Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 333*x^7 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749)
gp: K = bnfinit(y^9 - 333*y^7 + 36963*y^5 - 1519590*y^3 + 16867449*y - 21932749, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 333*x^7 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 333*x^7 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749)
\( x^{9} - 333x^{7} + 36963x^{5} - 1519590x^{3} + 16867449x - 21932749 \)
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: |
\(80515213381214514081\)
\(\medspace = 3^{22}\cdot 37^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(162.84\) | 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: | $3^{22/9}37^{2/3}\approx 162.84117089348263$ | ||
| Ramified primes: |
\(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $9$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(999=3^{3}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(322,·)$, $\chi_{999}(454,·)$, $\chi_{999}(334,·)$, $\chi_{999}(655,·)$, $\chi_{999}(787,·)$, $\chi_{999}(121,·)$, $\chi_{999}(667,·)$, $\chi_{999}(988,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{37}a^{3}$, $\frac{1}{37}a^{4}$, $\frac{1}{2627}a^{5}-\frac{26}{2627}a^{4}+\frac{28}{2627}a^{3}+\frac{33}{71}a^{2}-\frac{28}{71}a-\frac{7}{71}$, $\frac{1}{97199}a^{6}-\frac{6}{2627}a^{4}-\frac{26}{2627}a^{3}+\frac{9}{71}a^{2}+\frac{7}{71}a-\frac{3}{71}$, $\frac{1}{97199}a^{7}+\frac{31}{2627}a^{4}+\frac{4}{2627}a^{3}-\frac{8}{71}a^{2}-\frac{29}{71}a+\frac{29}{71}$, $\frac{1}{97199}a^{8}+\frac{29}{2627}a^{4}-\frac{28}{2627}a^{3}+\frac{13}{71}a^{2}-\frac{26}{71}a+\frac{4}{71}$
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
$C_{3}$, which has order $3$ (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: |
$\frac{4}{97199}a^{6}-\frac{24}{2627}a^{4}-\frac{33}{2627}a^{3}+\frac{36}{71}a^{2}+\frac{99}{71}a-\frac{296}{71}$, $\frac{3}{97199}a^{6}-\frac{18}{2627}a^{4}-\frac{7}{2627}a^{3}+\frac{27}{71}a^{2}+\frac{21}{71}a-\frac{151}{71}$, $\frac{5}{97199}a^{8}+\frac{39}{97199}a^{7}-\frac{1265}{97199}a^{6}-\frac{255}{2627}a^{5}+\frac{2297}{2627}a^{4}+\frac{15613}{2627}a^{3}-\frac{806}{71}a^{2}-\frac{4112}{71}a+\frac{6944}{71}$, $\frac{9}{97199}a^{8}+\frac{85}{97199}a^{7}-\frac{2116}{97199}a^{6}-\frac{533}{2627}a^{5}+\frac{3393}{2627}a^{4}+\frac{30524}{2627}a^{3}-\frac{1057}{71}a^{2}-\frac{8338}{71}a+\frac{12793}{71}$, $\frac{1}{97199}a^{8}-\frac{17}{97199}a^{7}-\frac{189}{97199}a^{6}+\frac{92}{2627}a^{5}+\frac{232}{2627}a^{4}-\frac{4747}{2627}a^{3}-\frac{78}{71}a^{2}+\frac{1609}{71}a-\frac{1986}{71}$, $\frac{6}{97199}a^{8}-\frac{15}{97199}a^{7}-\frac{1883}{97199}a^{6}+\frac{104}{2627}a^{5}+\frac{5037}{2627}a^{4}-\frac{6649}{2627}a^{3}-\frac{4229}{71}a^{2}-\frac{549}{71}a+\frac{13456}{71}$, $\frac{3}{97199}a^{8}+\frac{23}{97199}a^{7}-\frac{780}{97199}a^{6}-\frac{150}{2627}a^{5}+\frac{1570}{2627}a^{4}+\frac{9556}{2627}a^{3}-\frac{897}{71}a^{2}-\frac{3567}{71}a+\frac{7264}{71}$, $\frac{1}{97199}a^{8}+\frac{29}{97199}a^{7}-\frac{404}{97199}a^{6}-\frac{230}{2627}a^{5}+\frac{1593}{2627}a^{4}+\frac{20340}{2627}a^{3}-\frac{2570}{71}a^{2}-\frac{13514}{71}a+\frac{39948}{71}$
| 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: | \( 2734333.08957 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 2734333.08957 \cdot 3}{2\cdot\sqrt{80515213381214514081}}\cr\approx \mathstrut & 0.234031148563 \end{aligned}\] (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 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749)
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 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749, 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 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749);
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 + 36963*x^5 - 1519590*x^3 + 16867449*x - 21932749);
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 cyclic group of order 9 |
| The 9 conjugacy class representatives for $C_9$ |
| Character table for $C_9$ |
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]
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 | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | R | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.6 | $x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$ | $9$ | $1$ | $22$ | $C_9$ | $[2, 3]$ |
|
\(37\)
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.2 | $x^{3} + 74$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |