Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149)
gp: K = bnfinit(y^9 - 27*y^7 - 3*y^6 + 198*y^5 + 549*y^4 + 2244*y^3 + 2592*y^2 + 1053*y + 149, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149)
\( x^{9} - 27x^{7} - 3x^{6} + 198x^{5} + 549x^{4} + 2244x^{3} + 2592x^{2} + 1053x + 149 \)
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: |
\(-2136545302048171875\)
\(\medspace = -\,3^{19}\cdot 5^{6}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(108.80\) | 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: | $3^{53/24}5^{2/3}7^{3/4}\approx 142.37911979255588$ | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{1301601839}a^{8}+\frac{168168278}{1301601839}a^{7}+\frac{149390689}{1301601839}a^{6}-\frac{563364392}{1301601839}a^{5}+\frac{68197811}{1301601839}a^{4}+\frac{473795393}{1301601839}a^{3}+\frac{539045543}{1301601839}a^{2}-\frac{17292263}{1301601839}a+\frac{102021281}{1301601839}$
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: | $5$ | 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: |
$\frac{2553388}{1301601839}a^{8}+\frac{416339764}{1301601839}a^{7}-\frac{1550342203}{1301601839}a^{6}-\frac{8288169017}{1301601839}a^{5}+\frac{29305443511}{1301601839}a^{4}+\frac{55197747299}{1301601839}a^{3}+\frac{31778724880}{1301601839}a^{2}+\frac{9493560226}{1301601839}a+\frac{1227398085}{1301601839}$, $\frac{313946820}{1301601839}a^{8}+\frac{813803727}{1301601839}a^{7}-\frac{12269304564}{1301601839}a^{6}-\frac{7686740403}{1301601839}a^{5}+\frac{101431425558}{1301601839}a^{4}+\frac{175635036540}{1301601839}a^{3}+\frac{1029565623723}{1301601839}a^{2}+\frac{1066142737581}{1301601839}a+\frac{262602128507}{1301601839}$, $\frac{351697282}{1301601839}a^{8}-\frac{505888598}{1301601839}a^{7}-\frac{8268251539}{1301601839}a^{6}+\frac{8692910778}{1301601839}a^{5}+\frac{51390549825}{1301601839}a^{4}+\frac{147768353173}{1301601839}a^{3}+\frac{568665372524}{1301601839}a^{2}+\frac{325682727515}{1301601839}a+\frac{56107943819}{1301601839}$, $\frac{279973446}{1301601839}a^{8}-\frac{988732518}{1301601839}a^{7}-\frac{4095237813}{1301601839}a^{6}+\frac{13704234197}{1301601839}a^{5}+\frac{7822511591}{1301601839}a^{4}+\frac{126438533246}{1301601839}a^{3}+\frac{183242502634}{1301601839}a^{2}+\frac{79358318170}{1301601839}a+\frac{11159434128}{1301601839}$, $\frac{5740170069}{1301601839}a^{8}-\frac{3915763179}{1301601839}a^{7}-\frac{184981732497}{1301601839}a^{6}+\frac{114134103154}{1301601839}a^{5}+\frac{2114144331073}{1301601839}a^{4}+\frac{1612961741371}{1301601839}a^{3}-\frac{301311435295}{1301601839}a^{2}-\frac{431526217063}{1301601839}a-\frac{84060122352}{1301601839}$
| 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: | \( 1257376.6678 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1257376.6678 \cdot 1}{2\cdot\sqrt{2136545302048171875}}\cr\approx \mathstrut & 0.85351064247 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149)
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 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149, 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 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149);
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 - 27*x^7 - 3*x^6 + 198*x^5 + 549*x^4 + 2244*x^3 + 2592*x^2 + 1053*x + 149);
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 144 |
| The 9 conjugacy class representatives for $(C_3^2:C_8):C_2$ |
| Character table for $(C_3^2:C_8):C_2$ |
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 siblings: | data not computed |
| Degree 24 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | R | R | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.19.57 | $x^{9} + 18 x^{3} + 18 x^{2} + 21$ | $9$ | $1$ | $19$ | $(C_3^2:C_8):C_2$ | $[19/8, 19/8]_{8}^{2}$ |
|
\(5\)
| 5.9.6.1 | $x^{9} + 9 x^{7} + 24 x^{6} + 27 x^{5} + 9 x^{4} - 186 x^{3} + 216 x^{2} - 504 x + 647$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.8.6.1 | $x^{8} + 14 x^{4} - 245$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |