Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721)
gp: K = bnfinit(y^9 - y^8 - 198*y^7 - 107*y^6 + 10319*y^5 + 24533*y^4 - 118922*y^3 - 460626*y^2 - 367949*y + 57721, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721)
\( x^{9} - x^{8} - 198x^{7} - 107x^{6} + 10319x^{5} + 24533x^{4} - 118922x^{3} - 460626x^{2} - 367949x + 57721 \)
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: |
\(15072974715383053921\)
\(\medspace = 19^{8}\cdot 31^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(135.18\) | 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: | $19^{8/9}31^{2/3}\approx 135.17955010958042$ | ||
| Ramified primes: |
\(19\), \(31\)
| 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: | \(589=19\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(397,·)$, $\chi_{589}(366,·)$, $\chi_{589}(125,·)$, $\chi_{589}(149,·)$, $\chi_{589}(311,·)$, $\chi_{589}(408,·)$, $\chi_{589}(346,·)$, $\chi_{589}(253,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{25\!\cdots\!29}a^{8}-\frac{34\!\cdots\!91}{25\!\cdots\!29}a^{7}-\frac{46\!\cdots\!88}{25\!\cdots\!29}a^{6}+\frac{10\!\cdots\!34}{25\!\cdots\!29}a^{5}+\frac{78\!\cdots\!26}{25\!\cdots\!29}a^{4}+\frac{24\!\cdots\!26}{25\!\cdots\!29}a^{3}+\frac{68\!\cdots\!71}{25\!\cdots\!29}a^{2}-\frac{34\!\cdots\!73}{25\!\cdots\!29}a-\frac{23\!\cdots\!48}{12\!\cdots\!57}$
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 $3$ (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: | $8$ | 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: |
$\frac{2335439503002}{12\!\cdots\!57}a^{8}-\frac{3947967663468}{12\!\cdots\!57}a^{7}-\frac{460304001254696}{12\!\cdots\!57}a^{6}+\frac{83481472660399}{12\!\cdots\!57}a^{5}+\frac{24\!\cdots\!00}{12\!\cdots\!57}a^{4}+\frac{38\!\cdots\!58}{12\!\cdots\!57}a^{3}-\frac{29\!\cdots\!21}{12\!\cdots\!57}a^{2}-\frac{79\!\cdots\!56}{12\!\cdots\!57}a-\frac{46\!\cdots\!64}{12\!\cdots\!57}$, $\frac{3580472426003}{12\!\cdots\!57}a^{8}-\frac{9896722714953}{12\!\cdots\!57}a^{7}-\frac{698362349306116}{12\!\cdots\!57}a^{6}+\frac{912063065824864}{12\!\cdots\!57}a^{5}+\frac{36\!\cdots\!94}{12\!\cdots\!57}a^{4}+\frac{14\!\cdots\!96}{12\!\cdots\!57}a^{3}-\frac{50\!\cdots\!65}{12\!\cdots\!57}a^{2}-\frac{60\!\cdots\!84}{12\!\cdots\!57}a+\frac{25\!\cdots\!26}{12\!\cdots\!57}$, $\frac{14\!\cdots\!98}{25\!\cdots\!29}a^{8}-\frac{59\!\cdots\!35}{25\!\cdots\!29}a^{7}-\frac{28\!\cdots\!09}{25\!\cdots\!29}a^{6}+\frac{68\!\cdots\!08}{25\!\cdots\!29}a^{5}+\frac{14\!\cdots\!81}{25\!\cdots\!29}a^{4}-\frac{42\!\cdots\!62}{25\!\cdots\!29}a^{3}-\frac{20\!\cdots\!07}{25\!\cdots\!29}a^{2}-\frac{25\!\cdots\!72}{25\!\cdots\!29}a+\frac{18\!\cdots\!51}{12\!\cdots\!57}$, $\frac{51311707260832}{25\!\cdots\!29}a^{8}-\frac{644701892170253}{25\!\cdots\!29}a^{7}-\frac{24\!\cdots\!02}{25\!\cdots\!29}a^{6}+\frac{38\!\cdots\!60}{25\!\cdots\!29}a^{5}+\frac{20\!\cdots\!53}{25\!\cdots\!29}a^{4}+\frac{71\!\cdots\!33}{25\!\cdots\!29}a^{3}-\frac{32\!\cdots\!93}{25\!\cdots\!29}a^{2}-\frac{44\!\cdots\!35}{25\!\cdots\!29}a+\frac{37\!\cdots\!63}{12\!\cdots\!57}$, $\frac{11\!\cdots\!81}{25\!\cdots\!29}a^{8}-\frac{91\!\cdots\!62}{25\!\cdots\!29}a^{7}-\frac{15\!\cdots\!27}{25\!\cdots\!29}a^{6}+\frac{11\!\cdots\!50}{25\!\cdots\!29}a^{5}+\frac{39\!\cdots\!17}{25\!\cdots\!29}a^{4}-\frac{20\!\cdots\!66}{25\!\cdots\!29}a^{3}-\frac{47\!\cdots\!38}{25\!\cdots\!29}a^{2}+\frac{68\!\cdots\!71}{25\!\cdots\!29}a+\frac{72\!\cdots\!38}{12\!\cdots\!57}$, $\frac{49\!\cdots\!21}{25\!\cdots\!29}a^{8}-\frac{19\!\cdots\!02}{25\!\cdots\!29}a^{7}-\frac{91\!\cdots\!29}{25\!\cdots\!29}a^{6}+\frac{21\!\cdots\!91}{25\!\cdots\!29}a^{5}+\frac{43\!\cdots\!51}{25\!\cdots\!29}a^{4}-\frac{62\!\cdots\!54}{25\!\cdots\!29}a^{3}-\frac{54\!\cdots\!59}{25\!\cdots\!29}a^{2}-\frac{70\!\cdots\!58}{25\!\cdots\!29}a+\frac{52\!\cdots\!13}{12\!\cdots\!57}$, $\frac{23\!\cdots\!95}{25\!\cdots\!29}a^{8}-\frac{17\!\cdots\!54}{25\!\cdots\!29}a^{7}-\frac{39\!\cdots\!53}{25\!\cdots\!29}a^{6}+\frac{24\!\cdots\!91}{25\!\cdots\!29}a^{5}+\frac{15\!\cdots\!71}{25\!\cdots\!29}a^{4}-\frac{64\!\cdots\!41}{25\!\cdots\!29}a^{3}-\frac{15\!\cdots\!92}{25\!\cdots\!29}a^{2}+\frac{40\!\cdots\!10}{25\!\cdots\!29}a-\frac{13\!\cdots\!65}{12\!\cdots\!57}$, $\frac{56\!\cdots\!44}{25\!\cdots\!29}a^{8}-\frac{24\!\cdots\!78}{25\!\cdots\!29}a^{7}-\frac{10\!\cdots\!60}{25\!\cdots\!29}a^{6}+\frac{28\!\cdots\!22}{25\!\cdots\!29}a^{5}+\frac{48\!\cdots\!61}{25\!\cdots\!29}a^{4}-\frac{17\!\cdots\!55}{25\!\cdots\!29}a^{3}-\frac{58\!\cdots\!36}{25\!\cdots\!29}a^{2}-\frac{75\!\cdots\!84}{25\!\cdots\!29}a+\frac{51\!\cdots\!14}{12\!\cdots\!57}$
| 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: | \( 785284.164362 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 785284.164362 \cdot 3}{2\cdot\sqrt{15072974715383053921}}\cr\approx \mathstrut & 0.155341886320 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721)
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 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721, 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 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721);
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 - 198*x^7 - 107*x^6 + 10319*x^5 + 24533*x^4 - 118922*x^3 - 460626*x^2 - 367949*x + 57721);
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
| 3.3.361.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.9.0.1}{9} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | R | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| 19.9.8.6 | $x^{9} + 133$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
|
\(31\)
| 31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 31.3.2.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |