Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1)
gp: K = bnfinit(y^9 - 3*y^8 + 3*y^7 - 6*y^6 + 12*y^5 + 3*y^4 - 21*y^3 + 9*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1)
\( x^{9} - 3x^{8} + 3x^{7} - 6x^{6} + 12x^{5} + 3x^{4} - 21x^{3} + 9x^{2} + 1 \)
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: |
\(-98673100992\)
\(\medspace = -\,2^{6}\cdot 3^{7}\cdot 89^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.66\) | 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: |
\(2\), \(3\), \(89\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-267}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{123}a^{8}+\frac{50}{123}a^{7}-\frac{53}{123}a^{6}+\frac{14}{123}a^{5}+\frac{16}{123}a^{4}-\frac{10}{123}a^{3}-\frac{59}{123}a^{2}-\frac{43}{123}a+\frac{58}{123}$
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 $2$)
| 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: |
$a^{8}-3a^{7}+3a^{6}-6a^{5}+12a^{4}+3a^{3}-21a^{2}+9a$, $\frac{52}{123}a^{8}-\frac{106}{123}a^{7}+\frac{73}{123}a^{6}-\frac{256}{123}a^{5}+\frac{340}{123}a^{4}+\frac{464}{123}a^{3}-\frac{731}{123}a^{2}+\frac{224}{123}a-\frac{59}{123}$, $a-1$, $\frac{166}{123}a^{8}-\frac{433}{123}a^{7}+\frac{427}{123}a^{6}-\frac{997}{123}a^{5}+\frac{1672}{123}a^{4}+\frac{677}{123}a^{3}-\frac{2660}{123}a^{2}+\frac{1472}{123}a-\frac{212}{123}$, $\frac{329}{123}a^{8}-\frac{524}{123}a^{7}+\frac{275}{123}a^{6}-\frac{1667}{123}a^{5}+\frac{1697}{123}a^{4}+\frac{3229}{123}a^{3}-\frac{2191}{123}a^{2}+\frac{121}{123}a-\frac{352}{123}$
| 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: | \( 212.318104201 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 212.318104201 \cdot 1}{2\cdot\sqrt{98673100992}}\cr\approx \mathstrut & 0.670636575693 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 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 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 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 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 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 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 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
$C_3^2:\GL(2,3)$ (as 9T26):
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 432 |
| The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$ |
| Character table for $((C_3^2:Q_8):C_3):C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
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 | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
|
\(89\)
| $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 89.2.0.1 | $x^{2} + 82 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.1.1 | $x^{2} + 89$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.267.2t1.a.a | $1$ | $ 3 \cdot 89 $ | \(\Q(\sqrt{-267}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.1068.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 89 $ | 3.1.1068.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.3204.24t22.d.a | $2$ | $ 2^{2} \cdot 3^{2} \cdot 89 $ | 8.2.24668275248.4 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.3204.24t22.d.b | $2$ | $ 2^{2} \cdot 3^{2} \cdot 89 $ | 8.2.24668275248.4 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.9612.4t5.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 89 $ | 4.2.9612.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.285156.6t8.a.a | $3$ | $ 2^{2} \cdot 3^{2} \cdot 89^{2}$ | 4.2.9612.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 4.2566404.8t23.d.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 89^{2}$ | 8.2.24668275248.4 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
| * | 8.98673100992.9t26.a.a | $8$ | $ 2^{6} \cdot 3^{7} \cdot 89^{3}$ | 9.3.98673100992.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
| 8.781...632.18t157.a.a | $8$ | $ 2^{6} \cdot 3^{7} \cdot 89^{5}$ | 9.3.98673100992.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
| 16.192...736.24t1334.a.a | $16$ | $ 2^{10} \cdot 3^{14} \cdot 89^{8}$ | 9.3.98673100992.1 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |
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 *.