Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531)
gp: K = bnfinit(y^9 - y^8 - 27*y^7 + 45*y^6 + 192*y^5 - 376*y^4 - 267*y^3 + 276*y^2 + 423*y + 531, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - x^8 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531)
\( x^{9} - x^{8} - 27x^{7} + 45x^{6} + 192x^{5} - 376x^{4} - 267x^{3} + 276x^{2} + 423x + 531 \)
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: | $[5, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(17982892873895481\)
\(\medspace = 3^{2}\cdot 7^{6}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(63.99\) | 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: | $3^{1/2}7^{2/3}19^{8/9}\approx 86.82188052060228$ | ||
| Ramified primes: |
\(3\), \(7\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{6}+\frac{2}{9}a^{5}-\frac{2}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{144171}a^{8}+\frac{103}{16019}a^{7}-\frac{20816}{144171}a^{6}-\frac{62365}{144171}a^{5}+\frac{2119}{144171}a^{4}+\frac{27757}{144171}a^{3}+\frac{1744}{16019}a^{2}+\frac{1631}{48057}a+\frac{2641}{16019}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
$C_{9}$, which has order $9$
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: | $6$ | 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: |
$\frac{964}{48057}a^{8}+\frac{5711}{144171}a^{7}-\frac{64546}{144171}a^{6}-\frac{65735}{144171}a^{5}+\frac{425384}{144171}a^{4}+\frac{194263}{144171}a^{3}-\frac{167201}{48057}a^{2}-\frac{217000}{48057}a-\frac{67367}{16019}$, $\frac{826}{144171}a^{8}-\frac{1070}{48057}a^{7}-\frac{37667}{144171}a^{6}+\frac{51671}{144171}a^{5}+\frac{356641}{144171}a^{4}-\frac{188135}{144171}a^{3}-\frac{211745}{48057}a^{2}-\frac{238675}{48057}a-\frac{45175}{16019}$, $\frac{998}{144171}a^{8}-\frac{3956}{144171}a^{7}-\frac{3307}{16019}a^{6}+\frac{11742}{16019}a^{5}+\frac{19607}{16019}a^{4}-\frac{636125}{144171}a^{3}+\frac{47399}{48057}a^{2}-\frac{70276}{48057}a+\frac{40640}{16019}$, $\frac{207}{16019}a^{8}-\frac{339}{16019}a^{7}-\frac{15422}{48057}a^{6}+\frac{37315}{48057}a^{5}+\frac{82436}{48057}a^{4}-\frac{271670}{48057}a^{3}+\frac{55721}{48057}a^{2}+\frac{35692}{16019}a+\frac{66426}{16019}$, $\frac{1640}{144171}a^{8}-\frac{1525}{144171}a^{7}-\frac{49808}{144171}a^{6}+\frac{66791}{144171}a^{5}+\frac{463588}{144171}a^{4}-\frac{209753}{48057}a^{3}-\frac{534331}{48057}a^{2}+\frac{160079}{16019}a+\frac{246395}{16019}$, $\frac{3623}{48057}a^{8}+\frac{15631}{144171}a^{7}-\frac{253052}{144171}a^{6}-\frac{129352}{144171}a^{5}+\frac{1726141}{144171}a^{4}+\frac{197234}{144171}a^{3}-\frac{785773}{48057}a^{2}-\frac{934796}{48057}a-\frac{257323}{16019}$
| 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: | \( 27843.7285838 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 27843.7285838 \cdot 9}{2\cdot\sqrt{17982892873895481}}\cr\approx \mathstrut & 1.18037456301 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531)
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 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531, 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 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531);
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 - 27*x^7 + 45*x^6 + 192*x^5 - 376*x^4 - 267*x^3 + 276*x^2 + 423*x + 531);
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
$C_3^3:A_4$ (as 9T25):
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 324 |
| The 13 conjugacy class representatives for $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ |
| Character table for $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ |
Intermediate fields
| 3.3.17689.1 |
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 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.9.0.1}{9} }$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{5}$ | R | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
|
\(19\)
| 19.9.8.2 | $x^{9} + 57$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |