Properties

Label 10.4.585682963101963.1
Degree $10$
Signature $[4, 3]$
Discriminant $-5.857\times 10^{14}$
Root discriminant \(29.98\)
Ramified primes $3,167$
Class number $1$
Class group trivial
Galois group $S_{6}$ (as 10T32)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7)
 
gp: K = bnfinit(y^10 - 9*y^8 - 2*y^7 + 35*y^6 + 8*y^5 - 63*y^4 - 9*y^3 + 30*y^2 + 8*y - 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7)
 

\( x^{10} - 9x^{8} - 2x^{7} + 35x^{6} + 8x^{5} - 63x^{4} - 9x^{3} + 30x^{2} + 8x - 7 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-585682963101963\) \(\medspace = -\,3^{3}\cdot 167^{6}\) 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:  \(29.98\)
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:  $3^{1/2}167^{2/3}\approx 52.52567088234156$
Ramified primes:   \(3\), \(167\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1929}a^{9}+\frac{374}{1929}a^{8}-\frac{950}{1929}a^{7}-\frac{122}{643}a^{6}+\frac{110}{1929}a^{5}+\frac{213}{643}a^{4}-\frac{91}{643}a^{3}+\frac{42}{643}a^{2}+\frac{286}{643}a+\frac{686}{1929}$ 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

Trivial group, which has order $1$

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:  $6$
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:   $\frac{629}{1929}a^{9}-\frac{92}{1929}a^{8}-\frac{5347}{1929}a^{7}-\frac{221}{643}a^{6}+\frac{19036}{1929}a^{5}+\frac{876}{643}a^{4}-\frac{9014}{643}a^{3}+\frac{55}{643}a^{2}-\frac{789}{643}a+\frac{3256}{1929}$, $\frac{491}{643}a^{9}+\frac{379}{643}a^{8}-\frac{4133}{643}a^{7}-\frac{4167}{643}a^{6}+\frac{14144}{643}a^{5}+\frac{14754}{643}a^{4}-\frac{20232}{643}a^{3}-\frac{19795}{643}a^{2}+\frac{113}{643}a+\frac{4395}{643}$, $\frac{1498}{1929}a^{9}+\frac{842}{1929}a^{8}-\frac{13001}{1929}a^{7}-\frac{3359}{643}a^{6}+\frac{47111}{1929}a^{5}+\frac{12363}{643}a^{4}-\frac{25079}{643}a^{3}-\frac{17459}{643}a^{2}+\frac{6620}{643}a+\frac{22619}{1929}$, $\frac{2597}{1929}a^{9}+\frac{991}{1929}a^{8}-\frac{23107}{1929}a^{7}-\frac{4979}{643}a^{6}+\frac{85054}{1929}a^{5}+\frac{19471}{643}a^{4}-\frac{45999}{643}a^{3}-\frac{29814}{643}a^{2}+\frac{11651}{643}a+\frac{41584}{1929}$, $\frac{60}{643}a^{9}-\frac{65}{643}a^{8}-\frac{416}{643}a^{7}+\frac{545}{643}a^{6}+\frac{1456}{643}a^{5}-\frac{2169}{643}a^{4}-\frac{2234}{643}a^{3}+\frac{4345}{643}a^{2}+\frac{40}{643}a-\frac{635}{643}$, $\frac{82}{1929}a^{9}-\frac{196}{1929}a^{8}-\frac{740}{1929}a^{7}+\frac{284}{643}a^{6}+\frac{3233}{1929}a^{5}-\frac{538}{643}a^{4}-\frac{1675}{643}a^{3}-\frac{414}{643}a^{2}+\frac{304}{643}a+\frac{311}{1929}$ 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:  \( 13527.5655241 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 13527.5655241 \cdot 1}{2\cdot\sqrt{585682963101963}}\cr\approx \mathstrut & 1.10922064148 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7)
 
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^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7, 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^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7);
 
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^10 - 9*x^8 - 2*x^7 + 35*x^6 + 8*x^5 - 63*x^4 - 9*x^3 + 30*x^2 + 8*x - 7);
 
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

$S_6$ (as 10T32):

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 non-solvable group of order 720
The 11 conjugacy class representatives for $S_{6}$
Character table for $S_{6}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 6 siblings: 6.0.83667.1, 6.4.21000500667.1
Degree 12 siblings: data not computed
Degree 15 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.0.83667.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ R ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(167\) Copy content Toggle raw display $\Q_{167}$$x + 162$$1$$1$$0$Trivial$[\ ]$
167.3.2.1$x^{3} + 167$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
167.3.2.1$x^{3} + 167$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
167.3.2.1$x^{3} + 167$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$