Properties

Label 10.0.5910009391872.1
Degree $10$
Signature $[0, 5]$
Discriminant $-5.910\times 10^{12}$
Root discriminant \(18.93\)
Ramified primes $2,3,19$
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 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169)
 
gp: K = bnfinit(y^10 - y^9 + 3*y^8 + 6*y^7 + 9*y^6 + 3*y^5 + 3*y^4 + 66*y^3 + 75*y^2 - 91*y + 169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169)
 

\( x^{10} - x^{9} + 3x^{8} + 6x^{7} + 9x^{6} + 3x^{5} + 3x^{4} + 66x^{3} + 75x^{2} - 91x + 169 \) 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:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-5910009391872\) \(\medspace = -\,2^{8}\cdot 3^{11}\cdot 19^{4}\) 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:  \(18.93\)
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:   \(2\), \(3\), \(19\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{2}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{27}a^{8}+\frac{1}{27}a^{7}+\frac{1}{27}a^{6}+\frac{1}{27}a^{5}+\frac{1}{27}a^{4}+\frac{1}{27}a^{3}-\frac{2}{27}a^{2}-\frac{2}{27}a-\frac{2}{27}$, $\frac{1}{9477}a^{9}-\frac{22}{3159}a^{8}+\frac{1}{117}a^{7}+\frac{2}{3159}a^{6}-\frac{127}{3159}a^{5}-\frac{173}{1053}a^{4}-\frac{311}{3159}a^{3}+\frac{230}{3159}a^{2}+\frac{524}{1053}a-\frac{97}{729}$ 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:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{80}{9477} a^{9} + \frac{5}{3159} a^{8} - \frac{2}{117} a^{7} - \frac{160}{3159} a^{6} - \frac{370}{3159} a^{5} + \frac{34}{1053} a^{4} + \frac{310}{3159} a^{3} - \frac{850}{3159} a^{2} - \frac{736}{1053} a + \frac{794}{729} \)  (order $6$) 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:   $\frac{5}{729}a^{9}-\frac{2}{243}a^{8}+\frac{10}{243}a^{6}-\frac{14}{243}a^{5}-\frac{10}{81}a^{4}-\frac{151}{243}a^{3}+\frac{16}{243}a^{2}-\frac{26}{81}a-\frac{797}{729}$, $\frac{68}{9477}a^{9}-\frac{92}{3159}a^{8}+\frac{1}{39}a^{7}+\frac{136}{3159}a^{6}-\frac{914}{3159}a^{5}+\frac{287}{1053}a^{4}-\frac{3247}{3159}a^{3}+\frac{6514}{3159}a^{2}-\frac{3212}{1053}a+\frac{208}{729}$, $\frac{16}{3159}a^{9}-\frac{1}{1053}a^{8}-\frac{4}{117}a^{7}+\frac{32}{1053}a^{6}+\frac{74}{1053}a^{5}-\frac{233}{351}a^{4}-\frac{62}{1053}a^{3}+\frac{170}{1053}a^{2}-\frac{313}{351}a-\frac{13}{243}$, $\frac{4}{1053}a^{9}+\frac{1}{117}a^{8}+\frac{4}{351}a^{7}-\frac{2}{39}a^{6}+\frac{4}{117}a^{5}+\frac{4}{351}a^{4}+\frac{95}{351}a^{3}-\frac{53}{117}a^{2}-\frac{17}{351}a+\frac{11}{81}$ 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:  \( 1657.63006945 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1657.63006945 \cdot 1}{6\cdot\sqrt{5910009391872}}\cr\approx \mathstrut & 1.11286279173 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169)
 
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 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169, 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 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169);
 
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 - x^9 + 3*x^8 + 6*x^7 + 9*x^6 + 3*x^5 + 3*x^4 + 66*x^3 + 75*x^2 - 91*x + 169);
 
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

\(\Q(\sqrt{-3}) \), 5.3.1403568.1

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: 5.3.1403568.1
Degree 6 sibling: 6.0.113689008.4
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: 5.3.1403568.1

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 ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ R ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{5}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) 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 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.9.9$x^{6} + 6 x^{4} + 21$$6$$1$$9$$C_6$$[2]_{2}$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$[\ ]$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$