Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 18*x^5 - 12*x^3 + 9*x - 8)
gp: K = bnfinit(y^9 - 18*y^5 - 12*y^3 + 9*y - 8, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 18*x^5 - 12*x^3 + 9*x - 8);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 18*x^5 - 12*x^3 + 9*x - 8)
\( x^{9} - 18x^{5} - 12x^{3} + 9x - 8 \)
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: |
\(-4760622968832\)
\(\medspace = -\,2^{12}\cdot 3^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.62\) | 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: | $2^{19/12}3^{13/6}\approx 32.3887020988129$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{292}a^{8}-\frac{17}{146}a^{7}-\frac{3}{73}a^{6}+\frac{43}{292}a^{5}+\frac{53}{292}a^{4}+\frac{24}{73}a^{3}-\frac{16}{73}a^{2}-\frac{87}{292}a+\frac{30}{73}$
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 $3$
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: |
$\frac{5}{73}a^{8}-\frac{23}{292}a^{7}-\frac{21}{292}a^{6}-\frac{4}{73}a^{5}-\frac{100}{73}a^{4}+\frac{387}{292}a^{3}+\frac{107}{292}a^{2}+\frac{225}{146}a+\frac{235}{73}$, $\frac{9}{292}a^{8}+\frac{59}{292}a^{7}-\frac{35}{292}a^{6}-\frac{51}{292}a^{5}-\frac{107}{292}a^{4}-\frac{961}{292}a^{3}+\frac{373}{292}a^{2}+\frac{385}{292}a-\frac{95}{73}$, $\frac{15}{73}a^{8}-\frac{69}{292}a^{7}+\frac{5}{146}a^{6}+\frac{25}{292}a^{5}-\frac{527}{146}a^{4}+\frac{1161}{292}a^{3}-\frac{230}{73}a^{2}+\frac{547}{292}a-\frac{25}{73}$, $\frac{13}{292}a^{8}-\frac{1}{73}a^{7}-\frac{5}{146}a^{6}-\frac{25}{292}a^{5}-\frac{187}{292}a^{4}+\frac{113}{146}a^{3}-\frac{51}{146}a^{2}-\frac{255}{292}a+\frac{25}{73}$, $\frac{187}{292}a^{8}+\frac{139}{292}a^{7}+\frac{23}{73}a^{6}+\frac{21}{73}a^{5}-\frac{3375}{292}a^{4}-\frac{2561}{292}a^{3}-\frac{1969}{146}a^{2}-\frac{764}{73}a+\frac{135}{73}$
| 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: | \( 1319.99130745 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1319.99130745 \cdot 3}{2\cdot\sqrt{4760622968832}}\cr\approx \mathstrut & 1.80077738748 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 18*x^5 - 12*x^3 + 9*x - 8)
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 - 18*x^5 - 12*x^3 + 9*x - 8, 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 - 18*x^5 - 12*x^3 + 9*x - 8);
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 - 18*x^5 - 12*x^3 + 9*x - 8);
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
$C_3^2:\GL(2,3)$ (as 9T26):
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 432 |
| The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$ |
| Character table for $((C_3^2:Q_8):C_3):C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 sibling: | 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 | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.12.29 | $x^{8} + 2 x^{5} + 2 x^{2} + 6$ | $8$ | $1$ | $12$ | $\textrm{GL(2,3)}$ | $[4/3, 4/3, 2]_{3}^{2}$ | |
|
\(3\)
| 3.9.19.46 | $x^{9} + 6 x^{6} + 18 x^{4} + 18 x^{2} + 12$ | $9$ | $1$ | $19$ | $S_3\times C_3$ | $[2, 5/2]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.972.3t2.d.a | $2$ | $ 2^{2} \cdot 3^{5}$ | 3.1.972.2 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.3888.24t22.b.a | $2$ | $ 2^{4} \cdot 3^{5}$ | 8.2.725594112.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.3888.24t22.b.b | $2$ | $ 2^{4} \cdot 3^{5}$ | 8.2.725594112.1 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.3888.4t5.b.a | $3$ | $ 2^{4} \cdot 3^{5}$ | 4.2.3888.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.11664.6t8.b.a | $3$ | $ 2^{4} \cdot 3^{6}$ | 4.2.3888.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 4.186624.8t23.b.a | $4$ | $ 2^{8} \cdot 3^{6}$ | 8.2.725594112.1 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ | |
| * | 8.476...832.9t26.a.a | $8$ | $ 2^{12} \cdot 3^{19}$ | 9.3.4760622968832.3 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $2$ |
| 8.476...832.18t157.a.a | $8$ | $ 2^{12} \cdot 3^{19}$ | 9.3.4760622968832.3 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $-2$ | |
| 16.111...616.24t1334.a.a | $16$ | $ 2^{26} \cdot 3^{34}$ | 9.3.4760622968832.3 | $((C_3^2:Q_8):C_3):C_2$ (as 9T26) | $1$ | $0$ |
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 *.