Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*x - 1)
gp: K = bnfinit(y^9 - y^8 + 2*y^7 - 4*y^2 + 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*x - 1)
\( x^{9} - x^{8} + 2x^{7} - 4x^{2} + 4x - 1 \)
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: |
\(-962721767\)
\(\medspace = -\,7^{8}\cdot 167\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.96\) | 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: |
\(7\), \(167\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-167}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: |
$a^{8}-a^{7}+2a^{6}-4a+4$, $9a^{8}-3a^{7}+16a^{6}+11a^{5}+7a^{4}+5a^{3}+4a^{2}-34a+14$, $a^{7}+2a^{5}+2a^{4}+2a^{3}+2a^{2}+a-2$, $a^{8}-a^{7}+2a^{6}-4a+3$, $8a^{8}-3a^{7}+14a^{6}+9a^{5}+5a^{4}+4a^{3}+2a^{2}-30a+13$
| 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: | \( 7.77137506101 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 7.77137506101 \cdot 1}{2\cdot\sqrt{962721767}}\cr\approx \mathstrut & 0.248511831519 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 2*x^7 - 4*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 - x^8 + 2*x^7 - 4*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 - x^8 + 2*x^7 - 4*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 - x^8 + 2*x^7 - 4*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
$S_3\wr C_3$ (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
| \(\Q(\zeta_{7})^+\) |
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 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
|
\(167\)
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.2.0.1 | $x^{2} + 166 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 167.2.1.1 | $x^{2} + 835$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.167.2t1.a.a | $1$ | $ 167 $ | \(\Q(\sqrt{-167}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.1169.6t1.b.a | $1$ | $ 7 \cdot 167 $ | 6.0.11182568663.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.1169.6t1.b.b | $1$ | $ 7 \cdot 167 $ | 6.0.11182568663.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 3.8183.6t6.a.a | $3$ | $ 7^{2} \cdot 167 $ | 6.0.400967.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-3$ | |
| 3.1366561.4t4.a.a | $3$ | $ 7^{2} \cdot 167^{2}$ | 4.4.1366561.1 | $A_4$ (as 4T4) | $1$ | $3$ | |
| 6.547945864487.18t197.a.a | $6$ | $ 7^{6} \cdot 167^{3}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
| * | 6.19647383.9t28.a.a | $6$ | $ 7^{6} \cdot 167 $ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ |
| 6.152...943.18t202.a.a | $6$ | $ 7^{6} \cdot 167^{5}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
| 6.547945864487.18t197.b.a | $6$ | $ 7^{6} \cdot 167^{3}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
| 8.448...121.12t176.a.a | $8$ | $ 7^{8} \cdot 167^{4}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
| 8.640...303.24t1539.a.a | $8$ | $ 7^{7} \cdot 167^{4}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
| 8.640...303.24t1539.a.b | $8$ | $ 7^{7} \cdot 167^{4}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $0$ | $0$ | |
| 12.219...929.18t206.a.a | $12$ | $ 7^{10} \cdot 167^{4}$ | 9.3.962721767.1 | $S_3 \wr C_3 $ (as 9T28) | $1$ | $0$ | |
| 12.170...209.36t1101.a.a | $12$ | $ 7^{10} \cdot 167^{8}$ | 9.3.962721767.1 | $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 *.