Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5)
gp: K = bnfinit(y^9 - 3*y^8 + 4*y^7 - 6*y^6 - y^5 + 9*y^4 - 21*y^3 + 7*y^2 + 25*y + 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5)
\( x^{9} - 3x^{8} + 4x^{7} - 6x^{6} - x^{5} + 9x^{4} - 21x^{3} + 7x^{2} + 25x + 5 \)
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: |
\(354866144000\)
\(\medspace = 2^{8}\cdot 5^{3}\cdot 223^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.20\) | 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: | $2^{4/3}5^{1/2}223^{1/2}\approx 84.14159905646059$ | ||
| Ramified primes: |
\(2\), \(5\), \(223\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{1115}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{14018}a^{8}-\frac{2819}{7009}a^{7}+\frac{2673}{7009}a^{6}-\frac{17}{7009}a^{5}-\frac{4663}{14018}a^{4}+\frac{3141}{7009}a^{3}-\frac{3641}{14018}a^{2}-\frac{2655}{7009}a-\frac{6555}{14018}$
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 $1$
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: |
$\frac{2627}{14018}a^{8}-\frac{4009}{7009}a^{7}+\frac{5962}{7009}a^{6}-\frac{9614}{7009}a^{5}+\frac{2031}{14018}a^{4}+\frac{8823}{7009}a^{3}-\frac{60703}{14018}a^{2}+\frac{13288}{7009}a+\frac{50191}{14018}$, $\frac{1393}{14018}a^{8}-\frac{1827}{7009}a^{7}+\frac{1710}{7009}a^{6}-\frac{2654}{7009}a^{5}-\frac{5225}{14018}a^{4}+\frac{8806}{7009}a^{3}-\frac{25433}{14018}a^{2}+\frac{2337}{7009}a+\frac{22639}{14018}$, $\frac{480}{7009}a^{8}-\frac{766}{7009}a^{7}+\frac{786}{7009}a^{6}-\frac{2302}{7009}a^{5}-\frac{2369}{7009}a^{4}+\frac{1490}{7009}a^{3}-\frac{9448}{7009}a^{2}-\frac{4533}{7009}a+\frac{641}{7009}$, $\frac{1382}{7009}a^{8}-\frac{4717}{7009}a^{7}+\frac{7695}{7009}a^{6}-\frac{11943}{7009}a^{5}+\frac{4014}{7009}a^{4}+\frac{11591}{7009}a^{3}-\frac{34445}{7009}a^{2}+\frac{21030}{7009}a+\frac{24654}{7009}$, $\frac{1188}{7009}a^{8}-\frac{4349}{7009}a^{7}+\frac{7903}{7009}a^{6}-\frac{12356}{7009}a^{5}+\frac{4475}{7009}a^{4}+\frac{12449}{7009}a^{3}-\frac{36000}{7009}a^{2}+\frac{27856}{7009}a+\frac{13677}{7009}$, $\frac{1005}{14018}a^{8}-\frac{1459}{7009}a^{7}+\frac{1918}{7009}a^{6}-\frac{3067}{7009}a^{5}-\frac{4303}{14018}a^{4}+\frac{9664}{7009}a^{3}-\frac{28543}{14018}a^{2}+\frac{2154}{7009}a+\frac{42739}{14018}$
| 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: | \( 591.374538482 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 591.374538482 \cdot 1}{2\cdot\sqrt{354866144000}}\cr\approx \mathstrut & 0.627061385591 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5)
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 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5, 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 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5);
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 - 3*x^8 + 4*x^7 - 6*x^6 - x^5 + 9*x^4 - 21*x^3 + 7*x^2 + 25*x + 5);
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
$S_3\wr S_3$ (as 9T31):
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 1296 |
| The 22 conjugacy class representatives for $S_3\wr S_3$ |
| Character table for $S_3\wr S_3$ is not computed |
Intermediate fields
| 3.3.892.1 |
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 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 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 | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(223\)
| $\Q_{223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.4460.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 223 $ | \(\Q(\sqrt{1115}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.892.2t1.a.a | $1$ | $ 2^{2} \cdot 223 $ | \(\Q(\sqrt{223}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.22300.6t3.c.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 223 $ | 6.6.88716536000.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
| * | 2.892.3t2.a.a | $2$ | $ 2^{2} \cdot 223 $ | 3.3.892.1 | $S_3$ (as 3T2) | $1$ | $2$ |
| 3.795664.6t8.a.a | $3$ | $ 2^{4} \cdot 223^{2}$ | 4.0.892.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.892.4t5.a.a | $3$ | $ 2^{2} \cdot 223 $ | 4.0.892.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.99458000.6t11.a.a | $3$ | $ 2^{4} \cdot 5^{3} \cdot 223^{2}$ | 6.2.99458000.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
| 3.111500.6t11.a.a | $3$ | $ 2^{2} \cdot 5^{3} \cdot 223 $ | 6.2.99458000.3 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
| 6.791...000.18t319.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 223^{4}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
| 6.397832000.18t311.a.a | $6$ | $ 2^{6} \cdot 5^{3} \cdot 223^{2}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
| * | 6.397832000.9t31.a.a | $6$ | $ 2^{6} \cdot 5^{3} \cdot 223^{2}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ |
| 6.791...000.18t300.a.a | $6$ | $ 2^{8} \cdot 5^{3} \cdot 223^{4}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $2$ | |
| 8.125...000.24t2893.a.a | $8$ | $ 2^{14} \cdot 5^{4} \cdot 223^{6}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 8.31826560000.12t213.a.a | $8$ | $ 2^{10} \cdot 5^{4} \cdot 223^{2}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.112...000.36t2217.a.a | $12$ | $ 2^{18} \cdot 5^{6} \cdot 223^{7}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.314...000.36t2210.a.a | $12$ | $ 2^{14} \cdot 5^{6} \cdot 223^{6}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $-4$ | |
| 12.141...000.36t2214.a.a | $12$ | $ 2^{14} \cdot 5^{6} \cdot 223^{5}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.112...000.36t2216.a.a | $12$ | $ 2^{18} \cdot 5^{6} \cdot 223^{7}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 12.141...000.18t315.a.a | $12$ | $ 2^{14} \cdot 5^{6} \cdot 223^{5}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ | |
| 16.400...000.24t2912.a.a | $16$ | $ 2^{24} \cdot 5^{8} \cdot 223^{8}$ | 9.5.354866144000.1 | $S_3\wr S_3$ (as 9T31) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.