Properties

Label 9.9.165...401.1
Degree $9$
Signature $[9, 0]$
Discriminant $1.652\times 10^{20}$
Root discriminant \(176.38\)
Ramified primes $19,37$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $C_9$ (as 9T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139)
 
gp: K = bnfinit(y^9 - y^8 - 238*y^7 - 285*y^6 + 15680*y^5 + 55764*y^4 - 208901*y^3 - 1310052*y^2 - 1827386*y - 343139, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139)
 

\( x^{9} - x^{8} - 238 x^{7} - 285 x^{6} + 15680 x^{5} + 55764 x^{4} - 208901 x^{3} - 1310052 x^{2} - 1827386 x - 343139 \) 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:   \(165247690404112349401\) \(\medspace = 19^{6}\cdot 37^{8}\) 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:  \(176.38\)
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:  $19^{2/3}37^{8/9}\approx 176.3841224409683$
Ramified primes:   \(19\), \(37\) 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:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(292,·)$, $\chi_{703}(7,·)$, $\chi_{703}(201,·)$, $\chi_{703}(330,·)$, $\chi_{703}(49,·)$, $\chi_{703}(343,·)$, $\chi_{703}(248,·)$, $\chi_{703}(638,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{23}a^{6}+\frac{11}{23}a^{5}-\frac{9}{23}a^{4}+\frac{3}{23}a^{3}+\frac{10}{23}a^{2}-\frac{1}{23}a+\frac{11}{23}$, $\frac{1}{23}a^{7}+\frac{8}{23}a^{5}+\frac{10}{23}a^{4}+\frac{4}{23}a^{2}-\frac{1}{23}a-\frac{6}{23}$, $\frac{1}{41\!\cdots\!11}a^{8}+\frac{48254617162527}{41\!\cdots\!11}a^{7}-\frac{838626189770321}{41\!\cdots\!11}a^{6}+\frac{56\!\cdots\!99}{41\!\cdots\!11}a^{5}-\frac{93\!\cdots\!64}{41\!\cdots\!11}a^{4}-\frac{11\!\cdots\!47}{41\!\cdots\!11}a^{3}+\frac{10\!\cdots\!58}{41\!\cdots\!11}a^{2}-\frac{78\!\cdots\!22}{41\!\cdots\!11}a-\frac{232564917655428}{41\!\cdots\!11}$ 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

$C_{3}$, 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:   $\frac{5429443608}{136429331284573}a^{8}-\frac{19390499196}{136429331284573}a^{7}-\frac{1253808385689}{136429331284573}a^{6}+\frac{1745433927252}{136429331284573}a^{5}+\frac{82902824765339}{136429331284573}a^{4}+\frac{80935463803145}{136429331284573}a^{3}-\frac{14\!\cdots\!37}{136429331284573}a^{2}-\frac{143552732372253}{5931710055851}a-\frac{497993322433033}{136429331284573}$, $\frac{1737174800}{136429331284573}a^{8}-\frac{4366854593}{136429331284573}a^{7}-\frac{405865064899}{136429331284573}a^{6}+\frac{139986984207}{136429331284573}a^{5}+\frac{26744444979705}{136429331284573}a^{4}+\frac{52990292145576}{136429331284573}a^{3}-\frac{427850976639639}{136429331284573}a^{2}-\frac{15\!\cdots\!82}{136429331284573}a-\frac{10\!\cdots\!24}{136429331284573}$, $\frac{801490494845}{41\!\cdots\!11}a^{8}+\frac{1628206113851}{41\!\cdots\!11}a^{7}-\frac{254127878057272}{41\!\cdots\!11}a^{6}+\frac{52284668358347}{41\!\cdots\!11}a^{5}+\frac{16\!\cdots\!21}{41\!\cdots\!11}a^{4}+\frac{21\!\cdots\!16}{41\!\cdots\!11}a^{3}-\frac{27\!\cdots\!07}{41\!\cdots\!11}a^{2}-\frac{70\!\cdots\!64}{41\!\cdots\!11}a-\frac{14\!\cdots\!95}{41\!\cdots\!11}$, $\frac{95229275886245}{41\!\cdots\!11}a^{8}-\frac{427791677791928}{41\!\cdots\!11}a^{7}-\frac{21\!\cdots\!90}{41\!\cdots\!11}a^{6}+\frac{46\!\cdots\!03}{41\!\cdots\!11}a^{5}+\frac{13\!\cdots\!90}{41\!\cdots\!11}a^{4}+\frac{66\!\cdots\!92}{41\!\cdots\!11}a^{3}-\frac{22\!\cdots\!36}{41\!\cdots\!11}a^{2}-\frac{47\!\cdots\!38}{41\!\cdots\!11}a-\frac{93\!\cdots\!69}{41\!\cdots\!11}$, $\frac{11033487345372}{41\!\cdots\!11}a^{8}-\frac{67666956296394}{41\!\cdots\!11}a^{7}-\frac{22\!\cdots\!60}{41\!\cdots\!11}a^{6}+\frac{81\!\cdots\!28}{41\!\cdots\!11}a^{5}+\frac{12\!\cdots\!95}{41\!\cdots\!11}a^{4}-\frac{33\!\cdots\!54}{41\!\cdots\!11}a^{3}-\frac{18\!\cdots\!46}{41\!\cdots\!11}a^{2}-\frac{39\!\cdots\!66}{41\!\cdots\!11}a-\frac{16\!\cdots\!33}{41\!\cdots\!11}$, $\frac{5479649104696}{41\!\cdots\!11}a^{8}+\frac{25236428537757}{41\!\cdots\!11}a^{7}-\frac{11\!\cdots\!23}{41\!\cdots\!11}a^{6}-\frac{81\!\cdots\!67}{41\!\cdots\!11}a^{5}+\frac{43\!\cdots\!10}{41\!\cdots\!11}a^{4}+\frac{55\!\cdots\!53}{41\!\cdots\!11}a^{3}+\frac{18\!\cdots\!72}{41\!\cdots\!11}a^{2}+\frac{21\!\cdots\!62}{41\!\cdots\!11}a+\frac{39\!\cdots\!38}{41\!\cdots\!11}$, $\frac{2503450171354}{41\!\cdots\!11}a^{8}-\frac{17456224595367}{41\!\cdots\!11}a^{7}-\frac{537903926239154}{41\!\cdots\!11}a^{6}+\frac{24\!\cdots\!74}{41\!\cdots\!11}a^{5}+\frac{34\!\cdots\!24}{41\!\cdots\!11}a^{4}-\frac{37\!\cdots\!86}{41\!\cdots\!11}a^{3}-\frac{78\!\cdots\!38}{41\!\cdots\!11}a^{2}-\frac{14\!\cdots\!47}{41\!\cdots\!11}a-\frac{28\!\cdots\!73}{41\!\cdots\!11}$, $\frac{4268301291196}{41\!\cdots\!11}a^{8}-\frac{41835080448008}{41\!\cdots\!11}a^{7}-\frac{983313901346462}{41\!\cdots\!11}a^{6}+\frac{58\!\cdots\!45}{41\!\cdots\!11}a^{5}+\frac{71\!\cdots\!50}{41\!\cdots\!11}a^{4}-\frac{44\!\cdots\!24}{41\!\cdots\!11}a^{3}-\frac{13\!\cdots\!89}{41\!\cdots\!11}a^{2}-\frac{24\!\cdots\!37}{41\!\cdots\!11}a-\frac{45\!\cdots\!10}{41\!\cdots\!11}$ 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:  \( 5458915.67791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 5458915.67791 \cdot 3}{2\cdot\sqrt{165247690404112349401}}\cr\approx \mathstrut & 0.326136875995 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139)
 
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 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139, 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 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139);
 
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 - 238*x^7 - 285*x^6 + 15680*x^5 + 55764*x^4 - 208901*x^3 - 1310052*x^2 - 1827386*x - 343139);
 
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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ R ${\href{/padicField/23.1.0.1}{1} }^{9}$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.1.0.1}{1} }^{9}$ R ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.1.0.1}{1} }^{9}$ ${\href{/padicField/53.9.0.1}{9} }$ ${\href{/padicField/59.9.0.1}{9} }$

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 19.9.6.1$x^{9} + 1444 x^{3} - 116603$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
\(37\) Copy content Toggle raw display 37.9.8.1$x^{9} + 37$$9$$1$$8$$C_9$$[\ ]_{9}$