Properties

Label 10.2.131220000000.1
Degree $10$
Signature $[2, 4]$
Discriminant $131220000000$
Root discriminant \(12.94\)
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 - 3*x^9 + 5*x^8 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^10 - 3*y^9 + 5*y^8 - 17*y^6 + 33*y^5 - 17*y^4 + 5*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 5*x^8 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 5*x^8 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1)
 

\( x^{10} - 3x^{9} + 5x^{8} - 17x^{6} + 33x^{5} - 17x^{4} + 5x^{2} - 3x + 1 \) 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:   \(131220000000\) \(\medspace = 2^{8}\cdot 3^{8}\cdot 5^{7}\) 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:  \(12.94\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{4}{9}a^{2}-\frac{2}{9}$, $\frac{1}{63}a^{9}+\frac{1}{21}a^{8}-\frac{5}{63}a^{7}-\frac{1}{7}a^{6}+\frac{2}{21}a^{5}-\frac{5}{21}a^{4}-\frac{23}{63}a^{3}+\frac{10}{21}a^{2}+\frac{10}{63}a+\frac{5}{21}$ 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:   $\frac{6}{7}a^{9}-\frac{17}{7}a^{8}+\frac{26}{7}a^{7}+\frac{20}{21}a^{6}-\frac{104}{7}a^{5}+\frac{176}{7}a^{4}-\frac{54}{7}a^{3}-\frac{30}{7}a^{2}+\frac{18}{7}a-\frac{10}{21}$, $\frac{31}{21}a^{9}-\frac{25}{7}a^{8}+\frac{104}{21}a^{7}+\frac{26}{7}a^{6}-\frac{169}{7}a^{5}+\frac{237}{7}a^{4}+\frac{1}{21}a^{3}-\frac{54}{7}a^{2}+\frac{65}{21}a-\frac{13}{7}$, $\frac{25}{63}a^{9}-\frac{86}{63}a^{8}+\frac{148}{63}a^{7}-\frac{29}{63}a^{6}-\frac{51}{7}a^{5}+\frac{110}{7}a^{4}-\frac{617}{63}a^{3}-\frac{83}{63}a^{2}+\frac{145}{63}a-\frac{17}{63}$, $\frac{80}{63}a^{9}-\frac{215}{63}a^{8}+\frac{314}{63}a^{7}+\frac{148}{63}a^{6}-\frac{463}{21}a^{5}+\frac{727}{21}a^{4}-\frac{370}{63}a^{3}-\frac{617}{63}a^{2}+\frac{317}{63}a-\frac{32}{63}$, $\frac{109}{63}a^{9}-\frac{94}{21}a^{8}+\frac{421}{63}a^{7}+\frac{65}{21}a^{6}-\frac{601}{21}a^{5}+\frac{946}{21}a^{4}-\frac{554}{63}a^{3}-\frac{142}{21}a^{2}+\frac{418}{63}a-\frac{50}{21}$ 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:  \( 46.2201509815 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 46.2201509815 \cdot 1}{2\cdot\sqrt{131220000000}}\cr\approx \mathstrut & 0.397723096684 \end{aligned}\]

# 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 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1)
 
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 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1, 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 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1);
 
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 - 17*x^6 + 33*x^5 - 17*x^4 + 5*x^2 - 3*x + 1);
 
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

\(\Q(\sqrt{5}) \), 5.1.162000.2

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: 5.1.162000.2
Degree 6 sibling: 6.2.328050000.2
Degree 10 sibling: 10.2.10628820000000.3
Degree 12 sibling: deg 12
Degree 15 sibling: deg 15
Degree 20 siblings: deg 20, deg 20, deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 5.1.162000.2

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 ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 3.2.0.1$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}$
3.6.8.4$x^{6} + 18 x^{5} + 114 x^{4} + 344 x^{3} + 732 x^{2} + 744 x + 296$$3$$2$$8$$C_6$$[2]^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.6.5.1$x^{6} + 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$