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Properties

Label 46.0.11629800537...1696.1
Degree $46$
Signature $[0, 23]$
Discriminant $-\,2^{46}\cdot 3^{23}\cdot 47^{45}$
Root discriminant $149.74$
Ramified primes $2, 3, 47$
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![4424729404869, 0, 135691701749316, 0, 1243840599368730, 0, 5389975930597830, 0, 13474939826494575, 0, 21723236326348830, 0, 24229763594773695, 0, 19614570529102515, 0, 11922582086317215, 0, 5577816180733200, 0, 2045199266268840, 0, 595505978715960, 0, 138951395033724, 0, 26127612741384, 0, 3968479767780, 0, 486458810244, 0, 47908822221, 0, 3757554684, 0, 231320934, 0, 10926090, 0, 381969, 0, 9306, 0, 141, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 + 141*x^44 + 9306*x^42 + 381969*x^40 + 10926090*x^38 + 231320934*x^36 + 3757554684*x^34 + 47908822221*x^32 + 486458810244*x^30 + 3968479767780*x^28 + 26127612741384*x^26 + 138951395033724*x^24 + 595505978715960*x^22 + 2045199266268840*x^20 + 5577816180733200*x^18 + 11922582086317215*x^16 + 19614570529102515*x^14 + 24229763594773695*x^12 + 21723236326348830*x^10 + 13474939826494575*x^8 + 5389975930597830*x^6 + 1243840599368730*x^4 + 135691701749316*x^2 + 4424729404869)
 
gp: K = bnfinit(x^46 + 141*x^44 + 9306*x^42 + 381969*x^40 + 10926090*x^38 + 231320934*x^36 + 3757554684*x^34 + 47908822221*x^32 + 486458810244*x^30 + 3968479767780*x^28 + 26127612741384*x^26 + 138951395033724*x^24 + 595505978715960*x^22 + 2045199266268840*x^20 + 5577816180733200*x^18 + 11922582086317215*x^16 + 19614570529102515*x^14 + 24229763594773695*x^12 + 21723236326348830*x^10 + 13474939826494575*x^8 + 5389975930597830*x^6 + 1243840599368730*x^4 + 135691701749316*x^2 + 4424729404869, 1)
 

Normalized defining polynomial

\( x^{46} + 141 x^{44} + 9306 x^{42} + 381969 x^{40} + 10926090 x^{38} + 231320934 x^{36} + 3757554684 x^{34} + 47908822221 x^{32} + 486458810244 x^{30} + 3968479767780 x^{28} + 26127612741384 x^{26} + 138951395033724 x^{24} + 595505978715960 x^{22} + 2045199266268840 x^{20} + 5577816180733200 x^{18} + 11922582086317215 x^{16} + 19614570529102515 x^{14} + 24229763594773695 x^{12} + 21723236326348830 x^{10} + 13474939826494575 x^{8} + 5389975930597830 x^{6} + 1243840599368730 x^{4} + 135691701749316 x^{2} + 4424729404869 \)

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:  \(-11629800537156905520110155516248308482128739308939556543099290959577415814793321701040599438192541696=-\,2^{46}\cdot 3^{23}\cdot 47^{45}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.74$
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:  $2, 3, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(564=2^{2}\cdot 3\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{564}(1,·)$, $\chi_{564}(515,·)$, $\chi_{564}(385,·)$, $\chi_{564}(11,·)$, $\chi_{564}(397,·)$, $\chi_{564}(527,·)$, $\chi_{564}(145,·)$, $\chi_{564}(275,·)$, $\chi_{564}(277,·)$, $\chi_{564}(23,·)$, $\chi_{564}(25,·)$, $\chi_{564}(539,·)$, $\chi_{564}(541,·)$, $\chi_{564}(287,·)$, $\chi_{564}(289,·)$, $\chi_{564}(419,·)$, $\chi_{564}(503,·)$, $\chi_{564}(37,·)$, $\chi_{564}(167,·)$, $\chi_{564}(553,·)$, $\chi_{564}(157,·)$, $\chi_{564}(49,·)$, $\chi_{564}(179,·)$, $\chi_{564}(311,·)$, $\chi_{564}(443,·)$, $\chi_{564}(61,·)$, $\chi_{564}(395,·)$, $\chi_{564}(407,·)$, $\chi_{564}(457,·)$, $\chi_{564}(203,·)$, $\chi_{564}(205,·)$, $\chi_{564}(337,·)$, $\chi_{564}(35,·)$, $\chi_{564}(563,·)$, $\chi_{564}(107,·)$, $\chi_{564}(97,·)$, $\chi_{564}(227,·)$, $\chi_{564}(529,·)$, $\chi_{564}(361,·)$, $\chi_{564}(323,·)$, $\chi_{564}(359,·)$, $\chi_{564}(241,·)$, $\chi_{564}(467,·)$, $\chi_{564}(169,·)$, $\chi_{564}(121,·)$, $\chi_{564}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$, $\frac{1}{387420489} a^{36}$, $\frac{1}{387420489} a^{37}$, $\frac{1}{1162261467} a^{38}$, $\frac{1}{1162261467} a^{39}$, $\frac{1}{3486784401} a^{40}$, $\frac{1}{3486784401} a^{41}$, $\frac{1}{10460353203} a^{42}$, $\frac{1}{10460353203} a^{43}$, $\frac{1}{31381059609} a^{44}$, $\frac{1}{31381059609} a^{45}$

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

\(\Q(\sqrt{-141}) \), \(\Q(\zeta_{47})^+\)

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 R $23^{2}$ $46$ $46$ $46$ $46$ $23^{2}$ $46$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ $23^{2}$ 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
3Data not computed
47Data not computed