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Properties

Label 9.1.3756492407660544.1
Degree $9$
Signature $[1, 4]$
Discriminant $3.756\times 10^{15}$
Root discriminant \(53.77\)
Ramified primes $2,3,19$
Class number $9$ (GRH)
Class group [9] (GRH)
Galois group $S_3\wr S_3$ (as 9T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995)
 
gp: K = bnfinit(y^9 - 3*y^8 + 4*y^7 + 18*y^6 + 10*y^5 - 320*y^4 + 316*y^3 + 1350*y^2 - 7019*y + 9995, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995)
 

\( x^{9} - 3x^{8} + 4x^{7} + 18x^{6} + 10x^{5} - 320x^{4} + 316x^{3} + 1350x^{2} - 7019x + 9995 \) 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:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3756492407660544\) \(\medspace = 2^{13}\cdot 3^{3}\cdot 19^{8}\) 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:  \(53.77\)
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:  $2^{13/6}3^{1/2}19^{8/9}\approx 106.52760230477175$
Ramified primes:   \(2\), \(3\), \(19\) 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:  \(\Q(\sqrt{6}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{24}a^{6}+\frac{1}{12}a^{5}-\frac{1}{24}a^{4}+\frac{1}{12}a^{3}+\frac{1}{24}a^{2}-\frac{1}{6}a+\frac{7}{24}$, $\frac{1}{48}a^{7}-\frac{1}{48}a^{6}+\frac{5}{48}a^{5}-\frac{7}{48}a^{4}-\frac{5}{48}a^{3}+\frac{17}{48}a^{2}-\frac{17}{48}a-\frac{3}{16}$, $\frac{1}{6666240}a^{8}+\frac{603}{222208}a^{7}+\frac{2407}{158720}a^{6}+\frac{503}{10416}a^{5}-\frac{12863}{666624}a^{4}-\frac{78515}{666624}a^{3}-\frac{1369397}{3333120}a^{2}-\frac{431383}{1666560}a+\frac{366613}{1333248}$ 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 $9$ (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:  $4$
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:   $\frac{293}{277760}a^{8}-\frac{65}{83328}a^{7}-\frac{1541}{59520}a^{6}+\frac{433}{2604}a^{5}-\frac{29521}{83328}a^{4}-\frac{26045}{83328}a^{3}+\frac{1153477}{416640}a^{2}-\frac{1310537}{208320}a+\frac{341129}{55552}$, $\frac{109}{222208}a^{8}-\frac{143}{111104}a^{7}-\frac{559}{15872}a^{6}+\frac{8}{217}a^{5}+\frac{3105}{111104}a^{4}+\frac{26029}{111104}a^{3}-\frac{204369}{111104}a^{2}+\frac{385941}{55552}a-\frac{1544139}{222208}$, $\frac{23019}{1111040}a^{8}+\frac{1553}{333312}a^{7}-\frac{17803}{238080}a^{6}-\frac{85}{2604}a^{5}+\frac{301825}{333312}a^{4}-\frac{621235}{333312}a^{3}-\frac{8889589}{1666560}a^{2}+\frac{16375049}{833280}a-\frac{5394873}{222208}$, $\frac{1951589}{333312}a^{8}-\frac{103389}{55552}a^{7}-\frac{615287}{23808}a^{6}+\frac{420635}{5208}a^{5}+\frac{24437939}{55552}a^{4}-\frac{78772009}{55552}a^{3}-\frac{265556803}{55552}a^{2}+\frac{646031453}{83328}a+\frac{787403151}{111104}$ 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:  \( 19868.0877124 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 19868.0877124 \cdot 9}{2\cdot\sqrt{3756492407660544}}\cr\approx \mathstrut & 4.54701926021 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995)
 
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 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995, 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 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995);
 
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 - 3*x^8 + 4*x^7 + 18*x^6 + 10*x^5 - 320*x^4 + 316*x^3 + 1350*x^2 - 7019*x + 9995);
 
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

$S_3\wr S_3$ (as 9T31):

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 1296
The 22 conjugacy class representatives for $S_3\wr S_3$
Character table for $S_3\wr S_3$

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ R ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
\(3\) Copy content Toggle raw display 3.3.0.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(19\) Copy content Toggle raw display 19.9.8.2$x^{9} + 57$$9$$1$$8$$C_9$$[\ ]_{9}$