Properties

Label 10.2.10628820000000.4
Degree $10$
Signature $[2, 4]$
Discriminant $1.063\times 10^{13}$
Root discriminant \(20.07\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $S_5$ (as 10T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11)
 
gp: K = bnfinit(y^10 - 5*y^9 + 11*y^8 - 14*y^7 + 33*y^6 - 71*y^5 + 101*y^4 - 90*y^3 + 107*y^2 - 73*y - 11, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11)
 

\( x^{10} - 5x^{9} + 11x^{8} - 14x^{7} + 33x^{6} - 71x^{5} + 101x^{4} - 90x^{3} + 107x^{2} - 73x - 11 \) 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:   \(10628820000000\) \(\medspace = 2^{8}\cdot 3^{12}\cdot 5^{7}\) 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:  \(20.07\)
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))
 
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{6}a^{5}-\frac{1}{6}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{204}a^{8}-\frac{1}{51}a^{7}+\frac{7}{102}a^{5}+\frac{47}{204}a^{4}+\frac{41}{102}a^{3}+\frac{29}{102}a^{2}+\frac{1}{34}a-\frac{89}{204}$, $\frac{1}{5100}a^{9}+\frac{2}{1275}a^{8}+\frac{19}{510}a^{7}+\frac{38}{425}a^{6}-\frac{839}{5100}a^{5}-\frac{17}{150}a^{4}+\frac{101}{425}a^{3}+\frac{283}{2550}a^{2}-\frac{7}{20}a+\frac{911}{2550}$ 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:   $\frac{22}{1275}a^{9}-\frac{33}{425}a^{8}+\frac{12}{85}a^{7}-\frac{56}{425}a^{6}+\frac{214}{425}a^{5}-\frac{472}{425}a^{4}+\frac{92}{75}a^{3}-\frac{316}{425}a^{2}+\frac{162}{85}a-\frac{1741}{1275}$, $\frac{7}{425}a^{9}-\frac{203}{5100}a^{8}-\frac{3}{170}a^{7}+\frac{226}{1275}a^{6}+\frac{287}{2550}a^{5}-\frac{427}{5100}a^{4}-\frac{657}{850}a^{3}+\frac{937}{425}a^{2}-\frac{152}{255}a+\frac{2273}{5100}$, $\frac{11}{1275}a^{9}-\frac{33}{850}a^{8}+\frac{6}{85}a^{7}-\frac{28}{425}a^{6}+\frac{107}{425}a^{5}-\frac{236}{425}a^{4}+\frac{46}{75}a^{3}+\frac{109}{850}a^{2}-\frac{4}{85}a-\frac{233}{1275}$, $\frac{227}{1700}a^{9}-\frac{3427}{5100}a^{8}+\frac{382}{255}a^{7}-\frac{4957}{2550}a^{6}+\frac{7547}{1700}a^{5}-\frac{48293}{5100}a^{4}+\frac{17818}{1275}a^{3}-\frac{34477}{2550}a^{2}+\frac{15853}{1020}a-\frac{18981}{1700}$, $\frac{127}{340}a^{9}-\frac{619}{340}a^{8}+\frac{395}{102}a^{7}-\frac{2297}{510}a^{6}+\frac{10831}{1020}a^{5}-\frac{7791}{340}a^{4}+\frac{17201}{510}a^{3}-\frac{7691}{255}a^{2}+\frac{7901}{204}a-\frac{25783}{1020}$ 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:  \( 755.722725676 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 755.722725676 \cdot 1}{2\cdot\sqrt{10628820000000}}\cr\approx \mathstrut & 0.722552528636 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11)
 
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 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11, 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 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11);
 
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 - 5*x^9 + 11*x^8 - 14*x^7 + 33*x^6 - 71*x^5 + 101*x^4 - 90*x^3 + 107*x^2 - 73*x - 11);
 
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

$S_5$ (as 10T12):

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 120
The 7 conjugacy class representatives for $S_5$
Character table for $S_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.1.1458000.1

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 5 sibling: 5.1.1458000.1
Degree 6 sibling: 6.2.118098000.6
Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.1.1458000.1

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 R ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{5}$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.10.8.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.10.1$x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.2.1.1$x^{2} + 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$