Properties

Label 42.0.74252462132...3504.1
Degree $42$
Signature $[0, 21]$
Discriminant $-\,2^{42}\cdot 7^{76}$
Root discriminant $67.65$
Ramified primes $2, 7$
Class number $1923461$ (GRH)
Class group $[1923461]$ (GRH)
Galois group $C_{42}$ (as 42T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 42*x^40 + 819*x^38 + 9842*x^36 + 81585*x^34 + 494802*x^32 + 2272424*x^30 + 8069423*x^28 + 22428224*x^26 + 49085050*x^24 + 84669739*x^22 + 114704933*x^20 + 121049551*x^18 + 98190708*x^16 + 60019861*x^14 + 26865216*x^12 + 8444436*x^10 + 1749188*x^8 + 215453*x^6 + 13181*x^4 + 294*x^2 + 1)
 
gp: K = bnfinit(x^42 + 42*x^40 + 819*x^38 + 9842*x^36 + 81585*x^34 + 494802*x^32 + 2272424*x^30 + 8069423*x^28 + 22428224*x^26 + 49085050*x^24 + 84669739*x^22 + 114704933*x^20 + 121049551*x^18 + 98190708*x^16 + 60019861*x^14 + 26865216*x^12 + 8444436*x^10 + 1749188*x^8 + 215453*x^6 + 13181*x^4 + 294*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 294, 0, 13181, 0, 215453, 0, 1749188, 0, 8444436, 0, 26865216, 0, 60019861, 0, 98190708, 0, 121049551, 0, 114704933, 0, 84669739, 0, 49085050, 0, 22428224, 0, 8069423, 0, 2272424, 0, 494802, 0, 81585, 0, 9842, 0, 819, 0, 42, 0, 1]);
 

Normalized defining polynomial

\( x^{42} + 42 x^{40} + 819 x^{38} + 9842 x^{36} + 81585 x^{34} + 494802 x^{32} + 2272424 x^{30} + 8069423 x^{28} + 22428224 x^{26} + 49085050 x^{24} + 84669739 x^{22} + 114704933 x^{20} + 121049551 x^{18} + 98190708 x^{16} + 60019861 x^{14} + 26865216 x^{12} + 8444436 x^{10} + 1749188 x^{8} + 215453 x^{6} + 13181 x^{4} + 294 x^{2} + 1 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-74252462132603256348231837398371002884673933378885582779211491265789772693504=-\,2^{42}\cdot 7^{76}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $67.65$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(196=2^{2}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(51,·)$, $\chi_{196}(135,·)$, $\chi_{196}(9,·)$, $\chi_{196}(11,·)$, $\chi_{196}(141,·)$, $\chi_{196}(15,·)$, $\chi_{196}(149,·)$, $\chi_{196}(23,·)$, $\chi_{196}(25,·)$, $\chi_{196}(155,·)$, $\chi_{196}(29,·)$, $\chi_{196}(163,·)$, $\chi_{196}(37,·)$, $\chi_{196}(65,·)$, $\chi_{196}(39,·)$, $\chi_{196}(169,·)$, $\chi_{196}(43,·)$, $\chi_{196}(177,·)$, $\chi_{196}(179,·)$, $\chi_{196}(53,·)$, $\chi_{196}(137,·)$, $\chi_{196}(57,·)$, $\chi_{196}(151,·)$, $\chi_{196}(191,·)$, $\chi_{196}(193,·)$, $\chi_{196}(67,·)$, $\chi_{196}(71,·)$, $\chi_{196}(183,·)$, $\chi_{196}(79,·)$, $\chi_{196}(81,·)$, $\chi_{196}(85,·)$, $\chi_{196}(93,·)$, $\chi_{196}(95,·)$, $\chi_{196}(165,·)$, $\chi_{196}(99,·)$, $\chi_{196}(107,·)$, $\chi_{196}(109,·)$, $\chi_{196}(113,·)$, $\chi_{196}(121,·)$, $\chi_{196}(123,·)$, $\chi_{196}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{1923461}$, which has order $1923461$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a^{35} + 35 a^{33} + 560 a^{31} + 5425 a^{29} + 35525 a^{27} + 166257 a^{25} + 573300 a^{23} + 1480049 a^{21} + 2877854 a^{19} + 4205936 a^{17} + 4575312 a^{15} + 3637270 a^{13} + 2051777 a^{11} + 784343 a^{9} + 188653 a^{7} + 25060 a^{5} + 1414 a^{3} + 21 a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1776855897760068.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 6.0.153664.1, 7.7.13841287201.1, 14.0.3138866894939200133545984.1, \(\Q(\zeta_{49})^+\)

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 $42$ $21^{2}$ R $42$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $21^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ $42$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
7Data not computed