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Properties

Label 10.4.15382321408.1
Degree $10$
Signature $[4, 3]$
Discriminant $-15382321408$
Root discriminant \(10.44\)
Ramified primes $2,59,89,11443$
Class number $1$
Class group trivial
Galois group $S_{10}$ (as 10T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{10} - x^{9} - x^{8} - 3x^{6} - x^{5} + 7x^{4} - 3x^{2} + 3x - 1 \) 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:  $[4, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-15382321408\) \(\medspace = -\,2^{8}\cdot 59\cdot 89\cdot 11443\) 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:  \(10.44\)
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:  $2^{15/8}59^{1/2}89^{1/2}11443^{1/2}\approx 28432.968243602874$
Ramified primes:   \(2\), \(59\), \(89\), \(11443\) 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:  $\Q(\sqrt{-60087193}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$ 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:  $6$
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:   $a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{3}{4}a^{4}-a^{3}+a^{2}+a-\frac{5}{4}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+a^{4}+\frac{7}{2}a^{3}-\frac{1}{2}a^{2}-\frac{11}{4}a+\frac{1}{2}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{5}{4}a^{5}-\frac{7}{4}a^{4}+\frac{1}{2}a^{3}+2a^{2}+\frac{1}{4}a+\frac{3}{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{3}{2}a^{5}-\frac{1}{2}a^{4}+4a^{3}-2a+\frac{3}{2}$, $\frac{1}{2}a^{9}-a^{8}-\frac{1}{2}a^{7}+a^{6}-a^{5}+a^{4}+\frac{11}{2}a^{3}-3a^{2}-\frac{11}{2}a+3$ 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:  \( 14.634461215941819 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 14.634461215941819 \cdot 1}{2\cdot\sqrt{15382321408}}\cr\approx \mathstrut & 0.234150661649348 \end{aligned}\]

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

$S_{10}$ (as 10T45):

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 3628800
The 42 conjugacy class representatives for $S_{10}$
Character table for $S_{10}$

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 20 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }$ ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ R

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.8.9$x^{8} + 8 x^{7} + 56 x^{6} + 216 x^{5} + 680 x^{4} + 1296 x^{3} + 2016 x^{2} + 1728 x + 1296$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
\(59\) Copy content Toggle raw display 59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.8.0.1$x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
\(89\) Copy content Toggle raw display 89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.8.0.1$x^{8} + 65 x^{3} + 40 x^{2} + 79 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
\(11443\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$