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Properties

Label 46.0.10362709124...5184.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,2^{69}\cdot 47^{45}$
Root discriminant $122.26$
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![394264576, 0, 18136170496, 0, 249372344320, 0, 1620920238080, 0, 6078450892800, 0, 14698799431680, 0, 24592222126080, 0, 29861984010240, 0, 27227103068160, 0, 19106738995200, 0, 10508706447360, 0, 4589770997760, 0, 1606419849216, 0, 453092777984, 0, 103229265920, 0, 18980865024, 0, 2803991424, 0, 329881344, 0, 30462016, 0, 2158240, 0, 113176, 0, 4136, 0, 94, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576)
 
gp: K = bnfinit(x^46 + 94*x^44 + 4136*x^42 + 113176*x^40 + 2158240*x^38 + 30462016*x^36 + 329881344*x^34 + 2803991424*x^32 + 18980865024*x^30 + 103229265920*x^28 + 453092777984*x^26 + 1606419849216*x^24 + 4589770997760*x^22 + 10508706447360*x^20 + 19106738995200*x^18 + 27227103068160*x^16 + 29861984010240*x^14 + 24592222126080*x^12 + 14698799431680*x^10 + 6078450892800*x^8 + 1620920238080*x^6 + 249372344320*x^4 + 18136170496*x^2 + 394264576, 1)
 

Normalized defining polynomial

\( x^{46} + 94 x^{44} + 4136 x^{42} + 113176 x^{40} + 2158240 x^{38} + 30462016 x^{36} + 329881344 x^{34} + 2803991424 x^{32} + 18980865024 x^{30} + 103229265920 x^{28} + 453092777984 x^{26} + 1606419849216 x^{24} + 4589770997760 x^{22} + 10508706447360 x^{20} + 19106738995200 x^{18} + 27227103068160 x^{16} + 29861984010240 x^{14} + 24592222126080 x^{12} + 14698799431680 x^{10} + 6078450892800 x^{8} + 1620920238080 x^{6} + 249372344320 x^{4} + 18136170496 x^{2} + 394264576 \)

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:  \(-1036270912454247829848884601705469565826954223470060599867947317935733537595789906053570110685184=-\,2^{69}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.26$
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$$\chi_{376}(1,·)$, $\chi_{376}(5,·)$, $\chi_{376}(81,·)$, $\chi_{376}(145,·)$, $\chi_{376}(9,·)$, $\chi_{376}(13,·)$, $\chi_{376}(45,·)$, $\chi_{376}(17,·)$, $\chi_{376}(89,·)$, $\chi_{376}(25,·)$, $\chi_{376}(29,·)$, $\chi_{376}(69,·)$, $\chi_{376}(261,·)$, $\chi_{376}(133,·)$, $\chi_{376}(289,·)$, $\chi_{376}(293,·)$, $\chi_{376}(49,·)$, $\chi_{376}(169,·)$, $\chi_{376}(301,·)$, $\chi_{376}(93,·)$, $\chi_{376}(177,·)$, $\chi_{376}(181,·)$, $\chi_{376}(117,·)$, $\chi_{376}(317,·)$, $\chi_{376}(245,·)$, $\chi_{376}(65,·)$, $\chi_{376}(325,·)$, $\chi_{376}(97,·)$, $\chi_{376}(77,·)$, $\chi_{376}(209,·)$, $\chi_{376}(85,·)$, $\chi_{376}(121,·)$, $\chi_{376}(345,·)$, $\chi_{376}(349,·)$, $\chi_{376}(353,·)$, $\chi_{376}(229,·)$, $\chi_{376}(337,·)$, $\chi_{376}(361,·)$, $\chi_{376}(225,·)$, $\chi_{376}(109,·)$, $\chi_{376}(241,·)$, $\chi_{376}(221,·)$, $\chi_{376}(373,·)$, $\chi_{376}(153,·)$, $\chi_{376}(249,·)$, $\chi_{376}(125,·)$$\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

\(\Q(\sqrt{-94}) \), \(\Q(\zeta_{47})^+\)

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 $46$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $46$ $23^{2}$ $46$ $46$ $46$ $23^{2}$ 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