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Properties

Label 9.9.150...921.1
Degree $9$
Signature $[9, 0]$
Discriminant $1.507\times 10^{19}$
Root discriminant \(135.18\)
Ramified primes $19,31$
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 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547)
 
gp: K = bnfinit(y^9 - y^8 - 198*y^7 - 107*y^6 + 10319*y^5 + 973*y^4 - 189602*y^3 + 170782*y^2 + 815941*y - 1009547, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547)
 

\( x^{9} - x^{8} - 198x^{7} - 107x^{6} + 10319x^{5} + 973x^{4} - 189602x^{3} + 170782x^{2} + 815941x - 1009547 \) 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:   \(15072974715383053921\) \(\medspace = 19^{8}\cdot 31^{6}\) 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:  \(135.18\)
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:  $19^{8/9}31^{2/3}\approx 135.17955010958042$
Ramified primes:   \(19\), \(31\) 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:  \(589=19\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{589}(1,·)$, $\chi_{589}(36,·)$, $\chi_{589}(5,·)$, $\chi_{589}(180,·)$, $\chi_{589}(118,·)$, $\chi_{589}(311,·)$, $\chi_{589}(25,·)$, $\chi_{589}(125,·)$, $\chi_{589}(377,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{77}a^{6}-\frac{3}{77}a^{5}+\frac{37}{77}a^{4}-\frac{34}{77}a^{3}-\frac{10}{77}a^{2}+\frac{30}{77}a$, $\frac{1}{77}a^{7}-\frac{5}{77}a^{5}+\frac{1}{7}a^{4}+\frac{31}{77}a^{3}-\frac{1}{7}a^{2}-\frac{20}{77}a$, $\frac{1}{296127980083859}a^{8}-\frac{204901758108}{42303997154837}a^{7}+\frac{1453221995007}{296127980083859}a^{6}-\frac{11025878218788}{296127980083859}a^{5}-\frac{23750577604970}{296127980083859}a^{4}-\frac{87961651417616}{296127980083859}a^{3}+\frac{96774401398772}{296127980083859}a^{2}-\frac{69273911285653}{296127980083859}a+\frac{1639347761804}{3845817923167}$ 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:  $7$

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:   $\frac{2709219}{410718418979}a^{8}+\frac{150621685}{2875028932853}a^{7}-\frac{4148848936}{2875028932853}a^{6}-\frac{30551023512}{2875028932853}a^{5}+\frac{191430252330}{2875028932853}a^{4}+\frac{1062126291368}{2875028932853}a^{3}-\frac{2858986244157}{2875028932853}a^{2}-\frac{4919143222550}{2875028932853}a+\frac{108388105378}{37338038089}$, $\frac{1526912}{2875028932853}a^{8}-\frac{60952054}{2875028932853}a^{7}-\frac{210484556}{2875028932853}a^{6}+\frac{1532992111}{410718418979}a^{5}+\frac{17515275566}{2875028932853}a^{4}-\frac{457240165708}{2875028932853}a^{3}-\frac{259325324669}{2875028932853}a^{2}+\frac{5266689599100}{2875028932853}a-\frac{63532605412}{37338038089}$, $\frac{295670748272}{296127980083859}a^{8}-\frac{330213742886}{42303997154837}a^{7}-\frac{6204833541400}{42303997154837}a^{6}+\frac{267667681613175}{296127980083859}a^{5}+\frac{13\!\cdots\!27}{296127980083859}a^{4}-\frac{90\!\cdots\!03}{296127980083859}a^{3}+\frac{20\!\cdots\!12}{296127980083859}a^{2}+\frac{45\!\cdots\!58}{296127980083859}a-\frac{639366699879544}{3845817923167}$, $\frac{696912598941}{296127980083859}a^{8}+\frac{276601023136}{296127980083859}a^{7}-\frac{137489862909051}{296127980083859}a^{6}-\frac{266996390094723}{296127980083859}a^{5}+\frac{68\!\cdots\!59}{296127980083859}a^{4}+\frac{14\!\cdots\!47}{42303997154837}a^{3}-\frac{11\!\cdots\!80}{296127980083859}a^{2}-\frac{46\!\cdots\!04}{296127980083859}a+\frac{65\!\cdots\!99}{3845817923167}$, $\frac{45055820958}{42303997154837}a^{8}+\frac{1634357380471}{296127980083859}a^{7}-\frac{52845330229611}{296127980083859}a^{6}-\frac{364121045178459}{296127980083859}a^{5}+\frac{153910113634484}{42303997154837}a^{4}+\frac{77\!\cdots\!87}{296127980083859}a^{3}-\frac{11\!\cdots\!21}{296127980083859}a^{2}-\frac{45\!\cdots\!39}{42303997154837}a+\frac{585360471556046}{3845817923167}$, $\frac{1336018557638}{296127980083859}a^{8}-\frac{10544272927303}{296127980083859}a^{7}-\frac{189892472648009}{296127980083859}a^{6}+\frac{11\!\cdots\!93}{296127980083859}a^{5}+\frac{55\!\cdots\!79}{296127980083859}a^{4}-\frac{50\!\cdots\!21}{42303997154837}a^{3}+\frac{17\!\cdots\!84}{296127980083859}a^{2}+\frac{17\!\cdots\!28}{296127980083859}a-\frac{21\!\cdots\!75}{3845817923167}$, $\frac{1409585290}{296127980083859}a^{8}-\frac{70959404689}{296127980083859}a^{7}+\frac{548729445193}{296127980083859}a^{6}+\frac{4339035032166}{296127980083859}a^{5}-\frac{34704635759749}{296127980083859}a^{4}-\frac{27013915447700}{296127980083859}a^{3}+\frac{328865101962457}{296127980083859}a^{2}-\frac{280035747427042}{296127980083859}a-\frac{1492260719209}{3845817923167}$, $\frac{99016423410}{296127980083859}a^{8}-\frac{937949820376}{296127980083859}a^{7}-\frac{11331367398763}{296127980083859}a^{6}+\frac{82206114005828}{296127980083859}a^{5}+\frac{291422938422661}{296127980083859}a^{4}-\frac{304704131483880}{42303997154837}a^{3}+\frac{188896529281300}{296127980083859}a^{2}+\frac{10\!\cdots\!69}{296127980083859}a-\frac{116806622424686}{3845817923167}$ 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:  \( 6898775.79927 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 6898775.79927 \cdot 3}{2\cdot\sqrt{15072974715383053921}}\cr\approx \mathstrut & 1.36468923556 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547)
 
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 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547, 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 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547);
 
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 - 198*x^7 - 107*x^6 + 10319*x^5 + 973*x^4 - 189602*x^3 + 170782*x^2 + 815941*x - 1009547);
 
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

3.3.361.1

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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.1.0.1}{1} }^{9}$ ${\href{/padicField/11.1.0.1}{1} }^{9}$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ R ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.9.0.1}{9} }$ R ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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
\(19\) Copy content Toggle raw display 19.9.8.3$x^{9} + 152$$9$$1$$8$$C_9$$[\ ]_{9}$
\(31\) Copy content Toggle raw display 31.3.2.3$x^{3} + 155$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} + 155$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} + 155$$3$$1$$2$$C_3$$[\ ]_{3}$