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Properties

Label 10.2.13601422721673.1
Degree $10$
Signature $[2, 4]$
Discriminant $1.360\times 10^{13}$
Root discriminant \(20.58\)
Ramified primes $3,73$
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 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439)
 
gp: K = bnfinit(y^10 - 5*y^9 + 17*y^8 - 17*y^7 - 17*y^6 + 96*y^5 - 109*y^4 - 109*y^3 + 163*y^2 - 202*y - 439, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 5*x^9 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 5*x^9 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439)
 

\( x^{10} - 5x^{9} + 17x^{8} - 17x^{7} - 17x^{6} + 96x^{5} - 109x^{4} - 109x^{3} + 163x^{2} - 202x - 439 \) 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:   \(13601422721673\) \(\medspace = 3^{8}\cdot 73^{5}\) 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:  \(20.58\)
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:  $3^{4/3}73^{1/2}\approx 36.967757191167294$
Ramified primes:   \(3\), \(73\) 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:  \(\Q(\sqrt{73}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{479012499}a^{9}-\frac{16052455}{479012499}a^{8}-\frac{189484790}{479012499}a^{7}+\frac{237485411}{479012499}a^{6}+\frac{67844843}{159670833}a^{5}+\frac{1865263}{159670833}a^{4}+\frac{30961331}{68430357}a^{3}+\frac{119747806}{479012499}a^{2}-\frac{124762471}{479012499}a+\frac{212497231}{479012499}$ 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:   $\frac{5902423}{479012499}a^{9}-\frac{26637931}{479012499}a^{8}+\frac{76568986}{479012499}a^{7}-\frac{33239503}{479012499}a^{6}-\frac{58601089}{159670833}a^{5}+\frac{107626066}{159670833}a^{4}-\frac{22972408}{68430357}a^{3}-\frac{537123854}{479012499}a^{2}-\frac{114267064}{479012499}a+\frac{140588284}{479012499}$, $\frac{2456735}{479012499}a^{9}-\frac{8004254}{479012499}a^{8}+\frac{11217530}{479012499}a^{7}+\frac{60373588}{479012499}a^{6}-\frac{69442769}{159670833}a^{5}+\frac{63660038}{159670833}a^{4}+\frac{22190935}{68430357}a^{3}-\frac{732187432}{479012499}a^{2}+\frac{272617939}{479012499}a+\frac{486840629}{479012499}$, $\frac{17727287}{479012499}a^{9}-\frac{571154}{479012499}a^{8}-\frac{55852180}{479012499}a^{7}+\frac{642211312}{479012499}a^{6}-\frac{58170758}{159670833}a^{5}-\frac{499616155}{159670833}a^{4}+\frac{283415170}{68430357}a^{3}-\frac{999323551}{479012499}a^{2}-\frac{10326038363}{479012499}a-\frac{7582868593}{479012499}$, $\frac{7089214}{479012499}a^{9}+\frac{30008726}{479012499}a^{8}-\frac{122021867}{479012499}a^{7}+\frac{468575819}{479012499}a^{6}+\frac{60169148}{159670833}a^{5}-\frac{530144945}{159670833}a^{4}+\frac{104650979}{68430357}a^{3}+\frac{56395042}{479012499}a^{2}-\frac{7202621719}{479012499}a-\frac{6466430336}{479012499}$, $\frac{13721}{53223611}a^{9}+\frac{3925400}{159670833}a^{8}-\frac{629851}{53223611}a^{7}-\frac{21882988}{159670833}a^{6}+\frac{54403239}{53223611}a^{5}-\frac{75073415}{53223611}a^{4}-\frac{2836978}{7603373}a^{3}+\frac{406441723}{159670833}a^{2}-\frac{194534831}{53223611}a-\frac{1234973195}{159670833}$ 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:  \( 671.469092505 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 671.469092505 \cdot 1}{2\cdot\sqrt{13601422721673}}\cr\approx \mathstrut & 0.567522817160 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 5*x^9 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439)
 
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 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439, 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 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439);
 
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 + 17*x^8 - 17*x^7 - 17*x^6 + 96*x^5 - 109*x^4 - 109*x^3 + 163*x^2 - 202*x - 439);
 
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

\(\Q(\sqrt{73}) \), 5.1.5913.1

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: 5.1.5913.1
Degree 6 sibling: 6.2.2552340537.2
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: 5.1.5913.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}$ R ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
\(73\) Copy content Toggle raw display 73.2.1.1$x^{2} + 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 8024 x^{3} + 16372240 x^{2} + 1107697152 x + 99582464$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$