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Properties

Label 42.0.17067911533...7584.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,2^{42}\cdot 47\cdot 29989\cdot 133117577\cdot 2068356136392292742885048864559553903077433124109604781071241999887231$
Root discriminant $195.54$
Ramified primes $2, 47, 29989, 133117577, 2068356136392292742885048864559553903077433124109604781071241999887231$
Class number Not computed
Class group Not computed
Galois group $S_{42}$ (as 42T9491)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -2, 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, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 2*x + 5)
 
gp: K = bnfinit(x^42 - 2*x + 5, 1)
 

Normalized defining polynomial

\( x^{42} - 2 x + 5 \)

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

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 21]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1706791153370620261603760192550683915841711308180357561779039638906972822017958629996426464067584=-\,2^{42}\cdot 47\cdot 29989\cdot 133117577\cdot 2068356136392292742885048864559553903077433124109604781071241999887231\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.54$
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:  $2, 47, 29989, 133117577, 2068356136392292742885048864559553903077433124109604781071241999887231$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{2} a^{41} - \frac{1}{2} a^{40} - \frac{1}{2} a^{39} - \frac{1}{2} a^{38} - \frac{1}{2} a^{37} - \frac{1}{2} a^{36} - \frac{1}{2} a^{35} - \frac{1}{2} a^{34} - \frac{1}{2} a^{33} - \frac{1}{2} a^{32} - \frac{1}{2} a^{31} - \frac{1}{2} a^{30} - \frac{1}{2} a^{29} - \frac{1}{2} a^{28} - \frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$

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

$S_{42}$ (as 42T9491):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1405006117752879898543142606244511569936384000000000
The 53174 conjugacy class representatives for $S_{42}$ are not computed
Character table for $S_{42}$ 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 $37{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $20^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $37{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $40{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $39{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $29{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18{,}\,15{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }$ $21{,}\,20{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $24{,}\,{\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $30{,}\,{\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $26{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ R $20{,}\,18{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $22{,}\,15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.12.1$x^{12} - 48 x^{10} + 49 x^{8} + 8 x^{6} + 19 x^{4} - 24 x^{2} + 59$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
2.12.12.16$x^{12} - 16 x^{10} - 23 x^{8} + 24 x^{6} - 29 x^{4} - 8 x^{2} - 13$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
47Data not computed
29989Data not computed
133117577Data not computed
2068356136392292742885048864559553903077433124109604781071241999887231Data not computed