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Properties

Label 47.1.312...512.1
Degree $47$
Signature $[1, 23]$
Discriminant $-3.128\times 10^{91}$
Root discriminant $88.45$
Ramified primes see page
Class number not computed
Class group not computed
Galois group $S_{47}$ (as 47T6)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^47 + 4*x - 4)
 
gp: K = bnfinit(x^47 + 4*x - 4, 1)
 
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, 0, 0, 0, 0, 0, 0, 1]);
 

\(x^{47} + 4 x - 4\)  Toggle raw display

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

Invariants

Degree:  $47$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 23]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-31\!\cdots\!512\)\(\medspace = -\,2^{46}\cdot 3\cdot 521\cdot 11213\cdot 282143\cdot 89895405677639647952981987260243770129838848465517664225292339799\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $88.45$
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:  $2, 3, 521, 11213, 282143, 89895405677639647952981987260243770129838848465517664225292339799$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{2} a^{26}$, $\frac{1}{2} a^{27}$, $\frac{1}{2} a^{28}$, $\frac{1}{2} a^{29}$, $\frac{1}{2} a^{30}$, $\frac{1}{2} a^{31}$, $\frac{1}{2} a^{32}$, $\frac{1}{2} a^{33}$, $\frac{1}{2} a^{34}$, $\frac{1}{2} a^{35}$, $\frac{1}{2} a^{36}$, $\frac{1}{2} a^{37}$, $\frac{1}{2} a^{38}$, $\frac{1}{2} a^{39}$, $\frac{1}{2} a^{40}$, $\frac{1}{2} a^{41}$, $\frac{1}{2} a^{42}$, $\frac{1}{2} a^{43}$, $\frac{1}{2} a^{44}$, $\frac{1}{2} a^{45}$, $\frac{1}{6} a^{46} - \frac{1}{6} a^{45} + \frac{1}{6} a^{44} - \frac{1}{6} a^{43} + \frac{1}{6} a^{42} - \frac{1}{6} a^{41} + \frac{1}{6} a^{40} - \frac{1}{6} a^{39} + \frac{1}{6} a^{38} - \frac{1}{6} a^{37} + \frac{1}{6} a^{36} - \frac{1}{6} a^{35} + \frac{1}{6} a^{34} - \frac{1}{6} a^{33} + \frac{1}{6} a^{32} - \frac{1}{6} a^{31} + \frac{1}{6} a^{30} - \frac{1}{6} a^{29} + \frac{1}{6} a^{28} - \frac{1}{6} a^{27} + \frac{1}{6} a^{26} - \frac{1}{6} a^{25} + \frac{1}{6} a^{24} + \frac{1}{3} a^{23} - \frac{1}{3} a^{22} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$  Toggle raw display

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:  $23$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

$S_{47}$ (as 47T6):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 258623241511168180642964355153611979969197632389120000000000
The 124754 conjugacy class representatives for $S_{47}$ are not computed
Character table for $S_{47}$ 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 R $46{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $19{,}\,{\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $19{,}\,15{,}\,{\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $29{,}\,{\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $30{,}\,16{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $29{,}\,{\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $19{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ $32{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $37{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $35{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ $25{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $17{,}\,15{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $46{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $16{,}\,{\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $34{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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
2Data not computed
3Data not computed
521Data not computed
11213Data not computed
282143Data not computed
89895405677639647952981987260243770129838848465517664225292339799Data not computed