Properties

Label 9.9.1476349596018920529.1
Degree $9$
Signature $[9, 0]$
Discriminant $1.476\times 10^{18}$
Root discriminant \(104.42\)
Ramified primes $3,19$
Class number $3$ (GRH)
Class group [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 - 171*x^7 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017)
 
gp: K = bnfinit(y^9 - 171*y^7 + 9747*y^5 - 205770*y^3 + 1172889*y - 1118017, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 171*x^7 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 171*x^7 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017)
 

\( x^{9} - 171x^{7} + 9747x^{5} - 205770x^{3} + 1172889x - 1118017 \) 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:   \(1476349596018920529\) \(\medspace = 3^{22}\cdot 19^{6}\) 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:  \(104.42\)
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:  $3^{22/9}19^{2/3}\approx 104.4236335758831$
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$$\chi_{513}(448,·)$, $\chi_{513}(1,·)$, $\chi_{513}(292,·)$, $\chi_{513}(106,·)$, $\chi_{513}(172,·)$, $\chi_{513}(463,·)$, $\chi_{513}(277,·)$, $\chi_{513}(343,·)$, $\chi_{513}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{19}a^{3}$, $\frac{1}{19}a^{4}$, $\frac{1}{323}a^{5}+\frac{6}{323}a^{4}+\frac{7}{323}a^{3}-\frac{7}{17}a^{2}-\frac{7}{17}a+\frac{7}{17}$, $\frac{1}{6137}a^{6}-\frac{6}{323}a^{4}+\frac{6}{323}a^{3}-\frac{8}{17}a^{2}-\frac{1}{17}a-\frac{4}{17}$, $\frac{1}{6137}a^{7}+\frac{8}{323}a^{4}-\frac{8}{323}a^{3}+\frac{8}{17}a^{2}+\frac{5}{17}a+\frac{8}{17}$, $\frac{1}{6137}a^{8}-\frac{5}{323}a^{4}-\frac{6}{323}a^{3}-\frac{7}{17}a^{2}-\frac{4}{17}a-\frac{5}{17}$ 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

$C_{3}$, 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 $2$) 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:   $\frac{3}{6137}a^{6}-\frac{18}{323}a^{4}-\frac{16}{323}a^{3}+\frac{27}{17}a^{2}+\frac{48}{17}a-\frac{114}{17}$, $\frac{2}{6137}a^{6}-\frac{12}{323}a^{4}-\frac{5}{323}a^{3}+\frac{18}{17}a^{2}+\frac{15}{17}a-\frac{59}{17}$, $\frac{1}{6137}a^{8}-\frac{13}{6137}a^{7}-\frac{182}{6137}a^{6}+\frac{100}{323}a^{5}+\frac{529}{323}a^{4}-\frac{4204}{323}a^{3}-\frac{392}{17}a^{2}+\frac{2184}{17}a-\frac{1792}{17}$, $\frac{1}{6137}a^{8}-\frac{16}{6137}a^{7}-\frac{106}{6137}a^{6}+\frac{87}{323}a^{5}+\frac{141}{323}a^{4}-\frac{2353}{323}a^{3}-\frac{49}{17}a^{2}+\frac{841}{17}a-\frac{783}{17}$, $\frac{1}{6137}a^{8}+\frac{11}{6137}a^{7}-\frac{115}{6137}a^{6}-\frac{75}{323}a^{5}+\frac{9}{19}a^{4}+\frac{151}{19}a^{3}-\frac{4}{17}a^{2}-\frac{890}{17}a+\frac{868}{17}$, $\frac{3}{6137}a^{8}-\frac{40}{6137}a^{7}-\frac{126}{6137}a^{6}+\frac{177}{323}a^{5}-\frac{387}{323}a^{4}-\frac{2377}{323}a^{3}+\frac{465}{17}a^{2}-\frac{458}{17}a+\frac{99}{17}$, $\frac{2}{6137}a^{8}-\frac{3}{6137}a^{7}-\frac{297}{6137}a^{6}+\frac{10}{323}a^{5}+\frac{771}{323}a^{4}+\frac{11}{19}a^{3}-\frac{741}{17}a^{2}-39a+168$, $\frac{1}{6137}a^{8}+\frac{5}{6137}a^{7}-\frac{145}{6137}a^{6}-\frac{36}{323}a^{5}+\frac{332}{323}a^{4}+\frac{1365}{323}a^{3}-14a^{2}-\frac{619}{17}a+\frac{1298}{17}$ 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:  \( 784138.972462 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 784138.972462 \cdot 3}{2\cdot\sqrt{1476349596018920529}}\cr\approx \mathstrut & 0.495632360735 \end{aligned}\] (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 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017)
 
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 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017, 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 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017);
 
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 + 9747*x^5 - 205770*x^3 + 1172889*x - 1118017);
 
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

\(\Q(\zeta_{9})^+\)

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 ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.1.0.1}{1} }^{9}$ R ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.9.0.1}{9} }$

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 3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
\(19\) Copy content Toggle raw display 19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} + 38$$3$$1$$2$$C_3$$[\ ]_{3}$