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Properties

Label 9.9.428...721.1
Degree $9$
Signature $[9, 0]$
Discriminant $4.286\times 10^{20}$
Root discriminant \(196.09\)
Ramified primes $3,73$
Class number $9$ (GRH)
Class group [3, 3] (GRH)
Galois group $C_9$ (as 9T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081)
 
gp: K = bnfinit(y^9 - 219*y^7 - 730*y^6 + 9855*y^5 + 36792*y^4 - 148482*y^3 - 484866*y^2 + 609039*y + 810081, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081)
 

\( x^{9} - 219x^{7} - 730x^{6} + 9855x^{5} + 36792x^{4} - 148482x^{3} - 484866x^{2} + 609039x + 810081 \) 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:   \(428585957696282300721\) \(\medspace = 3^{12}\cdot 73^{8}\) 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:  \(196.09\)
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:  $3^{4/3}73^{8/9}\approx 196.08693569504558$
Ramified primes:   \(3\), \(73\) 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:  \(657=3^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{657}(256,·)$, $\chi_{657}(1,·)$, $\chi_{657}(4,·)$, $\chi_{657}(64,·)$, $\chi_{657}(493,·)$, $\chi_{657}(367,·)$, $\chi_{657}(16,·)$, $\chi_{657}(616,·)$, $\chi_{657}(154,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{27}a^{6}+\frac{2}{9}a^{4}+\frac{8}{27}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{81}a^{7}+\frac{2}{27}a^{5}+\frac{35}{81}a^{4}+\frac{1}{3}a^{3}+\frac{1}{9}a^{2}+\frac{4}{9}a$, $\frac{1}{413916568623801}a^{8}+\frac{192458090140}{137972189541267}a^{7}+\frac{1995748914698}{137972189541267}a^{6}-\frac{32833978338466}{413916568623801}a^{5}+\frac{32795724327563}{137972189541267}a^{4}+\frac{14660325714680}{45990729847089}a^{3}-\frac{14652602610068}{45990729847089}a^{2}-\frac{4762405218476}{15330243282363}a+\frac{1381028858582}{5110081094121}$ Copy content Toggle raw display

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

$C_{3}\times C_{3}$, 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:   $\frac{7911811492}{413916568623801}a^{8}+\frac{74594127098}{137972189541267}a^{7}-\frac{908692455283}{137972189541267}a^{6}-\frac{49907166813862}{413916568623801}a^{5}+\frac{36963862750744}{137972189541267}a^{4}+\frac{250204472549045}{45990729847089}a^{3}-\frac{230633230381868}{45990729847089}a^{2}-\frac{10\!\cdots\!13}{15330243282363}a+\frac{359833754029253}{5110081094121}$, $\frac{10034134732}{137972189541267}a^{8}-\frac{43821007774}{137972189541267}a^{7}-\frac{642502834945}{45990729847089}a^{6}+\frac{1592211733166}{137972189541267}a^{5}+\frac{75588447989050}{137972189541267}a^{4}-\frac{12757316730404}{15330243282363}a^{3}-\frac{28292804389505}{5110081094121}a^{2}+\frac{354591279675053}{15330243282363}a-\frac{39475113142669}{1703360364707}$, $\frac{163908409754}{413916568623801}a^{8}-\frac{312198410180}{137972189541267}a^{7}-\frac{10195942494917}{137972189541267}a^{6}+\frac{55682170475659}{413916568623801}a^{5}+\frac{434128245726422}{137972189541267}a^{4}-\frac{162119476282397}{45990729847089}a^{3}-\frac{17\!\cdots\!64}{45990729847089}a^{2}+\frac{494895269498458}{15330243282363}a+\frac{284343204023632}{5110081094121}$, $\frac{77102983939}{413916568623801}a^{8}-\frac{28404242210}{15330243282363}a^{7}-\frac{3190363041703}{137972189541267}a^{6}+\frac{43978910705132}{413916568623801}a^{5}+\frac{12362121073765}{15330243282363}a^{4}-\frac{85455045844999}{45990729847089}a^{3}-\frac{417489614221178}{45990729847089}a^{2}+\frac{47941725235958}{5110081094121}a+\frac{71097695363816}{5110081094121}$, $\frac{718261336}{2977817040459}a^{8}-\frac{1549718536}{992605680153}a^{7}-\frac{43595785978}{992605680153}a^{6}+\frac{310808311784}{2977817040459}a^{5}+\frac{1912950228712}{992605680153}a^{4}-\frac{643463038346}{330868560051}a^{3}-\frac{8271546068834}{330868560051}a^{2}+\frac{429829407110}{110289520017}a+\frac{225721010741}{12254391113}$, $\frac{55541637266}{137972189541267}a^{8}-\frac{388397015458}{137972189541267}a^{7}-\frac{3318086617541}{45990729847089}a^{6}+\frac{29932875846358}{137972189541267}a^{5}+\frac{433519199332306}{137972189541267}a^{4}-\frac{84227694011215}{15330243282363}a^{3}-\frac{632764027008608}{15330243282363}a^{2}+\frac{594000131585723}{15330243282363}a+\frac{105233022804719}{1703360364707}$, $\frac{2103238016}{137972189541267}a^{8}-\frac{16268200}{137972189541267}a^{7}-\frac{56394662156}{15330243282363}a^{6}-\frac{1196840523659}{137972189541267}a^{5}+\frac{27386766079981}{137972189541267}a^{4}+\frac{22029042442780}{45990729847089}a^{3}-\frac{57282414675902}{15330243282363}a^{2}-\frac{98690040594958}{15330243282363}a+\frac{105572005237367}{5110081094121}$, $\frac{2438725099}{137972189541267}a^{8}+\frac{46532797280}{137972189541267}a^{7}-\frac{257695918217}{45990729847089}a^{6}-\frac{8971001020342}{137972189541267}a^{5}+\frac{25938288410896}{137972189541267}a^{4}+\frac{69029527534102}{45990729847089}a^{3}-\frac{14681737769015}{5110081094121}a^{2}-\frac{18330516060235}{15330243282363}a+\frac{11024821907915}{5110081094121}$ 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:  \( 4758026.42602 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 4758026.42602 \cdot 9}{2\cdot\sqrt{428585957696282300721}}\cr\approx \mathstrut & 0.529529739145 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081)
 
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 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081, 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 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081);
 
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 - 219*x^7 - 730*x^6 + 9855*x^5 + 36792*x^4 - 148482*x^3 - 484866*x^2 + 609039*x + 810081);
 
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

3.3.5329.1

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 ${\href{/padicField/2.9.0.1}{9} }$ R ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.9.0.1}{9} }$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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 3.3.4.1$x^{3} + 6 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} + 6 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.3.4.1$x^{3} + 6 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
\(73\) Copy content Toggle raw display 73.9.8.1$x^{9} + 73$$9$$1$$8$$C_9$$[\ ]_{9}$