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Properties

Label 40.2.12379967386...3184.1
Degree $40$
Signature $[2, 19]$
Discriminant $-\,2^{40}\cdot 65581\cdot 130948229477800849\cdot 13111186908622089242206486448360790839011$
Root discriminant $71.17$
Ramified primes $2, 65581, 130948229477800849, 13111186908622089242206486448360790839011$
Class number Not computed
Class group Not computed
Galois group $S_{40}$ (as 40T315842)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 4*x - 1)
 
gp: K = bnfinit(x^40 - 4*x - 1, 1)
 

Normalized defining polynomial

\( x^{40} - 4 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 19]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-123799673865700652129832399822361287621723938537943027700441950831650013184=-\,2^{40}\cdot 65581\cdot 130948229477800849\cdot 13111186908622089242206486448360790839011\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.17$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 65581, 130948229477800849, 13111186908622089242206486448360790839011$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $20$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 815915283247897734345611269596115894272000000000
The 37338 conjugacy class representatives for $S_{40}$ are not computed
Character table for $S_{40}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $40$ $29{,}\,{\href{/LocalNumberField/5.11.0.1}{11} }$ $19{,}\,{\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $30{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $38{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $19{,}\,15{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ $33{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $20{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $24{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ $32{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $29{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $29{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $30{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
65581Data not computed
130948229477800849Data not computed
13111186908622089242206486448360790839011Data not computed