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Properties

Label 9.9.806460091894081.1
Degree $9$
Signature $[9, 0]$
Discriminant $8.065\times 10^{14}$
Root discriminant \(45.32\)
Ramified prime $73$
Class number $1$
Class group trivial
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 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^9 - y^8 - 32*y^7 + 11*y^6 + 278*y^5 + 34*y^4 - 427*y^3 - 150*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1)
 

\( x^{9} - x^{8} - 32x^{7} + 11x^{6} + 278x^{5} + 34x^{4} - 427x^{3} - 150x^{2} - 8x + 1 \) 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:   \(806460091894081\) \(\medspace = 73^{8}\) 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:  \(45.32\)
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:  $73^{8/9}\approx 45.319695872342585$
Ramified primes:   \(73\) 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:  \(73\)
Dirichlet character group:    $\lbrace$$\chi_{73}(32,·)$, $\chi_{73}(1,·)$, $\chi_{73}(2,·)$, $\chi_{73}(4,·)$, $\chi_{73}(37,·)$, $\chi_{73}(8,·)$, $\chi_{73}(64,·)$, $\chi_{73}(16,·)$, $\chi_{73}(55,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{49053}a^{8}-\frac{688}{49053}a^{7}-\frac{1555}{49053}a^{6}+\frac{1827}{16351}a^{5}+\frac{11912}{49053}a^{4}-\frac{24361}{49053}a^{3}-\frac{7844}{49053}a^{2}-\frac{2384}{16351}a-\frac{2745}{16351}$ 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

Trivial group, which has order $1$

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:   $a$, $\frac{15855}{16351}a^{8}-\frac{18474}{16351}a^{7}-\frac{504098}{16351}a^{6}+\frac{257306}{16351}a^{5}+\frac{4360076}{16351}a^{4}-\frac{180194}{16351}a^{3}-\frac{6704824}{16351}a^{2}-\frac{1227100}{16351}a+\frac{29512}{16351}$, $\frac{49784}{49053}a^{8}-\frac{61451}{49053}a^{7}-\frac{1578182}{49053}a^{6}+\frac{305424}{16351}a^{5}+\frac{13612972}{49053}a^{4}-\frac{1473242}{49053}a^{3}-\frac{20842288}{49053}a^{2}-\frac{908803}{16351}a+\frac{69982}{16351}$, $\frac{20442}{16351}a^{8}-\frac{72112}{49053}a^{7}-\frac{1948667}{49053}a^{6}+\frac{1014061}{49053}a^{5}+\frac{5614405}{16351}a^{4}-\frac{822068}{49053}a^{3}-\frac{25862006}{49053}a^{2}-\frac{4795171}{49053}a+\frac{59079}{16351}$, $\frac{9125}{16351}a^{8}-\frac{30350}{49053}a^{7}-\frac{291025}{16351}a^{6}+\frac{398023}{49053}a^{5}+\frac{7573520}{49053}a^{4}+\frac{74915}{49053}a^{3}-\frac{11731835}{49053}a^{2}-\frac{920815}{16351}a+\frac{128920}{49053}$, $\frac{3142}{16351}a^{8}-\frac{3364}{16351}a^{7}-\frac{301252}{49053}a^{6}+\frac{125554}{49053}a^{5}+\frac{2616355}{49053}a^{4}+\frac{45822}{16351}a^{3}-\frac{4020668}{49053}a^{2}-\frac{1225904}{49053}a-\frac{86668}{49053}$, $\frac{115423}{49053}a^{8}-\frac{141376}{49053}a^{7}-\frac{1220154}{16351}a^{6}+\frac{2089850}{49053}a^{5}+\frac{31586020}{49053}a^{4}-\frac{3143029}{49053}a^{3}-\frac{16129800}{16351}a^{2}-\frac{6385600}{49053}a+\frac{498457}{49053}$, $\frac{28781}{49053}a^{8}-\frac{32969}{49053}a^{7}-\frac{305808}{16351}a^{6}+\frac{452041}{49053}a^{5}+\frac{7970792}{49053}a^{4}-\frac{215624}{49053}a^{3}-\frac{4169474}{16351}a^{2}-\frac{2353517}{49053}a+\frac{192422}{49053}$ Copy content Toggle raw display
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:  \( 15512.1980357 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 15512.1980357 \cdot 1}{2\cdot\sqrt{806460091894081}}\cr\approx \mathstrut & 0.139836925010 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1)
 
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 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1, 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 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1);
 
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 - x^8 - 32*x^7 + 11*x^6 + 278*x^5 + 34*x^4 - 427*x^3 - 150*x^2 - 8*x + 1);
 
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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.3.0.1}{3} }^{3}$ ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.9.0.1}{9} }$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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
\(73\) Copy content Toggle raw display 73.9.8.1$x^{9} + 73$$9$$1$$8$$C_9$$[\ ]_{9}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.73.9t1.a.a$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$
* 1.73.9t1.a.b$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$
* 1.73.3t1.a.a$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
* 1.73.9t1.a.c$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$
* 1.73.9t1.a.d$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$
* 1.73.3t1.a.b$1$ $ 73 $ 3.3.5329.1 $C_3$ (as 3T1) $0$ $1$
* 1.73.9t1.a.e$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$
* 1.73.9t1.a.f$1$ $ 73 $ 9.9.806460091894081.1 $C_9$ (as 9T1) $0$ $1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.