Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231)
gp: K = bnfinit(y^9 - 333*y^7 - 630*y^6 + 30807*y^5 + 123228*y^4 - 600492*y^3 - 4304178*y^2 - 8369235*y - 5287231, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231)
\( x^{9} - 333x^{7} - 630x^{6} + 30807x^{5} + 123228x^{4} - 600492x^{3} - 4304178x^{2} - 8369235x - 5287231 \)
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: | $[9, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(22152259050635161449\)
\(\medspace = 3^{22}\cdot 163^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(141.09\) | 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: | $3^{22/9}163^{2/3}\approx 437.6109058848977$ | ||
| Ramified primes: |
\(3\), \(163\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{163}a^{6}-\frac{7}{163}a^{4}+\frac{22}{163}a^{3}$, $\frac{1}{163}a^{7}-\frac{7}{163}a^{5}+\frac{22}{163}a^{4}$, $\frac{1}{90\!\cdots\!17}a^{8}+\frac{14\!\cdots\!16}{90\!\cdots\!17}a^{7}+\frac{12\!\cdots\!94}{90\!\cdots\!17}a^{6}-\frac{39\!\cdots\!61}{90\!\cdots\!17}a^{5}-\frac{40\!\cdots\!79}{90\!\cdots\!17}a^{4}-\frac{15\!\cdots\!55}{90\!\cdots\!17}a^{3}-\frac{18\!\cdots\!23}{55\!\cdots\!59}a^{2}+\frac{26\!\cdots\!32}{55\!\cdots\!59}a+\frac{734951136682174}{55\!\cdots\!59}$
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 $1$ (assuming GRH)
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: | $8$ | 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: |
$\frac{134613360}{17998268979437}a^{8}-\frac{75553792428}{29\!\cdots\!31}a^{7}-\frac{7058344820770}{29\!\cdots\!31}a^{6}+\frac{10559774376540}{29\!\cdots\!31}a^{5}+\frac{644575934306046}{29\!\cdots\!31}a^{4}+\frac{467401036171305}{29\!\cdots\!31}a^{3}-\frac{94390932430584}{17998268979437}a^{2}-\frac{252993078168393}{17998268979437}a-\frac{166900227856119}{17998268979437}$, $\frac{134613360}{17998268979437}a^{8}-\frac{75553792428}{29\!\cdots\!31}a^{7}-\frac{7058344820770}{29\!\cdots\!31}a^{6}+\frac{10559774376540}{29\!\cdots\!31}a^{5}+\frac{644575934306046}{29\!\cdots\!31}a^{4}+\frac{467401036171305}{29\!\cdots\!31}a^{3}-\frac{94390932430584}{17998268979437}a^{2}-\frac{252993078168393}{17998268979437}a-\frac{148901958876682}{17998268979437}$, $\frac{384943811131327}{90\!\cdots\!17}a^{8}-\frac{12\!\cdots\!71}{90\!\cdots\!17}a^{7}-\frac{12\!\cdots\!81}{90\!\cdots\!17}a^{6}+\frac{15\!\cdots\!14}{90\!\cdots\!17}a^{5}+\frac{11\!\cdots\!72}{90\!\cdots\!17}a^{4}+\frac{64\!\cdots\!33}{55\!\cdots\!59}a^{3}-\frac{16\!\cdots\!66}{55\!\cdots\!59}a^{2}-\frac{48\!\cdots\!97}{55\!\cdots\!59}a-\frac{36\!\cdots\!52}{55\!\cdots\!59}$, $\frac{9022681396443}{90\!\cdots\!17}a^{8}+\frac{17865854287870}{90\!\cdots\!17}a^{7}-\frac{32\!\cdots\!11}{90\!\cdots\!17}a^{6}-\frac{86\!\cdots\!20}{90\!\cdots\!17}a^{5}+\frac{32\!\cdots\!67}{90\!\cdots\!17}a^{4}+\frac{66\!\cdots\!72}{55\!\cdots\!59}a^{3}-\frac{44\!\cdots\!30}{55\!\cdots\!59}a^{2}-\frac{21\!\cdots\!52}{55\!\cdots\!59}a-\frac{23\!\cdots\!50}{55\!\cdots\!59}$, $\frac{255701417609110}{90\!\cdots\!17}a^{8}-\frac{437444496179155}{90\!\cdots\!17}a^{7}-\frac{83\!\cdots\!00}{90\!\cdots\!17}a^{6}-\frac{84\!\cdots\!97}{90\!\cdots\!17}a^{5}+\frac{77\!\cdots\!09}{90\!\cdots\!17}a^{4}+\frac{15\!\cdots\!20}{90\!\cdots\!17}a^{3}-\frac{11\!\cdots\!11}{55\!\cdots\!59}a^{2}-\frac{43\!\cdots\!58}{55\!\cdots\!59}a-\frac{38\!\cdots\!59}{55\!\cdots\!59}$, $\frac{668706411884503}{90\!\cdots\!17}a^{8}-\frac{35\!\cdots\!49}{90\!\cdots\!17}a^{7}-\frac{20\!\cdots\!34}{90\!\cdots\!17}a^{6}+\frac{67\!\cdots\!36}{90\!\cdots\!17}a^{5}+\frac{17\!\cdots\!14}{90\!\cdots\!17}a^{4}-\frac{12\!\cdots\!02}{90\!\cdots\!17}a^{3}-\frac{23\!\cdots\!55}{55\!\cdots\!59}a^{2}-\frac{48\!\cdots\!91}{55\!\cdots\!59}a-\frac{24\!\cdots\!08}{55\!\cdots\!59}$, $\frac{52119367265601}{90\!\cdots\!17}a^{8}+\frac{23403638705038}{90\!\cdots\!17}a^{7}-\frac{14\!\cdots\!48}{90\!\cdots\!17}a^{6}-\frac{11\!\cdots\!57}{90\!\cdots\!17}a^{5}+\frac{11\!\cdots\!47}{90\!\cdots\!17}a^{4}+\frac{13\!\cdots\!75}{55\!\cdots\!59}a^{3}-\frac{15\!\cdots\!53}{55\!\cdots\!59}a^{2}-\frac{52\!\cdots\!84}{55\!\cdots\!59}a-\frac{42\!\cdots\!21}{55\!\cdots\!59}$, $\frac{10158489106173}{90\!\cdots\!17}a^{8}-\frac{12755455278979}{90\!\cdots\!17}a^{7}-\frac{30\!\cdots\!28}{90\!\cdots\!17}a^{6}-\frac{804998368052989}{90\!\cdots\!17}a^{5}+\frac{23\!\cdots\!44}{90\!\cdots\!17}a^{4}+\frac{26\!\cdots\!66}{55\!\cdots\!59}a^{3}-\frac{19\!\cdots\!37}{55\!\cdots\!59}a^{2}-\frac{62\!\cdots\!54}{55\!\cdots\!59}a-\frac{49\!\cdots\!04}{55\!\cdots\!59}$
| 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: | \( 4200335.96176 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 4200335.96176 \cdot 1}{2\cdot\sqrt{22152259050635161449}}\cr\approx \mathstrut & 0.228462530698 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231)
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 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231, 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 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231);
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 - 333*x^7 - 630*x^6 + 30807*x^5 + 123228*x^4 - 600492*x^3 - 4304178*x^2 - 8369235*x - 5287231);
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
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 27 |
| The 11 conjugacy class representatives for $C_9:C_3$ |
| Character table for $C_9:C_3$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
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 |
| 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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
|
\(163\)
| 163.3.0.1 | $x^{3} + 7 x + 161$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 163.3.2.3 | $x^{3} + 326$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 163.3.2.2 | $x^{3} + 489$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |