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Properties

Label 46.0.13951806204...6875.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,5^{43}\cdot 7\cdot 131063\cdot 210671\cdot 2644627\cdot 2978335879\cdot 603182222032259\cdot 1336443840257415421348497754512119$
Root discriminant $193.09$
Ramified primes $5, 7, 131063, 210671, 2644627, 2978335879, 603182222032259, 1336443840257415421348497754512119$
Class number Not computed
Class group Not computed
Galois group $S_{46}$ (as 46T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -3, 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, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - 3*x + 5)
 
gp: K = bnfinit(x^46 - 3*x + 5, 1)
 

Normalized defining polynomial

\( x^{46} - 3 x + 5 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 23]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1395180620453034809072318730770896284090869954075666943655311932200146112362517669680528342723846435546875=-\,5^{43}\cdot 7\cdot 131063\cdot 210671\cdot 2644627\cdot 2978335879\cdot 603182222032259\cdot 1336443840257415421348497754512119\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $193.09$
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:  $5, 7, 131063, 210671, 2644627, 2978335879, 603182222032259, 1336443840257415421348497754512119$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{5} a^{44} + \frac{1}{5} a^{43} + \frac{2}{5} a^{42} - \frac{2}{5} a^{41} - \frac{2}{5} a^{39} - \frac{2}{5} a^{38} + \frac{1}{5} a^{37} - \frac{1}{5} a^{36} - \frac{1}{5} a^{34} - \frac{1}{5} a^{33} - \frac{2}{5} a^{32} + \frac{2}{5} a^{31} + \frac{2}{5} a^{29} + \frac{2}{5} a^{28} - \frac{1}{5} a^{27} + \frac{1}{5} a^{26} + \frac{1}{5} a^{24} + \frac{1}{5} a^{23} + \frac{2}{5} a^{22} - \frac{2}{5} a^{21} - \frac{2}{5} a^{19} - \frac{2}{5} a^{18} + \frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{45} + \frac{1}{5} a^{43} + \frac{1}{5} a^{42} + \frac{2}{5} a^{41} - \frac{2}{5} a^{40} - \frac{2}{5} a^{38} - \frac{2}{5} a^{37} + \frac{1}{5} a^{36} - \frac{1}{5} a^{35} - \frac{1}{5} a^{33} - \frac{1}{5} a^{32} - \frac{2}{5} a^{31} + \frac{2}{5} a^{30} + \frac{2}{5} a^{28} + \frac{2}{5} a^{27} - \frac{1}{5} a^{26} + \frac{1}{5} a^{25} + \frac{1}{5} a^{23} + \frac{1}{5} a^{22} + \frac{2}{5} a^{21} - \frac{2}{5} a^{20} - \frac{2}{5} a^{18} - \frac{2}{5} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$

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:  $22$
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

$S_{46}$ (as 46T56):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5502622159812088949850305428800254892961651752960000000000
The 105558 conjugacy class representatives for $S_{46}$ are not computed
Character table for $S_{46}$ 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 $46$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R R $17{,}\,15{,}\,{\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ $24{,}\,17{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $27{,}\,18{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $32{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $23{,}\,{\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ $18{,}\,{\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ $20{,}\,{\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $40{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $26{,}\,{\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }$ $23{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $23{,}\,{\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $44{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
5Data not computed
7Data not computed
131063Data not computed
210671Data not computed
2644627Data not computed
2978335879Data not computed
603182222032259Data not computed
1336443840257415421348497754512119Data not computed