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Properties

Label 41.1.34387404758...2976.1
Degree $41$
Signature $[1, 20]$
Discriminant $2^{40}\cdot 63506090654183\cdot 49247500694181610257024215345813183672691438654247$
Root discriminant $69.56$
Ramified primes $2, 63506090654183, 49247500694181610257024215345813183672691438654247$
Class number Not computed
Class group Not computed
Galois group $S_{41}$ (as 41T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 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, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^41 + 4*x - 4)
 
gp: K = bnfinit(x^41 + 4*x - 4, 1)
 

Normalized defining polynomial

\( x^{41} + 4 x - 4 \)

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

Invariants

Degree:  $41$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3438740475870829672142694556791350302292427122659925573655422607717853822976=2^{40}\cdot 63506090654183\cdot 49247500694181610257024215345813183672691438654247\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.56$
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:  $2, 63506090654183, 49247500694181610257024215345813183672691438654247$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{2} a^{22}$, $\frac{1}{2} a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{2} a^{26}$, $\frac{1}{2} a^{27}$, $\frac{1}{2} a^{28}$, $\frac{1}{2} a^{29}$, $\frac{1}{2} a^{30}$, $\frac{1}{2} a^{31}$, $\frac{1}{2} a^{32}$, $\frac{1}{2} a^{33}$, $\frac{1}{2} a^{34}$, $\frac{1}{2} a^{35}$, $\frac{1}{2} a^{36}$, $\frac{1}{2} a^{37}$, $\frac{1}{2} a^{38}$, $\frac{1}{2} a^{39}$, $\frac{1}{42} a^{40} - \frac{5}{21} a^{39} - \frac{5}{42} a^{38} + \frac{4}{21} a^{37} + \frac{2}{21} a^{36} + \frac{1}{21} a^{35} + \frac{1}{42} a^{34} - \frac{5}{21} a^{33} - \frac{5}{42} a^{32} + \frac{4}{21} a^{31} + \frac{2}{21} a^{30} + \frac{1}{21} a^{29} + \frac{1}{42} a^{28} - \frac{5}{21} a^{27} - \frac{5}{42} a^{26} + \frac{4}{21} a^{25} + \frac{2}{21} a^{24} + \frac{1}{21} a^{23} + \frac{1}{42} a^{22} - \frac{5}{21} a^{21} + \frac{8}{21} a^{20} + \frac{4}{21} a^{19} + \frac{2}{21} a^{18} + \frac{1}{21} a^{17} - \frac{10}{21} a^{16} - \frac{5}{21} a^{15} + \frac{8}{21} a^{14} + \frac{4}{21} a^{13} + \frac{2}{21} a^{12} + \frac{1}{21} a^{11} - \frac{10}{21} a^{10} - \frac{5}{21} a^{9} + \frac{8}{21} a^{8} + \frac{4}{21} a^{7} + \frac{2}{21} a^{6} + \frac{1}{21} a^{5} - \frac{10}{21} a^{4} - \frac{5}{21} a^{3} + \frac{8}{21} a^{2} + \frac{4}{21} a + \frac{4}{21}$

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

$S_{41}$ (as 41T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 33452526613163807108170062053440751665152000000000
The 44583 conjugacy class representatives for $S_{41}$ are not computed
Character table for $S_{41}$ 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 $39{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ $31{,}\,{\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ $22{,}\,{\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ $30{,}\,{\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $19{,}\,18{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $37{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ $36{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $24{,}\,17$ $20{,}\,{\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ $19{,}\,{\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $34{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $23{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }$ $21{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
63506090654183Data not computed
49247500694181610257024215345813183672691438654247Data not computed