Properties

Label 10.2.30176086429637.1
Degree $10$
Signature $[2, 4]$
Discriminant $3.018\times 10^{13}$
Root discriminant \(22.28\)
Ramified primes $163,191$
Class number $2$
Class group [2]
Galois group $S_{6}$ (as 10T32)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*x - 7)
 
gp: K = bnfinit(y^10 - 3*y^9 + 5*y^8 - y^7 - 8*y^6 + 27*y^5 - 55*y^4 + 18*y^3 - 39*y^2 + 2*y - 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*x - 7);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*x - 7)
 

\( x^{10} - 3x^{9} + 5x^{8} - x^{7} - 8x^{6} + 27x^{5} - 55x^{4} + 18x^{3} - 39x^{2} + 2x - 7 \) 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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(30176086429637\) \(\medspace = 163^{3}\cdot 191^{3}\) 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:  \(22.28\)
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:  $163^{1/2}191^{1/2}\approx 176.44545899512406$
Ramified primes:   \(163\), \(191\) 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:  \(\Q(\sqrt{31133}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{8}+\frac{2}{7}a^{6}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{3101}a^{9}+\frac{130}{3101}a^{8}+\frac{18}{3101}a^{7}+\frac{178}{3101}a^{6}-\frac{699}{3101}a^{5}-\frac{353}{3101}a^{4}-\frac{1375}{3101}a^{3}+\frac{545}{3101}a^{2}-\frac{1535}{3101}a+\frac{10}{443}$ 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

$C_{2}$, which has order $2$

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:   $\frac{899}{3101}a^{9}-\frac{1854}{3101}a^{8}+\frac{1563}{3101}a^{7}+\frac{4086}{3101}a^{6}-\frac{8644}{3101}a^{5}+\frac{15346}{3101}a^{4}-\frac{22748}{3101}a^{3}-\frac{37658}{3101}a^{2}-\frac{10652}{3101}a-\frac{2085}{443}$, $\frac{547}{3101}a^{9}-\frac{1542}{3101}a^{8}+\frac{2315}{3101}a^{7}+\frac{349}{3101}a^{6}-\frac{4917}{3101}a^{5}+\frac{14233}{3101}a^{4}-\frac{26048}{3101}a^{3}+\frac{313}{443}a^{2}-\frac{11678}{3101}a-\frac{289}{443}$, $\frac{1035}{3101}a^{9}-\frac{3223}{3101}a^{8}+\frac{4897}{3101}a^{7}+\frac{55}{443}a^{6}-\frac{11121}{3101}a^{5}+\frac{28029}{3101}a^{4}-\frac{52040}{3101}a^{3}+\frac{7667}{3101}a^{2}-\frac{13417}{3101}a+\frac{161}{443}$, $\frac{995}{3101}a^{9}-\frac{2664}{3101}a^{8}+\frac{3734}{3101}a^{7}+\frac{177}{443}a^{6}-\frac{9298}{3101}a^{5}+\frac{24429}{3101}a^{4}-\frac{44441}{3101}a^{3}-\frac{8374}{3101}a^{2}-\frac{17138}{3101}a-\frac{1125}{443}$, $\frac{2252}{3101}a^{9}-\frac{895}{443}a^{8}+\frac{8640}{3101}a^{7}+\frac{3485}{3101}a^{6}-\frac{3315}{443}a^{5}+\frac{55604}{3101}a^{4}-\frac{98276}{3101}a^{3}-\frac{2372}{443}a^{2}-\frac{26671}{3101}a-\frac{959}{443}$ 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:  \( 822.418264941 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 822.418264941 \cdot 2}{2\cdot\sqrt{30176086429637}}\cr\approx \mathstrut & 0.933341881524 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*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 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*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 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*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 - 3*x^9 + 5*x^8 - x^7 - 8*x^6 + 27*x^5 - 55*x^4 + 18*x^3 - 39*x^2 + 2*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: 6.2.31133.1, 6.2.30176086429637.1
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: 6.2.31133.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(163\) Copy content Toggle raw display 163.2.1.2$x^{2} + 163$$2$$1$$1$$C_2$$[\ ]_{2}$
163.4.0.1$x^{4} + 8 x^{2} + 91 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
163.4.2.2$x^{4} - 25917 x^{2} + 53138$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
191.3.0.1$x^{3} + 4 x + 172$$1$$3$$0$$C_3$$[\ ]^{3}$
191.6.3.1$x^{6} + 145924 x^{2} - 1198473812$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$