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Properties

Label 40.2.489...959.1
Degree $40$
Signature $[2, 19]$
Discriminant $-4.899\times 10^{82}$
Root discriminant \(116.75\)
Ramified primes see page
Class number not computed
Class group not computed
Galois group $S_{40}$ (as 40T315842)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^40 - x - 3)
 
Copy content gp:K = bnfinit(y^40 - y - 3, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x - 3);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - x - 3)
 

\( x^{40} - x - 3 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $40$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 19]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(-489\!\cdots\!959\) \(\medspace = -\,3^{39}\cdot 47\cdot 31247\cdot 1492676502904837324513\cdot 55\!\cdots\!81\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(116.75\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(3\), \(47\), \(31247\), \(1492676502904837324513\), \(55147\!\cdots\!92581\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  $\Q(\sqrt{-36267\!\cdots\!34831}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $20$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{2}\cdot(2\pi)^{19}\cdot R \cdot h}{2\cdot\sqrt{48992385598969548352871113647963491615789709448988889059930192105219196517009951959}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^40 - x - 3) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^40 - x - 3, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x - 3); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^40 - x - 3); 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

$S_{40}$ (as 40T315842):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 815915283247897734345611269596115894272000000000
The 37338 conjugacy class representatives for $S_{40}$ are not computed
Character table for $S_{40}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.14.0.1}{14} }{,}\,{\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ R $29{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ $29{,}\,{\href{/padicField/7.11.0.1}{11} }$ $29{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ $20{,}\,17{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $25{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $17{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $40$ $15{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ $31{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ $24{,}\,16$ R $16{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $30{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.1.3.3a1.2$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
3.3.3.9a7.1$x^{9} + 6 x^{7} + 3 x^{6} + 18 x^{5} + 15 x^{4} + 26 x^{3} + 24 x^{2} + 15 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
3.3.3.9a4.1$x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 15 x^{4} + 11 x^{3} + 18 x^{2} + 9 x + 4$$3$$3$$9$$(C_3^3:C_3):C_2$$$[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
3.3.3.9a8.1$x^{9} + 6 x^{7} + 3 x^{6} + 15 x^{5} + 12 x^{4} + 20 x^{3} + 15 x^{2} + 12 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
3.3.3.9a7.1$x^{9} + 6 x^{7} + 3 x^{6} + 18 x^{5} + 15 x^{4} + 26 x^{3} + 24 x^{2} + 15 x + 7$$3$$3$$9$$C_3^2 : S_3 $$$[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.6.1.0a1.1$x^{6} + 2 x^{4} + 35 x^{3} + 9 x^{2} + 41 x + 5$$1$$6$$0$$C_6$$$[\ ]^{6}$$
47.8.1.0a1.1$x^{8} + x^{4} + 29 x^{3} + 19 x^{2} + 3 x + 5$$1$$8$$0$$C_8$$$[\ ]^{8}$$
47.9.1.0a1.1$x^{9} + x^{3} + 19 x^{2} + x + 42$$1$$9$$0$$C_9$$$[\ ]^{9}$$
47.13.1.0a1.1$x^{13} + 5 x + 42$$1$$13$$0$$C_{13}$$$[\ ]^{13}$$
\(31247\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $8$$1$$8$$0$$C_8$$$[\ ]^{8}$$
Deg $9$$1$$9$$0$$C_9$$$[\ ]^{9}$$
\(149\!\cdots\!513\) Copy content Toggle raw display $\Q_{14\!\cdots\!13}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{14\!\cdots\!13}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $36$$1$$36$$0$$C_{36}$$$[\ ]^{36}$$
\(551\!\cdots\!581\) Copy content Toggle raw display $\Q_{55\!\cdots\!81}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $7$$1$$7$$0$$C_7$$$[\ ]^{7}$$
Deg $24$$1$$24$$0$$C_{24}$$$[\ ]^{24}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)