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Properties

Label 9.9.147...289.5
Degree $9$
Signature $[9, 0]$
Discriminant $1.476\times 10^{26}$
Root discriminant \(808.47\)
Ramified primes $3,7,13$
Class number $27$ (GRH)
Class group [3, 9] (GRH)
Galois group $C_9:C_3$ (as 9T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 819*x^7 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259)
 
gp: K = bnfinit(y^9 - 819*y^7 + 223587*y^5 - 22607130*y^3 + 617174649*y - 1302924259, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 819*x^7 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 819*x^7 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259)
 

\( x^{9} - 819x^{7} + 223587x^{5} - 22607130x^{3} + 617174649x - 1302924259 \) 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:   \(147570226003740066617015289\) \(\medspace = 3^{22}\cdot 7^{8}\cdot 13^{8}\) 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:  \(808.47\)
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:   \(3\), \(7\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{91}a^{3}$, $\frac{1}{91}a^{4}$, $\frac{1}{8281}a^{5}$, $\frac{1}{8281}a^{6}$, $\frac{1}{753571}a^{7}$, $\frac{1}{753571}a^{8}$ 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

$C_{3}\times C_{9}$, 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 $2$) 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:   $\frac{1}{753571}a^{7}-\frac{6}{8281}a^{5}+\frac{10}{91}a^{3}-4a+10$, $\frac{1}{753571}a^{8}-\frac{10}{753571}a^{7}-\frac{8}{8281}a^{6}+\frac{79}{8281}a^{5}+\frac{19}{91}a^{4}-\frac{184}{91}a^{3}-11a^{2}+102a-172$, $\frac{1}{91}a^{3}-3a+10$, $\frac{1}{91}a^{3}-3a+9$, $\frac{17872}{753571}a^{8}-\frac{292314}{753571}a^{7}-\frac{102433}{8281}a^{6}+\frac{122397}{637}a^{5}+\frac{161026}{91}a^{4}-\frac{2182405}{91}a^{3}-81047a^{2}+734929a-987249$, $\frac{6722}{107653}a^{8}+\frac{137972}{753571}a^{7}-\frac{32075}{637}a^{6}-\frac{1253370}{8281}a^{5}+\frac{1217779}{91}a^{4}+\frac{3734632}{91}a^{3}-1269889a^{2}-3940574a+25774011$, $\frac{75039409}{753571}a^{8}+\frac{234343506}{753571}a^{7}-\frac{667312337}{8281}a^{6}-\frac{2083972862}{8281}a^{5}+\frac{1954545493}{91}a^{4}+\frac{6103908008}{91}a^{3}-2041707897a^{2}-6376102145a+41545222183$, $\frac{46656021}{753571}a^{8}+\frac{687633434}{753571}a^{7}-\frac{44076406}{1183}a^{6}-\frac{4547320853}{8281}a^{5}+\frac{523230016}{91}a^{4}+\frac{7711655729}{91}a^{3}-150711176a^{2}-2221315300a+5473435413$ 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:  \( 1102928823.35 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 1102928823.35 \cdot 27}{2\cdot\sqrt{147570226003740066617015289}}\cr\approx \mathstrut & 0.627555073150 \end{aligned}\] (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 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259)
 
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 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259, 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 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259);
 
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 + 223587*x^5 - 22607130*x^3 + 617174649*x - 1302924259);
 
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

$C_9:C_3$ (as 9T6):

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

3.3.670761.4

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 ${\href{/padicField/2.9.0.1}{9} }$ R ${\href{/padicField/5.9.0.1}{9} }$ R ${\href{/padicField/11.9.0.1}{9} }$ R ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.1.0.1}{1} }^{9}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.9.0.1}{9} }$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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 3.9.22.6$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$[2, 3]$
\(7\) Copy content Toggle raw display 7.9.8.1$x^{9} + 14$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
\(13\) Copy content Toggle raw display 13.9.8.1$x^{9} + 26$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$