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Properties

Label 9.9.3512479453921.1
Degree $9$
Signature $[9, 0]$
Discriminant $3.512\times 10^{12}$
Root discriminant \(24.77\)
Ramified prime $37$
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 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*x - 1)
 
gp: K = bnfinit(y^9 - y^8 - 16*y^7 + 11*y^6 + 66*y^5 - 32*y^4 - 73*y^3 + 7*y^2 + 7*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*x - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - x^8 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*x - 1)
 

\( x^{9} - x^{8} - 16x^{7} + 11x^{6} + 66x^{5} - 32x^{4} - 73x^{3} + 7x^{2} + 7x - 1 \) 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:   \(3512479453921\) \(\medspace = 37^{8}\) 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:  \(24.77\)
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:  $37^{8/9}\approx 24.771772796308063$
Ramified primes:   \(37\) 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:  \(37\)
Dirichlet character group:    $\lbrace$$\chi_{37}(1,·)$, $\chi_{37}(34,·)$, $\chi_{37}(33,·)$, $\chi_{37}(9,·)$, $\chi_{37}(10,·)$, $\chi_{37}(7,·)$, $\chi_{37}(12,·)$, $\chi_{37}(16,·)$, $\chi_{37}(26,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{1333}a^{8}-\frac{547}{1333}a^{7}+\frac{54}{1333}a^{6}-\frac{147}{1333}a^{5}+\frac{348}{1333}a^{4}+\frac{579}{1333}a^{3}-\frac{286}{1333}a^{2}+\frac{202}{1333}a+\frac{354}{1333}$ 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:   $\frac{762}{1333}a^{8}-\frac{918}{1333}a^{7}-\frac{12172}{1333}a^{6}+\frac{10622}{1333}a^{5}+\frac{50563}{1333}a^{4}-\frac{30684}{1333}a^{3}-\frac{55306}{1333}a^{2}+\frac{7294}{1333}a+\frac{3148}{1333}$, $\frac{549}{1333}a^{8}-\frac{378}{1333}a^{7}-\frac{9011}{1333}a^{6}+\frac{3276}{1333}a^{5}+\frac{39090}{1333}a^{4}-\frac{6048}{1333}a^{3}-\frac{49041}{1333}a^{2}-\frac{7739}{1333}a+\frac{5060}{1333}$, $\frac{571}{1333}a^{8}-\frac{415}{1333}a^{7}-\frac{9156}{1333}a^{6}+\frac{4041}{1333}a^{5}+\frac{37415}{1333}a^{4}-\frac{11972}{1333}a^{3}-\frac{42003}{1333}a^{2}+\frac{2037}{1333}a+\frac{4850}{1333}$, $\frac{1067}{1333}a^{8}-\frac{1128}{1333}a^{7}-\frac{17030}{1333}a^{6}+\frac{12442}{1333}a^{5}+\frac{70058}{1333}a^{4}-\frac{34044}{1333}a^{3}-\frac{75886}{1333}a^{2}-\frac{412}{1333}a+\frac{4478}{1333}$, $\frac{1330}{1333}a^{8}-\frac{1025}{1333}a^{7}-\frac{21490}{1333}a^{6}+\frac{9772}{1333}a^{5}+\frac{89600}{1333}a^{4}-\frac{23065}{1333}a^{3}-\frac{100450}{1333}a^{2}-\frac{11270}{1333}a+\frac{4270}{1333}$, $\frac{915}{1333}a^{8}-\frac{630}{1333}a^{7}-\frac{14574}{1333}a^{6}+\frac{5460}{1333}a^{5}+\frac{58485}{1333}a^{4}-\frac{11413}{1333}a^{3}-\frac{61740}{1333}a^{2}-\frac{11121}{1333}a+\frac{3990}{1333}$, $\frac{369}{1333}a^{8}-\frac{560}{1333}a^{7}-\frac{5401}{1333}a^{6}+\frac{7075}{1333}a^{5}+\frac{17773}{1333}a^{4}-\frac{24956}{1333}a^{3}-\frac{8225}{1333}a^{2}+\frac{15886}{1333}a-\frac{2674}{1333}$, $\frac{698}{1333}a^{8}-\frac{568}{1333}a^{7}-\frac{11629}{1333}a^{6}+\frac{5367}{1333}a^{5}+\frac{52285}{1333}a^{4}-\frac{9088}{1333}a^{3}-\frac{67661}{1333}a^{2}-\frac{16298}{1333}a+\frac{5819}{1333}$ 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:  \( 1598.24730552 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 1598.24730552 \cdot 1}{2\cdot\sqrt{3512479453921}}\cr\approx \mathstrut & 0.218311719331 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*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 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*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 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*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 - 16*x^7 + 11*x^6 + 66*x^5 - 32*x^4 - 73*x^3 + 7*x^2 + 7*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

3.3.1369.1

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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.1.0.1}{1} }^{9}$ R ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.1.0.1}{1} }^{9}$ ${\href{/padicField/47.3.0.1}{3} }^{3}$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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
\(37\) Copy content Toggle raw display 37.9.8.1$x^{9} + 37$$9$$1$$8$$C_9$$[\ ]_{9}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.37.9t1.a.a$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (as 9T1) $0$ $1$
* 1.37.9t1.a.b$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (as 9T1) $0$ $1$
* 1.37.3t1.a.a$1$ $ 37 $ 3.3.1369.1 $C_3$ (as 3T1) $0$ $1$
* 1.37.9t1.a.c$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (as 9T1) $0$ $1$
* 1.37.9t1.a.d$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (as 9T1) $0$ $1$
* 1.37.3t1.a.b$1$ $ 37 $ 3.3.1369.1 $C_3$ (as 3T1) $0$ $1$
* 1.37.9t1.a.e$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (as 9T1) $0$ $1$
* 1.37.9t1.a.f$1$ $ 37 $ 9.9.3512479453921.1 $C_9$ (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 *.