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Properties

Label 42.0.54369488606...3184.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,2^{42}\cdot 3^{21}\cdot 7^{77}$
Root discriminant $122.73$
Ramified primes $2, 3, 7$
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![73222472421, 0, 2391934099086, 0, 22723373941317, 0, 99265265112069, 0, 243471048667812, 0, 375225697314468, 0, 390547575000324, 0, 288556237574829, 0, 156850504352136, 0, 64374234331239, 0, 20324026948869, 0, 4999967638659, 0, 966154817250, 0, 147151945080, 0, 17647832475, 0, 1656597096, 0, 120236886, 0, 6608385, 0, 265734, 0, 7371, 0, 126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 126*x^40 + 7371*x^38 + 265734*x^36 + 6608385*x^34 + 120236886*x^32 + 1656597096*x^30 + 17647832475*x^28 + 147151945080*x^26 + 966154817250*x^24 + 4999967638659*x^22 + 20324026948869*x^20 + 64374234331239*x^18 + 156850504352136*x^16 + 288556237574829*x^14 + 390547575000324*x^12 + 375225697314468*x^10 + 243471048667812*x^8 + 99265265112069*x^6 + 22723373941317*x^4 + 2391934099086*x^2 + 73222472421)
 
gp: K = bnfinit(x^42 + 126*x^40 + 7371*x^38 + 265734*x^36 + 6608385*x^34 + 120236886*x^32 + 1656597096*x^30 + 17647832475*x^28 + 147151945080*x^26 + 966154817250*x^24 + 4999967638659*x^22 + 20324026948869*x^20 + 64374234331239*x^18 + 156850504352136*x^16 + 288556237574829*x^14 + 390547575000324*x^12 + 375225697314468*x^10 + 243471048667812*x^8 + 99265265112069*x^6 + 22723373941317*x^4 + 2391934099086*x^2 + 73222472421, 1)
 

Normalized defining polynomial

\( x^{42} + 126 x^{40} + 7371 x^{38} + 265734 x^{36} + 6608385 x^{34} + 120236886 x^{32} + 1656597096 x^{30} + 17647832475 x^{28} + 147151945080 x^{26} + 966154817250 x^{24} + 4999967638659 x^{22} + 20324026948869 x^{20} + 64374234331239 x^{18} + 156850504352136 x^{16} + 288556237574829 x^{14} + 390547575000324 x^{12} + 375225697314468 x^{10} + 243471048667812 x^{8} + 99265265112069 x^{6} + 22723373941317 x^{4} + 2391934099086 x^{2} + 73222472421 \)

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:  \(-5436948860695888782893198886016377149049148530413040928765325951335574011833955525853184=-\,2^{42}\cdot 3^{21}\cdot 7^{77}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.73$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(588=2^{2}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(131,·)$, $\chi_{588}(503,·)$, $\chi_{588}(395,·)$, $\chi_{588}(143,·)$, $\chi_{588}(529,·)$, $\chi_{588}(121,·)$, $\chi_{588}(277,·)$, $\chi_{588}(25,·)$, $\chi_{588}(541,·)$, $\chi_{588}(289,·)$, $\chi_{588}(419,·)$, $\chi_{588}(37,·)$, $\chi_{588}(551,·)$, $\chi_{588}(169,·)$, $\chi_{588}(299,·)$, $\chi_{588}(47,·)$, $\chi_{588}(563,·)$, $\chi_{588}(311,·)$, $\chi_{588}(59,·)$, $\chi_{588}(445,·)$, $\chi_{588}(193,·)$, $\chi_{588}(457,·)$, $\chi_{588}(587,·)$, $\chi_{588}(205,·)$, $\chi_{588}(335,·)$, $\chi_{588}(337,·)$, $\chi_{588}(83,·)$, $\chi_{588}(85,·)$, $\chi_{588}(215,·)$, $\chi_{588}(479,·)$, $\chi_{588}(227,·)$, $\chi_{588}(421,·)$, $\chi_{588}(167,·)$, $\chi_{588}(109,·)$, $\chi_{588}(467,·)$, $\chi_{588}(373,·)$, $\chi_{588}(361,·)$, $\chi_{588}(505,·)$, $\chi_{588}(251,·)$, $\chi_{588}(253,·)$, $\chi_{588}(383,·)$$\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

\(\Q(\sqrt{-21}) \), \(\Q(\zeta_{7})^+\), 6.0.29042496.1, 7.7.13841287201.1, 14.0.48052913294624214844455469056.1, \(\Q(\zeta_{49})^+\)

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 $21^{2}$ R $21^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$ $21^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $42$ $42$

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
7Data not computed