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Properties

Label 42.0.43153184068...0816.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,2^{42}\cdot 3^{21}\cdot 43^{41}$
Root discriminant $136.20$
Ramified primes $2, 3, 43$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![449795187729, 0, 11544743151711, 0, 88509697496451, 0, 320320809987156, 0, 667335020806575, 0, 893824482413655, 0, 825068752997220, 0, 550045835331480, 0, 273674766108555, 0, 104028419865825, 0, 30713152531815, 0, 7121890442160, 0, 1305679914396, 0, 189714175596, 0, 21806227080, 0, 1969594704, 0, 138020841, 0, 7345647, 0, 286767, 0, 7740, 0, 129, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729)
 
gp: K = bnfinit(x^42 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729, 1)
 

Normalized defining polynomial

\( x^{42} + 129 x^{40} + 7740 x^{38} + 286767 x^{36} + 7345647 x^{34} + 138020841 x^{32} + 1969594704 x^{30} + 21806227080 x^{28} + 189714175596 x^{26} + 1305679914396 x^{24} + 7121890442160 x^{22} + 30713152531815 x^{20} + 104028419865825 x^{18} + 273674766108555 x^{16} + 550045835331480 x^{14} + 825068752997220 x^{12} + 893824482413655 x^{10} + 667335020806575 x^{8} + 320320809987156 x^{6} + 88509697496451 x^{4} + 11544743151711 x^{2} + 449795187729 \)

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

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 21]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-431531840687485204395471525646239103130650561849883646892536309996173267675213047913250816=-\,2^{42}\cdot 3^{21}\cdot 43^{41}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $136.20$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(516=2^{2}\cdot 3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{516}(1,·)$, $\chi_{516}(515,·)$, $\chi_{516}(133,·)$, $\chi_{516}(263,·)$, $\chi_{516}(395,·)$, $\chi_{516}(13,·)$, $\chi_{516}(145,·)$, $\chi_{516}(131,·)$, $\chi_{516}(407,·)$, $\chi_{516}(25,·)$, $\chi_{516}(155,·)$, $\chi_{516}(287,·)$, $\chi_{516}(289,·)$, $\chi_{516}(419,·)$, $\chi_{516}(169,·)$, $\chi_{516}(385,·)$, $\chi_{516}(49,·)$, $\chi_{516}(179,·)$, $\chi_{516}(181,·)$, $\chi_{516}(347,·)$, $\chi_{516}(445,·)$, $\chi_{516}(191,·)$, $\chi_{516}(193,·)$, $\chi_{516}(323,·)$, $\chi_{516}(325,·)$, $\chi_{516}(71,·)$, $\chi_{516}(119,·)$, $\chi_{516}(397,·)$, $\chi_{516}(337,·)$, $\chi_{516}(467,·)$, $\chi_{516}(335,·)$, $\chi_{516}(97,·)$, $\chi_{516}(227,·)$, $\chi_{516}(229,·)$, $\chi_{516}(361,·)$, $\chi_{516}(491,·)$, $\chi_{516}(109,·)$, $\chi_{516}(371,·)$, $\chi_{516}(503,·)$, $\chi_{516}(121,·)$, $\chi_{516}(253,·)$, $\chi_{516}(383,·)$$\rbrace$
This is a CM field.

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

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

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:  $20$
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

$C_{42}$ (as 42T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-129}) \), 3.3.1849.1, 6.0.254030589504.1, 7.7.6321363049.1, 14.0.61568510194571181216996409344.1, \(\Q(\zeta_{43})^+\)

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 $21^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $21^{2}$ $42$ $21^{2}$ $21^{2}$ $21^{2}$ $42$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{6}$

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
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
3Data not computed
43Data not computed