Properties

Label 42.42.5197672349...4528.1
Degree $42$
Signature $[42, 0]$
Discriminant $2^{42}\cdot 7^{77}$
Root discriminant $70.86$
Ramified primes $2, 7$
Class number $1$ (GRH)
Class group Trivial (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 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7)
 
gp: K = bnfinit(x^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 686, 0, -19551, 0, 256221, 0, -1885324, 0, 8716708, 0, -27217932, 0, 60329941, 0, -98380632, 0, 121131479, 0, -114729727, 0, 84674891, 0, -49085750, 0, 22428280, 0, -8069425, 0, 2272424, 0, -494802, 0, 81585, 0, -9842, 0, 819, 0, -42, 0, 1]);
 

Normalized defining polynomial

\( x^{42} - 42 x^{40} + 819 x^{38} - 9842 x^{36} + 81585 x^{34} - 494802 x^{32} + 2272424 x^{30} - 8069425 x^{28} + 22428280 x^{26} - 49085750 x^{24} + 84674891 x^{22} - 114729727 x^{20} + 121131479 x^{18} - 98380632 x^{16} + 60329941 x^{14} - 27217932 x^{12} + 8716708 x^{10} - 1885324 x^{8} + 256221 x^{6} - 19551 x^{4} + 686 x^{2} - 7 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(519767234928222794437622861788597020192717533652199079454480438860528408854528=2^{42}\cdot 7^{77}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $70.86$
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$$\chi_{196}(1,·)$, $\chi_{196}(3,·)$, $\chi_{196}(65,·)$, $\chi_{196}(9,·)$, $\chi_{196}(139,·)$, $\chi_{196}(141,·)$, $\chi_{196}(143,·)$, $\chi_{196}(19,·)$, $\chi_{196}(149,·)$, $\chi_{196}(25,·)$, $\chi_{196}(27,·)$, $\chi_{196}(29,·)$, $\chi_{196}(31,·)$, $\chi_{196}(37,·)$, $\chi_{196}(167,·)$, $\chi_{196}(169,·)$, $\chi_{196}(171,·)$, $\chi_{196}(47,·)$, $\chi_{196}(177,·)$, $\chi_{196}(53,·)$, $\chi_{196}(137,·)$, $\chi_{196}(57,·)$, $\chi_{196}(159,·)$, $\chi_{196}(193,·)$, $\chi_{196}(195,·)$, $\chi_{196}(55,·)$, $\chi_{196}(81,·)$, $\chi_{196}(75,·)$, $\chi_{196}(83,·)$, $\chi_{196}(85,·)$, $\chi_{196}(87,·)$, $\chi_{196}(93,·)$, $\chi_{196}(165,·)$, $\chi_{196}(59,·)$, $\chi_{196}(131,·)$, $\chi_{196}(103,·)$, $\chi_{196}(109,·)$, $\chi_{196}(111,·)$, $\chi_{196}(113,·)$, $\chi_{196}(115,·)$, $\chi_{196}(187,·)$, $\chi_{196}(121,·)$$\rbrace$
This is not 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

Trivial group, which has order $1$ (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:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 40972406918644810000000000 \) (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

\(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{28})^+\), 7.7.13841287201.1, 14.14.21972068264574400934821888.1, \(\Q(\zeta_{49})^+\)

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 $21^{2}$ $42$ R $42$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ $42$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$ $21^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ $21^{2}$

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