↑

Properties

Label 9.9.532...969.6
Degree $9$
Signature $[9, 0]$
Discriminant $5.330\times 10^{20}$
Root discriminant \(200.89\)
Ramified primes $3,19$
Class number $9$ (GRH)
Class group [9] (GRH)
Galois group $C_9$ (as 9T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375)
 
gp: K = bnfinit(y^9 - 171*y^7 - 342*y^6 + 3591*y^5 + 5130*y^4 - 26904*y^3 - 18468*y^2 + 68913*y - 2375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375)
 

\( x^{9} - 171x^{7} - 342x^{6} + 3591x^{5} + 5130x^{4} - 26904x^{3} - 18468x^{2} + 68913x - 2375 \) Copy content Toggle raw display

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:   \(532962204162830310969\) \(\medspace = 3^{22}\cdot 19^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(200.89\)
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:  $3^{22/9}19^{8/9}\approx 200.8936770541996$
Ramified primes:   \(3\), \(19\) Copy content Toggle raw display
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:  \(513=3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{513}(64,·)$, $\chi_{513}(1,·)$, $\chi_{513}(139,·)$, $\chi_{513}(358,·)$, $\chi_{513}(427,·)$, $\chi_{513}(175,·)$, $\chi_{513}(340,·)$, $\chi_{513}(214,·)$, $\chi_{513}(505,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{7}+\frac{2}{25}a^{6}+\frac{2}{25}a^{5}-\frac{1}{5}a^{4}-\frac{1}{25}a^{3}+\frac{8}{25}a^{2}-\frac{7}{25}a$, $\frac{1}{743828021875}a^{8}+\frac{6870012161}{743828021875}a^{7}-\frac{94889092}{5950624175}a^{6}-\frac{27424038717}{743828021875}a^{5}-\frac{151862467596}{743828021875}a^{4}+\frac{135638788924}{743828021875}a^{3}-\frac{28414832928}{148765604375}a^{2}+\frac{360081196992}{743828021875}a+\frac{39870279}{5950624175}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

$C_{9}$, which has order $9$ (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 $2$) Copy content Toggle raw display
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:   $\frac{7393032}{29753120875}a^{8}+\frac{40764852}{29753120875}a^{7}-\frac{46605277}{1190124835}a^{6}-\frac{9015520644}{29753120875}a^{5}-\frac{3860438322}{29753120875}a^{4}+\frac{73005471868}{29753120875}a^{3}+\frac{23004221139}{5950624175}a^{2}+\frac{15675010944}{29753120875}a+\frac{222212006}{238024967}$, $\frac{7393032}{29753120875}a^{8}+\frac{40764852}{29753120875}a^{7}-\frac{46605277}{1190124835}a^{6}-\frac{9015520644}{29753120875}a^{5}-\frac{3860438322}{29753120875}a^{4}+\frac{73005471868}{29753120875}a^{3}+\frac{23004221139}{5950624175}a^{2}+\frac{15675010944}{29753120875}a-\frac{15812961}{238024967}$, $\frac{208210064}{148765604375}a^{8}-\frac{905846}{148765604375}a^{7}-\frac{1359058032}{5950624175}a^{6}-\frac{71855504438}{148765604375}a^{5}+\frac{477998378356}{148765604375}a^{4}+\frac{570474164036}{148765604375}a^{3}-\frac{341433552732}{29753120875}a^{2}-\frac{496044352087}{148765604375}a+\frac{79575641}{1190124835}$, $\frac{2184987589}{743828021875}a^{8}-\frac{7218423921}{743828021875}a^{7}-\frac{2703106748}{5950624175}a^{6}+\frac{338680837312}{743828021875}a^{5}+\frac{4886983177706}{743828021875}a^{4}-\frac{4521360513889}{743828021875}a^{3}-\frac{4072680961592}{148765604375}a^{2}+\frac{17609736721038}{743828021875}a+\frac{122938426506}{5950624175}$, $\frac{3632823554}{743828021875}a^{8}-\frac{15678717306}{743828021875}a^{7}-\frac{4412172038}{5950624175}a^{6}+\frac{1130753614157}{743828021875}a^{5}+\frac{7854598887566}{743828021875}a^{4}-\frac{14897936773479}{743828021875}a^{3}-\frac{5823601366862}{148765604375}a^{2}+\frac{53984832863943}{743828021875}a+\frac{15121109191}{5950624175}$, $\frac{1785325448}{743828021875}a^{8}+\frac{2315534003}{743828021875}a^{7}-\frac{2354281192}{5950624175}a^{6}-\frac{997912871591}{743828021875}a^{5}+\frac{3796993068867}{743828021875}a^{4}+\frac{12088690098952}{743828021875}a^{3}-\frac{2308274496844}{148765604375}a^{2}-\frac{33295825231834}{743828021875}a+\frac{9731216592}{5950624175}$, $\frac{21924565849}{743828021875}a^{8}-\frac{69763216686}{743828021875}a^{7}-\frac{5686115816}{1190124835}a^{6}+\frac{3824517931517}{743828021875}a^{5}+\frac{70929724090371}{743828021875}a^{4}-\frac{106371555449649}{743828021875}a^{3}-\frac{63966846684947}{148765604375}a^{2}+\frac{562512897638333}{743828021875}a-\frac{166068227804}{5950624175}$, $\frac{16156180261}{743828021875}a^{8}+\frac{46671062021}{743828021875}a^{7}-\frac{20991314824}{5950624175}a^{6}-\frac{13101795441637}{743828021875}a^{5}+\frac{19512141290569}{743828021875}a^{4}+\frac{137371399739914}{743828021875}a^{3}-\frac{5688379663533}{148765604375}a^{2}-\frac{367221924588588}{743828021875}a+\frac{102480467244}{5950624175}$ Copy content Toggle raw display (assuming GRH)
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:  \( 22548077.9628 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 22548077.9628 \cdot 9}{2\cdot\sqrt{532962204162830310969}}\cr\approx \mathstrut & 2.25031749605 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375)
 
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 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375, 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 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375);
 
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 - 171*x^7 - 342*x^6 + 3591*x^5 + 5130*x^4 - 26904*x^3 - 18468*x^2 + 68913*x - 2375);
 
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

$C_9$ (as 9T1):

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

3.3.29241.2

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.9.0.1}{9} }$ R ${\href{/padicField/5.1.0.1}{1} }^{9}$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ R ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.1.0.1}{1} }^{9}$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.1.0.1}{1} }^{9}$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.9.22.9$x^{9} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 75$$9$$1$$22$$C_9$$[2, 3]$
\(19\) Copy content Toggle raw display 19.9.8.3$x^{9} + 152$$9$$1$$8$$C_9$$[\ ]_{9}$