Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029)
gp: K = bnfinit(y^9 - 315*y^7 - 702*y^6 + 32238*y^5 + 127161*y^4 - 1030422*y^3 - 5346135*y^2 + 2138454*y + 24909029, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029)
\( x^{9} - 315x^{7} - 702x^{6} + 32238x^{5} + 127161x^{4} - 1030422x^{3} - 5346135x^{2} + 2138454x + 24909029 \)
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: |
\(49213007847751532409\)
\(\medspace = 3^{22}\cdot 199^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(154.17\) | 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: | $3^{22/9}199^{2/3}\approx 499.8792597608661$ | ||
| Ramified primes: |
\(3\), \(199\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{199}a^{6}+\frac{83}{199}a^{4}+\frac{94}{199}a^{3}$, $\frac{1}{3383}a^{7}+\frac{1}{3383}a^{6}+\frac{83}{3383}a^{5}+\frac{1172}{3383}a^{4}-\frac{1299}{3383}a^{3}-\frac{1}{17}a^{2}-\frac{6}{17}a$, $\frac{1}{34\!\cdots\!31}a^{8}-\frac{24099013812}{34\!\cdots\!31}a^{7}-\frac{6610528816368}{34\!\cdots\!31}a^{6}+\frac{635651486632785}{34\!\cdots\!31}a^{5}+\frac{850431756962977}{34\!\cdots\!31}a^{4}+\frac{13\!\cdots\!08}{34\!\cdots\!31}a^{3}-\frac{7114158829576}{17162297392769}a^{2}+\frac{2370752525197}{17162297392769}a+\frac{95554460632}{1009546905457}$
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: |
$\frac{325836}{87118260877}a^{8}-\frac{535860879}{17336533914523}a^{7}-\frac{16051498784}{17336533914523}a^{6}+\frac{87825179718}{17336533914523}a^{5}+\frac{1370444720574}{17336533914523}a^{4}-\frac{3135491666967}{17336533914523}a^{3}-\frac{202781601504}{87118260877}a^{2}-\frac{66901358793}{87118260877}a+\frac{46098816657}{5124603581}$, $\frac{43173}{5124603581}a^{8}-\frac{1264346019}{17336533914523}a^{7}-\frac{35209668103}{17336533914523}a^{6}+\frac{205995979647}{17336533914523}a^{5}+\frac{2957911626711}{17336533914523}a^{4}-\frac{7775914692002}{17336533914523}a^{3}-\frac{435330069750}{87118260877}a^{2}+\frac{27733814253}{87118260877}a+\frac{119919476683}{5124603581}$, $\frac{218274328044}{34\!\cdots\!31}a^{8}+\frac{645825435679}{34\!\cdots\!31}a^{7}-\frac{65470427279595}{34\!\cdots\!31}a^{6}-\frac{319739130013200}{34\!\cdots\!31}a^{5}+\frac{59\!\cdots\!06}{34\!\cdots\!31}a^{4}+\frac{204295385237595}{17162297392769}a^{3}-\frac{620967876035175}{17162297392769}a^{2}-\frac{408804442283089}{1009546905457}a-\frac{736183316432335}{1009546905457}$, $\frac{137760147714}{34\!\cdots\!31}a^{8}-\frac{737373789598}{34\!\cdots\!31}a^{7}-\frac{39686051695233}{34\!\cdots\!31}a^{6}+\frac{118099120674714}{34\!\cdots\!31}a^{5}+\frac{38\!\cdots\!80}{34\!\cdots\!31}a^{4}-\frac{18603479448507}{17162297392769}a^{3}-\frac{663502257335724}{17162297392769}a^{2}-\frac{124755577485853}{17162297392769}a+\frac{246585147550160}{1009546905457}$, $\frac{2980747108285}{34\!\cdots\!31}a^{8}-\frac{25199808537851}{34\!\cdots\!31}a^{7}-\frac{731449168397307}{34\!\cdots\!31}a^{6}+\frac{40\!\cdots\!41}{34\!\cdots\!31}a^{5}+\frac{62\!\cdots\!26}{34\!\cdots\!31}a^{4}-\frac{751414096034424}{17162297392769}a^{3}-\frac{95\!\cdots\!21}{17162297392769}a^{2}-\frac{14\!\cdots\!58}{17162297392769}a+\frac{26\!\cdots\!37}{1009546905457}$, $\frac{26947761886}{200899834185943}a^{8}+\frac{1839635625728}{34\!\cdots\!31}a^{7}-\frac{88312010425391}{34\!\cdots\!31}a^{6}-\frac{603357416398523}{34\!\cdots\!31}a^{5}+\frac{26\!\cdots\!51}{34\!\cdots\!31}a^{4}+\frac{149784612508408}{17162297392769}a^{3}+\frac{269802815365188}{17162297392769}a^{2}-\frac{545996944811087}{17162297392769}a-\frac{89665016672007}{1009546905457}$, $\frac{627419287723}{34\!\cdots\!31}a^{8}+\frac{8329509824899}{34\!\cdots\!31}a^{7}-\frac{85003780172532}{34\!\cdots\!31}a^{6}-\frac{15\!\cdots\!61}{34\!\cdots\!31}a^{5}-\frac{11\!\cdots\!51}{34\!\cdots\!31}a^{4}+\frac{61\!\cdots\!91}{34\!\cdots\!31}a^{3}+\frac{10\!\cdots\!76}{17162297392769}a^{2}-\frac{911111357572127}{17162297392769}a-\frac{312851931215937}{1009546905457}$, $\frac{100428505757853}{34\!\cdots\!31}a^{8}-\frac{513992517931617}{34\!\cdots\!31}a^{7}-\frac{28\!\cdots\!18}{34\!\cdots\!31}a^{6}+\frac{77\!\cdots\!03}{34\!\cdots\!31}a^{5}+\frac{28\!\cdots\!64}{34\!\cdots\!31}a^{4}-\frac{17\!\cdots\!89}{34\!\cdots\!31}a^{3}-\frac{47\!\cdots\!71}{17162297392769}a^{2}-\frac{27\!\cdots\!86}{17162297392769}a+\frac{14\!\cdots\!27}{1009546905457}$
| 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: | \( 11715069.3768 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 11715069.3768 \cdot 1}{2\cdot\sqrt{49213007847751532409}}\cr\approx \mathstrut & 0.427508619445 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029)
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 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029, 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 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029);
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 - 315*x^7 - 702*x^6 + 32238*x^5 + 127161*x^4 - 1030422*x^3 - 5346135*x^2 + 2138454*x + 24909029);
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.1.0.1}{1} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
|
\(199\)
| 199.3.0.1 | $x^{3} + x + 196$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 199.3.2.3 | $x^{3} + 796$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 199.3.2.2 | $x^{3} + 398$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |