Properties

Label 42.2.585...728.1
Degree $42$
Signature $[2, 20]$
Discriminant $5.853\times 10^{78}$
Root discriminant $75.06$
Ramified primes see page
Class number not computed
Class group not computed
Galois group $S_{42}$ (as 42T9491)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

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

\( x^{42} - 2 x - 1 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 20]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(585\!\cdots\!728\)\(\medspace = 2^{43}\cdot 151\cdot 4132611804876851\cdot 111869437509421489132043\cdot 9532243245509920520310487\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $75.06$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 151, 4132611804876851, 111869437509421489132043, 9532243245509920520310487$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 $37{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $22{,}\,17{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ $42$ $30{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ $25{,}\,{\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ $38{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $25{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $27{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ $31{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $23{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $21{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $20{,}\,15{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }$ $21{,}\,{\href{/LocalNumberField/59.11.0.1}{11} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.12.4$x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 38 x^{2} - 31$$2$$6$$12$12T87$[2, 2, 2, 2, 2]^{6}$
2.12.12.4$x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 38 x^{2} - 31$$2$$6$$12$12T87$[2, 2, 2, 2, 2]^{6}$
151Data not computed
4132611804876851Data not computed
111869437509421489132043Data not computed
9532243245509920520310487Data not computed