8027}{21097}$, $\frac{1}{4624}a^{8}+\frac{1}{4624}a^{7}+\frac{13}{4624}a^{6}+\frac{73}{4624}a^{5}+\frac{143}{2312}a^{4}+\frac{76}{289}a^{3}+\frac{273}{4624}a^{2}+\frac{72}{289}a-\frac{1}{289}$, $\frac{683}{168776}a^{8}-\frac{11863}{168776}a^{7}+\frac{24757}{168776}a^{6}-\frac{28919}{168776}a^{5}-\frac{38997}{21097}a^{4}-\frac{20491}{42194}a^{3}-\frac{207261}{168776}a^{2}-\frac{51193}{84388}a-\frac{28447}{21097}$, $\frac{8623}{168776}a^{8}-\frac{16435}{168776}a^{7}+\frac{39985}{168776}a^{6}+\frac{262653}{168776}a^{5}+\frac{17585}{21097}a^{4}+\frac{45773}{42194}a^{3}+\frac{158495}{168776}a^{2}+\frac{119247}{84388}a-\frac{2898}{21097}$
| 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: | \( 397.318629505 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 397.318629505 \cdot 6}{2\cdot\sqrt{32257648686537}}\cr\approx \mathstrut & 0.654174029380 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64)
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^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64, 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^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64);
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^8 + 9*x^7 + 21*x^6 - 6*x^5 + 72*x^4 + 33*x^3 + 60*x^2 + 64);
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
$S_3\wr S_3$ (as 9T31):
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 1296 |
| The 22 conjugacy class representatives for $S_3\wr S_3$ |
| Character table for $S_3\wr S_3$ is not computed |
Intermediate fields
| 3.1.10611.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.15.29 | $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 12$ | $9$ | $1$ | $15$ | $S_3\times C_3$ | $[3/2, 2]_{2}$ |
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.4.2.1 | $x^{4} + 254 x^{3} + 16395 x^{2} + 33782 x + 2129540$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.131.2t1.a.a | $1$ | $ 131 $ | \(\Q(\sqrt{-131}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.393.2t1.a.a | $1$ | $ 3 \cdot 131 $ | \(\Q(\sqrt{393}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.10611.6t3.a.a | $2$ | $ 3^{4} \cdot 131 $ | 6.2.44249175153.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.10611.3t2.a.a | $2$ | $ 3^{4} \cdot 131 $ | 3.1.10611.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.10611.4t5.a.a | $3$ | $ 3^{4} \cdot 131 $ | 4.2.10611.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.4170123.6t11.a.a | $3$ | $ 3^{5} \cdot 131^{2}$ | 6.0.337779963.3 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.1390041.6t8.b.a | $3$ | $ 3^{4} \cdot 131^{2}$ | 4.2.10611.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.31833.6t11.a.a | $3$ | $ 3^{5} \cdot 131 $ | 6.0.337779963.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
| * | 6.3040019667.9t31.a.a | $6$ | $ 3^{11} \cdot 131^{2}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
| 6.521...387.18t300.a.a | $6$ | $ 3^{11} \cdot 131^{4}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 6.521...387.18t319.a.a | $6$ | $ 3^{11} \cdot 131^{4}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 6.3040019667.18t311.a.a | $6$ | $ 3^{11} \cdot 131^{2}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 8.268...921.24t2893.a.a | $8$ | $ 3^{12} \cdot 131^{6}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 8.9120059001.12t213.a.a | $8$ | $ 3^{12} \cdot 131^{2}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.230...211.36t2217.a.a | $12$ | $ 3^{20} \cdot 131^{7}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
| 12.134...051.36t2214.a.a | $12$ | $ 3^{20} \cdot 131^{5}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
| 12.158...129.36t2210.a.a | $12$ | $ 3^{22} \cdot 131^{6}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.230...211.36t2216.a.a | $12$ | $ 3^{20} \cdot 131^{7}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-2$ | |
| 12.134...051.18t315.a.a | $12$ | $ 3^{20} \cdot 131^{5}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
| 16.178...409.24t2912.a.a | $16$ | $ 3^{30} \cdot 131^{8}$ | 9.1.32257648686537.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
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 *.