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Properties

Label 9.3.20878097849057792.3
Degree $9$
Signature $[3, 3]$
Discriminant $-2.088\times 10^{16}$
Root discriminant \(65.06\)
Ramified primes $2,7,19$
Class number $9$
Class group [9]
Galois group $C_3 \wr S_3 $ (as 9T20)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748)
 
gp: K = bnfinit(y^9 - 38*y^7 - 38*y^6 + 171*y^5 - 646*y^4 - 171*y^3 + 3914*y^2 - 4541*y + 1748, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748)
 

\( x^{9} - 38x^{7} - 38x^{6} + 171x^{5} - 646x^{4} - 171x^{3} + 3914x^{2} - 4541x + 1748 \) 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:  $[3, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-20878097849057792\) \(\medspace = -\,2^{9}\cdot 7^{4}\cdot 19^{8}\) 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:  \(65.06\)
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:  $2^{3/2}7^{2/3}19^{8/9}\approx 141.7795371895746$
Ramified primes:   \(2\), \(7\), \(19\) 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:  \(\Q(\sqrt{-2}) \)
$\card{ \Aut(K/\Q) }$:  $3$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{1852183755320}a^{8}+\frac{568263161381}{1852183755320}a^{7}-\frac{198915009677}{1852183755320}a^{6}+\frac{67961834013}{370436751064}a^{5}+\frac{82118358567}{231522969415}a^{4}-\frac{11224639959}{185218375532}a^{3}-\frac{410248015281}{1852183755320}a^{2}+\frac{462038110253}{1852183755320}a+\frac{156586408423}{463045938830}$ 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:  $2$

Class group and class number

$C_{9}$, which has order $9$

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:  $5$
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:   $\frac{15716925427}{926091877660}a^{8}-\frac{292869233}{926091877660}a^{7}-\frac{689366007539}{926091877660}a^{6}-\frac{149325263765}{185218375532}a^{5}+\frac{1620089523773}{231522969415}a^{4}+\frac{77660736351}{92609187766}a^{3}-\frac{21901193480167}{926091877660}a^{2}+\frac{23259935003631}{926091877660}a-\frac{2025496828184}{231522969415}$, $\frac{13228646}{1474668595}a^{8}+\frac{8793486}{1474668595}a^{7}-\frac{502891012}{1474668595}a^{6}-\frac{165702210}{294933719}a^{5}+\frac{1901832686}{1474668595}a^{4}-\frac{1420869510}{294933719}a^{3}-\frac{7938138136}{1474668595}a^{2}+\frac{47143951818}{1474668595}a-\frac{24983432963}{1474668595}$, $\frac{5489698}{3795458515}a^{8}+\frac{8722858}{3795458515}a^{7}-\frac{224292316}{3795458515}a^{6}-\frac{67753289}{759091703}a^{5}+\frac{246695483}{3795458515}a^{4}-\frac{874859310}{759091703}a^{3}-\frac{1441548953}{3795458515}a^{2}+\frac{7103859419}{3795458515}a-\frac{4231657689}{3795458515}$, $\frac{1515007111}{231522969415}a^{8}-\frac{877385904}{231522969415}a^{7}-\frac{49325712692}{231522969415}a^{6}-\frac{8311932652}{46304593883}a^{5}+\frac{30383220871}{231522969415}a^{4}-\frac{183078257227}{46304593883}a^{3}+\frac{753547321934}{231522969415}a^{2}+\frac{128849400643}{231522969415}a+\frac{2217384832317}{231522969415}$, $\frac{200062853}{463045938830}a^{8}-\frac{866208907}{463045938830}a^{7}-\frac{6873565101}{463045938830}a^{6}+\frac{3411858243}{92609187766}a^{5}+\frac{4218994544}{231522969415}a^{4}-\frac{21292836974}{46304593883}a^{3}+\frac{668159469067}{463045938830}a^{2}-\frac{799891353811}{463045938830}a+\frac{169169010963}{231522969415}$ 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:  \( 41799.8473316 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 41799.8473316 \cdot 9}{2\cdot\sqrt{20878097849057792}}\cr\approx \mathstrut & 2.58327914033 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748)
 
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 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748, 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 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748);
 
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 - 38*x^7 - 38*x^6 + 171*x^5 - 646*x^4 - 171*x^3 + 3914*x^2 - 4541*x + 1748);
 
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

$C_3\wr S_3$ (as 9T20):

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 solvable group of order 162
The 22 conjugacy class representatives for $C_3 \wr S_3 $
Character table for $C_3 \wr S_3 $

Intermediate fields

3.1.2888.1

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]
 

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 siblings: data not computed
Minimal sibling: 9.3.426083629572608.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ R ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.9.5$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$$2$$3$$9$$C_6$$[3]^{3}$
\(7\) Copy content Toggle raw display 7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.4.2$x^{6} - 42 x^{3} + 147$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(19\) Copy content Toggle raw display 19.9.8.2$x^{9} + 57$$9$$1$$8$$C_9$$[\ ]_{9}$