Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648)
gp: K = bnfinit(y^9 - 4*y^8 + 5*y^7 - 46*y^6 - 381*y^5 + 1056*y^4 - 49*y^3 + 2074*y^2 + 29152*y + 52648, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648)
\( x^{9} - 4x^{8} + 5x^{7} - 46x^{6} - 381x^{5} + 1056x^{4} - 49x^{3} + 2074x^{2} + 29152x + 52648 \)
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: |
\(-1023026794603831808\)
\(\medspace = -\,2^{9}\cdot 7^{6}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(100.25\) | 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: | $2^{3/2}7^{2/3}19^{8/9}\approx 141.7795371895746$ | ||
| Ramified primes: |
\(2\), \(7\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{12}a^{6}-\frac{1}{6}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{828}a^{7}-\frac{17}{414}a^{6}-\frac{1}{46}a^{5}-\frac{31}{414}a^{4}-\frac{11}{276}a^{3}-\frac{89}{207}a^{2}+\frac{169}{414}a+\frac{41}{207}$, $\frac{1}{9533397420}a^{8}-\frac{155474}{476669871}a^{7}-\frac{7496265}{211853276}a^{6}+\frac{197024357}{4766698710}a^{5}+\frac{4527793}{211853276}a^{4}-\frac{507703946}{2383349355}a^{3}-\frac{251800145}{1906679484}a^{2}+\frac{919162987}{4766698710}a+\frac{244391449}{794449785}$
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
$C_{3}\times C_{9}$, which has order $27$ (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: |
$\frac{189709}{1059266380}a^{8}-\frac{10531}{105926638}a^{7}+\frac{428197}{635559828}a^{6}-\frac{120439}{23027530}a^{5}-\frac{19493817}{211853276}a^{4}-\frac{159562949}{1588899570}a^{3}-\frac{318610589}{635559828}a^{2}-\frac{3110297011}{1588899570}a-\frac{465091681}{794449785}$, $\frac{17479}{317779914}a^{8}-\frac{771089}{953339742}a^{7}+\frac{1807315}{476669871}a^{6}-\frac{5108807}{317779914}a^{5}+\frac{2036579}{41449554}a^{4}+\frac{11351327}{317779914}a^{3}-\frac{45478459}{476669871}a^{2}+\frac{412655441}{953339742}a+\frac{810971224}{476669871}$, $\frac{9839}{4766698710}a^{8}-\frac{420023}{953339742}a^{7}+\frac{665765}{476669871}a^{6}+\frac{26722361}{4766698710}a^{5}-\frac{441257}{41449554}a^{4}+\frac{217291529}{4766698710}a^{3}+\frac{8681363}{476669871}a^{2}-\frac{128839993}{1588899570}a-\frac{920130539}{2383349355}$, $\frac{7684306}{2383349355}a^{8}-\frac{1648622}{158889957}a^{7}-\frac{33730349}{1906679484}a^{6}-\frac{262953161}{2383349355}a^{5}-\frac{525885406}{476669871}a^{4}+\frac{8973294751}{2383349355}a^{3}+\frac{8626572079}{635559828}a^{2}+\frac{6560661439}{2383349355}a-\frac{52397292592}{2383349355}$, $\frac{21794449}{9533397420}a^{8}+\frac{313295}{158889957}a^{7}+\frac{47355799}{1906679484}a^{6}+\frac{92949304}{2383349355}a^{5}-\frac{1159038307}{1906679484}a^{4}-\frac{687293564}{2383349355}a^{3}-\frac{1408982671}{635559828}a^{2}-\frac{33744918026}{2383349355}a-\frac{51232399057}{2383349355}$
| 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: | \( 38054.4761857 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 38054.4761857 \cdot 27}{2\cdot\sqrt{1023026794603831808}}\cr\approx \mathstrut & 1.00791893616 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648)
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 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648, 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 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648);
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 - 4*x^8 + 5*x^7 - 46*x^6 - 381*x^5 + 1056*x^4 - 49*x^3 + 2074*x^2 + 29152*x + 52648);
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
$C_3\wr S_3$ (as 9T20):
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 162 |
| The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
| Character table for $C_3 \wr S_3 $ |
Intermediate fields
| 3.1.2888.1 |
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 9 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 siblings: | data not computed |
| Minimal sibling: | 9.3.426083629572608.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.5 | $x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
|
\(7\)
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
|
\(19\)
| 19.9.8.2 | $x^{9} + 57$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |