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Properties

Label 46.0.22048317286...9472.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,2^{69}\cdot 47^{44}$
Root discriminant $112.45$
Ramified primes $2, 47$
Class number Not computed
Class group Not computed
Galois group $C_{46}$ (as 46T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8388608, 0, 1157627904, 0, 26528972800, 0, 241413652480, 0, 1163958681600, 0, 3440144547840, 0, 6802103992320, 0, 9530420428800, 0, 9848101109760, 0, 7724000870400, 0, 4695379476480, 0, 2246058147840, 0, 854478643200, 0, 260287340544, 0, 63694653440, 0, 12519293952, 0, 1968759936, 0, 245656320, 0, 23980736, 0, 1790880, 0, 98728, 0, 3784, 0, 90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 + 90*x^44 + 3784*x^42 + 98728*x^40 + 1790880*x^38 + 23980736*x^36 + 245656320*x^34 + 1968759936*x^32 + 12519293952*x^30 + 63694653440*x^28 + 260287340544*x^26 + 854478643200*x^24 + 2246058147840*x^22 + 4695379476480*x^20 + 7724000870400*x^18 + 9848101109760*x^16 + 9530420428800*x^14 + 6802103992320*x^12 + 3440144547840*x^10 + 1163958681600*x^8 + 241413652480*x^6 + 26528972800*x^4 + 1157627904*x^2 + 8388608)
 
gp: K = bnfinit(x^46 + 90*x^44 + 3784*x^42 + 98728*x^40 + 1790880*x^38 + 23980736*x^36 + 245656320*x^34 + 1968759936*x^32 + 12519293952*x^30 + 63694653440*x^28 + 260287340544*x^26 + 854478643200*x^24 + 2246058147840*x^22 + 4695379476480*x^20 + 7724000870400*x^18 + 9848101109760*x^16 + 9530420428800*x^14 + 6802103992320*x^12 + 3440144547840*x^10 + 1163958681600*x^8 + 241413652480*x^6 + 26528972800*x^4 + 1157627904*x^2 + 8388608, 1)
 

Normalized defining polynomial

\( x^{46} + 90 x^{44} + 3784 x^{42} + 98728 x^{40} + 1790880 x^{38} + 23980736 x^{36} + 245656320 x^{34} + 1968759936 x^{32} + 12519293952 x^{30} + 63694653440 x^{28} + 260287340544 x^{26} + 854478643200 x^{24} + 2246058147840 x^{22} + 4695379476480 x^{20} + 7724000870400 x^{18} + 9848101109760 x^{16} + 9530420428800 x^{14} + 6802103992320 x^{12} + 3440144547840 x^{10} + 1163958681600 x^{8} + 241413652480 x^{6} + 26528972800 x^{4} + 1157627904 x^{2} + 8388608 \)

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:  \(-22048317286260592124444353227775948209084132414256608507828666339058160374378508639437661929472=-\,2^{69}\cdot 47^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.45$
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:  $2, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(376=2^{3}\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{376}(1,·)$, $\chi_{376}(3,·)$, $\chi_{376}(97,·)$, $\chi_{376}(81,·)$, $\chi_{376}(9,·)$, $\chi_{376}(267,·)$, $\chi_{376}(337,·)$, $\chi_{376}(115,·)$, $\chi_{376}(145,·)$, $\chi_{376}(259,·)$, $\chi_{376}(89,·)$, $\chi_{376}(25,·)$, $\chi_{376}(27,·)$, $\chi_{376}(289,·)$, $\chi_{376}(291,·)$, $\chi_{376}(49,·)$, $\chi_{376}(169,·)$, $\chi_{376}(299,·)$, $\chi_{376}(307,·)$, $\chi_{376}(177,·)$, $\chi_{376}(51,·)$, $\chi_{376}(155,·)$, $\chi_{376}(283,·)$, $\chi_{376}(59,·)$, $\chi_{376}(65,·)$, $\chi_{376}(195,·)$, $\chi_{376}(147,·)$, $\chi_{376}(353,·)$, $\chi_{376}(75,·)$, $\chi_{376}(209,·)$, $\chi_{376}(83,·)$, $\chi_{376}(121,·)$, $\chi_{376}(345,·)$, $\chi_{376}(331,·)$, $\chi_{376}(347,·)$, $\chi_{376}(225,·)$, $\chi_{376}(131,·)$, $\chi_{376}(17,·)$, $\chi_{376}(361,·)$, $\chi_{376}(363,·)$, $\chi_{376}(241,·)$, $\chi_{376}(243,·)$, $\chi_{376}(153,·)$, $\chi_{376}(249,·)$, $\chi_{376}(251,·)$, $\chi_{376}(371,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{1048576} a^{40}$, $\frac{1}{1048576} a^{41}$, $\frac{1}{2097152} a^{42}$, $\frac{1}{2097152} a^{43}$, $\frac{1}{4194304} a^{44}$, $\frac{1}{4194304} a^{45}$

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

$C_{46}$ (as 46T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\zeta_{47})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $23^{2}$ $46$ $46$ $23^{2}$ $46$ $23^{2}$ $23^{2}$ $46$ $46$ $46$ $46$ $23^{2}$ $23^{2}$ R $46$ $23^{2}$

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
2Data not computed
47Data not computed