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Properties

Label 10.4.7941343942351.1
Degree $10$
Signature $[4, 3]$
Discriminant $-7.941\times 10^{12}$
Root discriminant \(19.50\)
Ramified primes $71,281$
Class number $1$
Class group trivial
Galois group $S_5$ (as 10T13)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*x - 1)
 
gp: K = bnfinit(y^10 - 2*y^9 + 5*y^7 - 7*y^6 + 20*y^5 - 26*y^4 + 2*y^3 - 30*y^2 - 21*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*x - 1)
 

\( x^{10} - 2x^{9} + 5x^{7} - 7x^{6} + 20x^{5} - 26x^{4} + 2x^{3} - 30x^{2} - 21x - 1 \) 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:   \(-7941343942351\) \(\medspace = -\,71^{3}\cdot 281^{3}\) 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:  \(19.50\)
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:  $71^{1/2}281^{1/2}\approx 141.24800883552305$
Ramified primes:   \(71\), \(281\) 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:  \(\Q(\sqrt{-19951}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{418331}a^{9}+\frac{144320}{418331}a^{8}-\frac{149450}{418331}a^{7}-\frac{194866}{418331}a^{6}+\frac{105609}{418331}a^{5}-\frac{187867}{418331}a^{4}-\frac{54097}{418331}a^{3}-\frac{75779}{418331}a^{2}-\frac{149535}{418331}a+\frac{87668}{418331}$ 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:   $\frac{36876}{418331}a^{9}-\frac{62662}{418331}a^{8}-\frac{25606}{418331}a^{7}+\frac{211302}{418331}a^{6}-\frac{224126}{418331}a^{5}+\frac{614530}{418331}a^{4}-\frac{697095}{418331}a^{3}-\frac{393655}{418331}a^{2}-\frac{650080}{418331}a-\frac{1271793}{418331}$, $\frac{53195}{418331}a^{9}-\frac{108112}{418331}a^{8}-\frac{30426}{418331}a^{7}+\frac{345310}{418331}a^{6}-\frac{314575}{418331}a^{5}+\frac{760856}{418331}a^{4}-\frac{1245959}{418331}a^{3}-\frac{26389}{418331}a^{2}-\frac{787022}{418331}a-\frac{473059}{418331}$, $\frac{8400}{418331}a^{9}-\frac{35238}{418331}a^{8}+\frac{31331}{418331}a^{7}+\frac{54803}{418331}a^{6}-\frac{164451}{418331}a^{5}+\frac{280063}{418331}a^{4}-\frac{525665}{418331}a^{3}+\frac{156182}{418331}a^{2}-\frac{264338}{418331}a+\frac{148640}{418331}$, $\frac{107923}{418331}a^{9}-\frac{270763}{418331}a^{8}+\frac{77686}{418331}a^{7}+\frac{649376}{418331}a^{6}-\frac{1042981}{418331}a^{5}+\frac{2169991}{418331}a^{4}-\frac{3429743}{418331}a^{3}+\frac{1329026}{418331}a^{2}-\frac{2402473}{418331}a-\frac{835325}{418331}$, $\frac{63714}{418331}a^{9}-\frac{129231}{418331}a^{8}-\frac{7078}{418331}a^{7}+\frac{371756}{418331}a^{6}-\frac{500640}{418331}a^{5}+\frac{1201858}{418331}a^{4}-\frac{1362142}{418331}a^{3}-\frac{225135}{418331}a^{2}-\frac{1239458}{418331}a-\frac{1131553}{418331}$, $\frac{19409}{418331}a^{9}-\frac{37496}{418331}a^{8}+\frac{32104}{418331}a^{7}-\frac{23623}{418331}a^{6}-\frac{56819}{418331}a^{5}+\frac{699055}{418331}a^{4}-\frac{1212856}{418331}a^{3}+\frac{893847}{418331}a^{2}-\frac{1199330}{418331}a-\frac{640627}{418331}$ 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:  \( 323.117320854 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 323.117320854 \cdot 1}{2\cdot\sqrt{7941343942351}}\cr\approx \mathstrut & 0.227532180255 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*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 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*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 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*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 - 2*x^9 + 5*x^7 - 7*x^6 + 20*x^5 - 26*x^4 + 2*x^3 - 30*x^2 - 21*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 10T13):

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

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 5 sibling: 5.3.19951.1
Degree 6 sibling: 6.0.7941343942351.1
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: 5.3.19951.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.5.0.1}{5} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.5.0.1}{5} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 64$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.0.1$x^{2} + 69 x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(281\) Copy content Toggle raw display $\Q_{281}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{281}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$