Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000)
gp: K = bnfinit(y^9 - 819*y^7 - 3276*y^6 + 194103*y^5 + 1199016*y^4 - 13831545*y^3 - 95085900*y^2 + 299344500*y + 2144870000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000)
\( x^{9} - 819 x^{7} - 3276 x^{6} + 194103 x^{5} + 1199016 x^{4} - 13831545 x^{3} - 95085900 x^{2} + \cdots + 2144870000 \)
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: |
\(147570226003740066617015289\)
\(\medspace = 3^{22}\cdot 7^{8}\cdot 13^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(808.47\) | 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: | $3^{22/9}7^{8/9}13^{8/9}\approx 808.4748732595972$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{80}a^{6}-\frac{1}{16}a^{5}+\frac{1}{80}a^{4}-\frac{11}{80}a^{3}-\frac{1}{40}a^{2}+\frac{1}{5}a$, $\frac{1}{6400}a^{7}-\frac{1}{320}a^{6}+\frac{153}{3200}a^{5}+\frac{13}{800}a^{4}+\frac{1073}{6400}a^{3}-\frac{361}{1600}a^{2}+\frac{123}{320}a-\frac{3}{16}$, $\frac{1}{40\!\cdots\!00}a^{8}+\frac{4803456212257}{81\!\cdots\!00}a^{7}+\frac{10\!\cdots\!03}{20\!\cdots\!00}a^{6}-\frac{71\!\cdots\!83}{20\!\cdots\!00}a^{5}+\frac{59\!\cdots\!93}{40\!\cdots\!00}a^{4}-\frac{84\!\cdots\!79}{40\!\cdots\!00}a^{3}-\frac{128935006826729}{10\!\cdots\!00}a^{2}-\frac{781576735113637}{40\!\cdots\!20}a+\frac{101318144140337}{204457217103136}$
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
$C_{3}\times C_{9}$, 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: | $8$ | 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: |
$\frac{5094449397}{421561272377600}a^{8}-\frac{13627061487}{84312254475520}a^{7}-\frac{1609602736909}{210780636188800}a^{6}+\frac{11671229681109}{210780636188800}a^{5}+\frac{636482156818701}{421561272377600}a^{4}-\frac{462571073587403}{421561272377600}a^{3}-\frac{203721257026287}{2107806361888}a^{2}-\frac{18\!\cdots\!53}{4215612723776}a-\frac{706319618526747}{1053903180944}$, $\frac{578784284271}{10\!\cdots\!00}a^{8}-\frac{348036234219}{52695159047200}a^{7}-\frac{191346248661337}{526951590472000}a^{6}+\frac{293879117176803}{131737897618000}a^{5}+\frac{78\!\cdots\!43}{10\!\cdots\!00}a^{4}-\frac{12\!\cdots\!51}{263475795236000}a^{3}-\frac{52\!\cdots\!07}{10539031809440}a^{2}-\frac{41\!\cdots\!01}{2634757952360}a+\frac{834659844317693}{65868948809}$, $\frac{17\!\cdots\!67}{20\!\cdots\!00}a^{8}+\frac{51\!\cdots\!53}{81\!\cdots\!40}a^{7}-\frac{65\!\cdots\!99}{10\!\cdots\!00}a^{6}-\frac{77\!\cdots\!71}{10\!\cdots\!00}a^{5}+\frac{21\!\cdots\!51}{20\!\cdots\!00}a^{4}+\frac{36\!\cdots\!97}{20\!\cdots\!00}a^{3}+\frac{47\!\cdots\!17}{25\!\cdots\!00}a^{2}-\frac{13\!\cdots\!57}{20\!\cdots\!60}a-\frac{24\!\cdots\!11}{102228608551568}$, $\frac{77\!\cdots\!31}{51\!\cdots\!00}a^{8}+\frac{25\!\cdots\!93}{10\!\cdots\!00}a^{7}-\frac{21\!\cdots\!07}{25\!\cdots\!00}a^{6}-\frac{47\!\cdots\!83}{25\!\cdots\!00}a^{5}-\frac{41\!\cdots\!97}{51\!\cdots\!00}a^{4}+\frac{86\!\cdots\!41}{51\!\cdots\!00}a^{3}+\frac{40\!\cdots\!59}{638928803447300}a^{2}-\frac{20\!\cdots\!61}{511143042757840}a-\frac{51\!\cdots\!91}{25557152137892}$, $\frac{84\!\cdots\!79}{10\!\cdots\!00}a^{8}+\frac{82\!\cdots\!89}{40\!\cdots\!20}a^{7}-\frac{96\!\cdots\!63}{51\!\cdots\!00}a^{6}-\frac{37\!\cdots\!27}{51\!\cdots\!00}a^{5}-\frac{17\!\cdots\!13}{10\!\cdots\!00}a^{4}+\frac{59\!\cdots\!89}{10\!\cdots\!00}a^{3}+\frac{34\!\cdots\!89}{12\!\cdots\!00}a^{2}-\frac{13\!\cdots\!77}{10\!\cdots\!80}a-\frac{37\!\cdots\!55}{51114304275784}$, $\frac{17\!\cdots\!53}{51\!\cdots\!00}a^{8}+\frac{80\!\cdots\!99}{10\!\cdots\!80}a^{7}-\frac{21\!\cdots\!91}{25\!\cdots\!00}a^{6}-\frac{36\!\cdots\!07}{12\!\cdots\!00}a^{5}-\frac{23\!\cdots\!91}{51\!\cdots\!00}a^{4}+\frac{58\!\cdots\!99}{25\!\cdots\!00}a^{3}+\frac{23\!\cdots\!97}{25\!\cdots\!00}a^{2}-\frac{13\!\cdots\!11}{255571521378920}a-\frac{32\!\cdots\!77}{12778576068946}$, $\frac{12\!\cdots\!87}{20\!\cdots\!00}a^{8}-\frac{10\!\cdots\!99}{40\!\cdots\!00}a^{7}-\frac{47\!\cdots\!39}{10\!\cdots\!00}a^{6}+\frac{13\!\cdots\!09}{10\!\cdots\!00}a^{5}+\frac{20\!\cdots\!31}{20\!\cdots\!00}a^{4}+\frac{20\!\cdots\!57}{20\!\cdots\!00}a^{3}-\frac{17\!\cdots\!07}{25\!\cdots\!00}a^{2}-\frac{10\!\cdots\!81}{20\!\cdots\!60}a+\frac{14\!\cdots\!97}{102228608551568}$, $\frac{21\!\cdots\!83}{10\!\cdots\!00}a^{8}-\frac{41\!\cdots\!81}{20\!\cdots\!00}a^{7}-\frac{19\!\cdots\!19}{12\!\cdots\!00}a^{6}+\frac{80\!\cdots\!99}{10\!\cdots\!00}a^{5}+\frac{33\!\cdots\!27}{10\!\cdots\!00}a^{4}-\frac{14\!\cdots\!37}{20\!\cdots\!00}a^{3}-\frac{11\!\cdots\!29}{51\!\cdots\!00}a^{2}+\frac{18\!\cdots\!01}{10\!\cdots\!80}a+\frac{24\!\cdots\!47}{51114304275784}$
| 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: | \( 562994976933 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 562994976933 \cdot 27}{2\cdot\sqrt{147570226003740066617015289}}\cr\approx \mathstrut & 320.338308739702 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^9 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000)
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 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000, 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 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000);
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 - 819*x^7 - 3276*x^6 + 194103*x^5 + 1199016*x^4 - 13831545*x^3 - 95085900*x^2 + 299344500*x + 2144870000);
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 solvable group of order 27 |
| The 11 conjugacy class representatives for $C_9:C_3$ |
| Character table for $C_9:C_3$ |
Intermediate fields
| 3.3.670761.2 |
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
| Galois closure: | 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 | ${\href{/padicField/2.1.0.1}{1} }^{9}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/11.9.0.1}{9} }$ | R | ${\href{/padicField/17.9.0.1}{9} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.9.0.1}{9} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.9.22.11 | $x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 3$ | $9$ | $1$ | $22$ | $C_9:C_3$ | $[2, 3]^{3}$ |
|
\(7\)
| 7.9.8.2 | $x^{9} + 7$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |
|
\(13\)
| 13.9.8.3 | $x^{9} + 52$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $[\ ]_{9}^{3}$ |