Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008)
gp: K = bnfinit(y^9 - y^8 - 48*y^7 + 73*y^6 + 660*y^5 - 1454*y^4 - 2149*y^3 + 8350*y^2 - 7432*y + 2008, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008)
\( x^{9} - x^{8} - 48x^{7} + 73x^{6} + 660x^{5} - 1454x^{4} - 2149x^{3} + 8350x^{2} - 7432x + 2008 \)
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: |
\(19925626416901921\)
\(\medspace = 109^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.72\) | 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: | $109^{8/9}\approx 64.72108301255068$ | ||
| Ramified primes: |
\(109\)
| 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: | \(109\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{109}(1,·)$, $\chi_{109}(66,·)$, $\chi_{109}(38,·)$, $\chi_{109}(105,·)$, $\chi_{109}(75,·)$, $\chi_{109}(45,·)$, $\chi_{109}(16,·)$, $\chi_{109}(27,·)$, $\chi_{109}(63,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{29632996}a^{8}-\frac{3673491}{29632996}a^{7}+\frac{1713649}{7408249}a^{6}-\frac{7104919}{29632996}a^{5}-\frac{2494475}{14816498}a^{4}-\frac{2305402}{7408249}a^{3}+\frac{6814941}{29632996}a^{2}+\frac{2078735}{14816498}a+\frac{1699681}{7408249}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $8$ | 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: |
$\frac{127785}{29632996}a^{8}-\frac{757799}{29632996}a^{7}-\frac{1794726}{7408249}a^{6}+\frac{38106527}{29632996}a^{5}+\frac{28753920}{7408249}a^{4}-\frac{287651383}{14816498}a^{3}-\frac{392479711}{29632996}a^{2}+\frac{693159197}{7408249}a-\frac{482843782}{7408249}$, $\frac{126003}{14816498}a^{8}+\frac{430343}{29632996}a^{7}-\frac{5866375}{14816498}a^{6}-\frac{3037425}{7408249}a^{5}+\frac{164177319}{29632996}a^{4}+\frac{15307909}{29632996}a^{3}-\frac{737311181}{29632996}a^{2}+\frac{379409821}{14816498}a-\frac{44780090}{7408249}$, $\frac{215233}{14816498}a^{8}-\frac{3009}{29632996}a^{7}-\frac{10119233}{14816498}a^{6}+\frac{2567851}{7408249}a^{5}+\frac{278057123}{29632996}a^{4}-\frac{300752871}{29632996}a^{3}-\frac{1125591101}{29632996}a^{2}+\frac{1026906909}{14816498}a-\frac{198629115}{7408249}$, $\frac{397171}{7408249}a^{8}+\frac{2163633}{29632996}a^{7}-\frac{18085343}{7408249}a^{6}-\frac{13882306}{7408249}a^{5}+\frac{968796449}{29632996}a^{4}-\frac{16049175}{29632996}a^{3}-\frac{4050085609}{29632996}a^{2}+\frac{1952165007}{14816498}a-\frac{176690929}{7408249}$, $\frac{40747}{14816498}a^{8}-\frac{33366}{7408249}a^{7}-\frac{790293}{7408249}a^{6}+\frac{1414089}{7408249}a^{5}+\frac{6211704}{7408249}a^{4}-\frac{10644197}{7408249}a^{3}+\frac{2501830}{7408249}a^{2}-\frac{29632789}{14816498}a+\frac{9180110}{7408249}$, $\frac{908477}{29632996}a^{8}+\frac{742811}{29632996}a^{7}-\frac{10600030}{7408249}a^{6}-\frac{11126141}{29632996}a^{5}+\frac{291717987}{14816498}a^{4}-\frac{63067458}{7408249}a^{3}-\frac{2419788597}{29632996}a^{2}+\frac{797254023}{7408249}a-\frac{254348446}{7408249}$, $\frac{105915}{14816498}a^{8}-\frac{531819}{29632996}a^{7}-\frac{1933330}{7408249}a^{6}+\frac{12437535}{14816498}a^{5}+\frac{44115591}{29632996}a^{4}-\frac{250598537}{29632996}a^{3}+\frac{252391111}{29632996}a^{2}-\frac{477261}{14816498}a-\frac{4883419}{7408249}$, $\frac{567457}{29632996}a^{8}-\frac{262269}{29632996}a^{7}-\frac{6867894}{7408249}a^{6}+\frac{25699291}{29632996}a^{5}+\frac{194267327}{14816498}a^{4}-\frac{149385033}{7408249}a^{3}-\frac{1535001245}{29632996}a^{2}+\frac{955395807}{7408249}a-\frac{524858270}{7408249}$
| 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: | \( 120939.78896 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 120939.78896 \cdot 1}{2\cdot\sqrt{19925626416901921}}\cr\approx \mathstrut & 0.21933259653 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008)
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 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008, 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 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008);
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 - 48*x^7 + 73*x^6 + 660*x^5 - 1454*x^4 - 2149*x^3 + 8350*x^2 - 7432*x + 2008);
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
| 3.3.11881.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.9.0.1}{9} }$ | ${\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.1.0.1}{1} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(109\)
| 109.9.8.1 | $x^{9} + 109$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.109.9t1.a.a | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.109.9t1.a.b | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.109.3t1.a.a | $1$ | $ 109 $ | 3.3.11881.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.109.9t1.a.c | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.109.9t1.a.d | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.109.3t1.a.b | $1$ | $ 109 $ | 3.3.11881.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.109.9t1.a.e | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.109.9t1.a.f | $1$ | $ 109 $ | 9.9.19925626416901921.1 | $C_9$ (as 9T1) | $0$ | $1$ |
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 *.