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Properties

Label 46.46.2204831728...9472.1
Degree $46$
Signature $[46, 0]$
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:  $[46, 0]$
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$$\chi_{376}(1,·)$, $\chi_{376}(277,·)$, $\chi_{376}(81,·)$, $\chi_{376}(9,·)$, $\chi_{376}(269,·)$, $\chi_{376}(337,·)$, $\chi_{376}(365,·)$, $\chi_{376}(145,·)$, $\chi_{376}(149,·)$, $\chi_{376}(89,·)$, $\chi_{376}(25,·)$, $\chi_{376}(285,·)$, $\chi_{376}(341,·)$, $\chi_{376}(289,·)$, $\chi_{376}(37,·)$, $\chi_{376}(49,·)$, $\chi_{376}(169,·)$, $\chi_{376}(173,·)$, $\chi_{376}(157,·)$, $\chi_{376}(177,·)$, $\chi_{376}(309,·)$, $\chi_{376}(61,·)$, $\chi_{376}(53,·)$, $\chi_{376}(65,·)$, $\chi_{376}(197,·)$, $\chi_{376}(97,·)$, $\chi_{376}(333,·)$, $\chi_{376}(205,·)$, $\chi_{376}(209,·)$, $\chi_{376}(213,·)$, $\chi_{376}(121,·)$, $\chi_{376}(345,·)$, $\chi_{376}(101,·)$, $\chi_{376}(165,·)$, $\chi_{376}(353,·)$, $\chi_{376}(357,·)$, $\chi_{376}(17,·)$, $\chi_{376}(361,·)$, $\chi_{376}(225,·)$, $\chi_{376}(237,·)$, $\chi_{376}(189,·)$, $\chi_{376}(241,·)$, $\chi_{376}(153,·)$, $\chi_{376}(249,·)$, $\chi_{376}(253,·)$, $\chi_{376}(21,·)$$\rbrace$
This is not 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:  $45$
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 $46$ $46$ $23^{2}$ $46$ $46$ $23^{2}$ $46$ $23^{2}$ $46$ $23^{2}$ $46$ $23^{2}$ $46$ R $46$ $46$

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