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Properties

Label 10.2.699092182315008.1
Degree $10$
Signature $[2, 4]$
Discriminant $6.991\times 10^{14}$
Root discriminant \(30.51\)
Ramified primes $2,3,43$
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 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34)
 
gp: K = bnfinit(y^10 - 2*y^9 - 6*y^8 + 2*y^7 + 21*y^6 + 10*y^5 - 67*y^4 + 30*y^3 + 76*y^2 - 96*y + 34, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34)
 

\( x^{10} - 2x^{9} - 6x^{8} + 2x^{7} + 21x^{6} + 10x^{5} - 67x^{4} + 30x^{3} + 76x^{2} - 96x + 34 \) 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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(699092182315008\) \(\medspace = 2^{12}\cdot 3^{3}\cdot 43^{6}\) 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:  \(30.51\)
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:  $2^{7/4}3^{1/2}43^{2/3}\approx 71.50593506160868$
Ramified primes:   \(2\), \(3\), \(43\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{35603}a^{9}-\frac{14762}{35603}a^{8}-\frac{3246}{35603}a^{7}-\frac{10676}{35603}a^{6}-\frac{1097}{35603}a^{5}-\frac{7635}{35603}a^{4}+\frac{9038}{35603}a^{3}+\frac{3591}{35603}a^{2}+\frac{9783}{35603}a+\frac{8592}{35603}$ 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:  $5$
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:   $\frac{13179}{35603}a^{9}-\frac{13606}{35603}a^{8}-\frac{91037}{35603}a^{7}-\frac{67154}{35603}a^{6}+\frac{211073}{35603}a^{5}+\frac{348443}{35603}a^{4}-\frac{514278}{35603}a^{3}-\frac{133010}{35603}a^{2}+\frac{794960}{35603}a-\frac{375205}{35603}$, $\frac{52132}{35603}a^{9}-\frac{84945}{35603}a^{8}-\frac{355443}{35603}a^{7}-\frac{15136}{35603}a^{6}+\frac{1164513}{35603}a^{5}+\frac{1010604}{35603}a^{4}-\frac{3312165}{35603}a^{3}+\frac{41041}{35603}a^{2}+\frac{4444756}{35603}a-\frac{3136263}{35603}$, $\frac{21072}{35603}a^{9}-\frac{37056}{35603}a^{8}-\frac{148761}{35603}a^{7}+\frac{10685}{35603}a^{6}+\frac{560011}{35603}a^{5}+\frac{468076}{35603}a^{4}-\frac{1522640}{35603}a^{3}-\frac{378456}{35603}a^{2}+\frac{1786156}{35603}a-\frac{773897}{35603}$, $\frac{7923}{35603}a^{9}-\frac{3471}{35603}a^{8}-\frac{83898}{35603}a^{7}-\frac{28823}{35603}a^{6}+\frac{209219}{35603}a^{5}+\frac{389025}{35603}a^{4}-\frac{559207}{35603}a^{3}-\frac{600555}{35603}a^{2}+\frac{1071068}{35603}a-\frac{390153}{35603}$, $\frac{7056}{35603}a^{9}-\frac{21897}{35603}a^{8}-\frac{11047}{35603}a^{7}+\frac{6092}{35603}a^{6}+\frac{127831}{35603}a^{5}-\frac{76427}{35603}a^{4}-\frac{277669}{35603}a^{3}+\frac{522805}{35603}a^{2}-\frac{325796}{35603}a+\frac{64449}{35603}$ 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:  \( 10748.9833723 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 10748.9833723 \cdot 1}{2\cdot\sqrt{699092182315008}}\cr\approx \mathstrut & 1.26721305190 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34)
 
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 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34, 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 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34);
 
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 - 2*x^9 - 6*x^8 + 2*x^7 + 21*x^6 + 10*x^5 - 67*x^4 + 30*x^3 + 76*x^2 - 96*x + 34);
 
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: 6.2.355008.1, 6.2.23630752512.11
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: 6.2.355008.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ R ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.6$x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
\(3\) Copy content Toggle raw display 3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$[\ ]$
43.3.2.1$x^{3} + 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.4.1$x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$