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Properties

Label 9.9.425...161.1
Degree $9$
Signature $[9, 0]$
Discriminant $4.257\times 10^{20}$
Root discriminant \(195.94\)
Ramified prime $379$
Class number $1$ (GRH)
Class group trivial (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 - x^8 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245)
 
gp: K = bnfinit(y^9 - y^8 - 168*y^7 + 591*y^6 + 5776*y^5 - 20434*y^4 - 69977*y^3 + 148399*y^2 + 389543*y + 65245, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245)
 

\( x^{9} - x^{8} - 168x^{7} + 591x^{6} + 5776x^{5} - 20434x^{4} - 69977x^{3} + 148399x^{2} + 389543x + 65245 \) 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:   \(425709831200577608161\) \(\medspace = 379^{8}\) 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:  \(195.94\)
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:  $379^{8/9}\approx 195.9402880627296$
Ramified primes:   \(379\) 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:  \(379\)
Dirichlet character group:    $\lbrace$$\chi_{379}(84,·)$, $\chi_{379}(1,·)$, $\chi_{379}(115,·)$, $\chi_{379}(51,·)$, $\chi_{379}(327,·)$, $\chi_{379}(234,·)$, $\chi_{379}(339,·)$, $\chi_{379}(180,·)$, $\chi_{379}(185,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{7}-\frac{1}{5}a^{3}$, $\frac{1}{33339274882175}a^{8}+\frac{2384396975508}{33339274882175}a^{7}-\frac{2693500521946}{33339274882175}a^{6}-\frac{1213107133993}{33339274882175}a^{5}+\frac{7429528544494}{33339274882175}a^{4}-\frac{13320900299313}{33339274882175}a^{3}-\frac{4409484713769}{33339274882175}a^{2}-\frac{7006629083702}{33339274882175}a+\frac{1605265949684}{6667854976435}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

Class group and class number

Trivial group, which has order $1$ (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:   $\frac{73932478408}{33339274882175}a^{8}+\frac{479190281304}{33339274882175}a^{7}-\frac{10024038521968}{33339274882175}a^{6}-\frac{34239279794669}{33339274882175}a^{5}+\frac{344499283655727}{33339274882175}a^{4}+\frac{928826168526006}{33339274882175}a^{3}-\frac{32\!\cdots\!02}{33339274882175}a^{2}-\frac{85\!\cdots\!66}{33339274882175}a-\frac{286357651772353}{6667854976435}$, $\frac{49866184449}{33339274882175}a^{8}+\frac{365207507347}{33339274882175}a^{7}-\frac{5910658248504}{33339274882175}a^{6}-\frac{20674805497357}{33339274882175}a^{5}+\frac{187817624052006}{33339274882175}a^{4}+\frac{364245545545758}{33339274882175}a^{3}-\frac{14\!\cdots\!81}{33339274882175}a^{2}-\frac{30\!\cdots\!48}{33339274882175}a-\frac{100568250743739}{6667854976435}$, $\frac{71648888346}{33339274882175}a^{8}-\frac{248172446577}{33339274882175}a^{7}-\frac{11514434053496}{33339274882175}a^{6}+\frac{70393450618932}{33339274882175}a^{5}+\frac{251745988561899}{33339274882175}a^{4}-\frac{20\!\cdots\!28}{33339274882175}a^{3}-\frac{130021440871544}{33339274882175}a^{2}+\frac{11\!\cdots\!48}{33339274882175}a+\frac{484866287013484}{6667854976435}$, $\frac{41683522346}{33339274882175}a^{8}-\frac{136526631792}{33339274882175}a^{7}-\frac{6702126790831}{33339274882175}a^{6}+\frac{39659554540987}{33339274882175}a^{5}+\frac{153818698721799}{33339274882175}a^{4}-\frac{11\!\cdots\!38}{33339274882175}a^{3}-\frac{303694784028359}{33339274882175}a^{2}+\frac{68\!\cdots\!68}{33339274882175}a+\frac{273359461495254}{6667854976435}$, $\frac{24993786106}{33339274882175}a^{8}+\frac{66734081263}{33339274882175}a^{7}-\frac{3426862213851}{33339274882175}a^{6}+\frac{3985165607027}{33339274882175}a^{5}+\frac{90515653240989}{33339274882175}a^{4}-\frac{159435957245768}{33339274882175}a^{3}-\frac{482012916294939}{33339274882175}a^{2}+\frac{399434270884578}{33339274882175}a+\frac{17627791232129}{6667854976435}$, $\frac{19659637108}{6667854976435}a^{8}+\frac{74896026564}{6667854976435}a^{7}-\frac{2766205391754}{6667854976435}a^{6}-\frac{1223754704209}{6667854976435}a^{5}+\frac{82306016863897}{6667854976435}a^{4}+\frac{13654953932661}{6667854976435}a^{3}-\frac{597985492437556}{6667854976435}a^{2}-\frac{682647154863606}{6667854976435}a-\frac{20792037545421}{1333570995287}$, $\frac{94756232018}{33339274882175}a^{8}+\frac{360507054019}{33339274882175}a^{7}-\frac{13717687169353}{33339274882175}a^{6}-\frac{7718998441634}{33339274882175}a^{5}+\frac{444356074621892}{33339274882175}a^{4}+\frac{105309565917566}{33339274882175}a^{3}-\frac{40\!\cdots\!17}{33339274882175}a^{2}-\frac{30\!\cdots\!01}{33339274882175}a+\frac{934434020130797}{6667854976435}$, $\frac{26041649866}{33339274882175}a^{8}-\frac{68274248067}{33339274882175}a^{7}-\frac{4235197015031}{33339274882175}a^{6}+\frac{22637525812667}{33339274882175}a^{5}+\frac{114251360494529}{33339274882175}a^{4}-\frac{719233195633588}{33339274882175}a^{3}-\frac{785890653699309}{33339274882175}a^{2}+\frac{47\!\cdots\!63}{33339274882175}a+\frac{916676023501034}{6667854976435}$ 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:  \( 67087578.0834 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 67087578.0834 \cdot 1}{2\cdot\sqrt{425709831200577608161}}\cr\approx \mathstrut & 0.832386904772 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245)
 
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 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245, 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 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245);
 
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 - 168*x^7 + 591*x^6 + 5776*x^5 - 20434*x^4 - 69977*x^3 + 148399*x^2 + 389543*x + 65245);
 
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

3.3.143641.1

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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.1.0.1}{1} }^{9}$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{3}$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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
\(379\) Copy content Toggle raw display Deg $9$$9$$1$$8$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.379.9t1.a.a$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (as 9T1) $0$ $1$
* 1.379.9t1.a.b$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (as 9T1) $0$ $1$
* 1.379.3t1.a.a$1$ $ 379 $ 3.3.143641.1 $C_3$ (as 3T1) $0$ $1$
* 1.379.9t1.a.c$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (as 9T1) $0$ $1$
* 1.379.9t1.a.d$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (as 9T1) $0$ $1$
* 1.379.3t1.a.b$1$ $ 379 $ 3.3.143641.1 $C_3$ (as 3T1) $0$ $1$
* 1.379.9t1.a.e$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (as 9T1) $0$ $1$
* 1.379.9t1.a.f$1$ $ 379 $ 9.9.425709831200577608161.1 $C_9$ (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 *.