Properties

Label 42.0.118...207.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.182\times 10^{65}$
Root discriminant $35.43$
Ramified prime $7$
Class number $43$ (GRH)
Class group $[43]$ (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1)
 
gp: K = bnfinit(x^42 - x^35 + x^28 - x^21 + x^14 - x^7 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1]);
 

\( x^{42} - x^{35} + x^{28} - x^{21} + x^{14} - x^{7} + 1 \)

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:  \(-11\!\cdots\!207\)\(\medspace = -\,7^{77}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.43$
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:  $7$
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:  \(49=7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{49}(1,·)$, $\chi_{49}(2,·)$, $\chi_{49}(3,·)$, $\chi_{49}(4,·)$, $\chi_{49}(5,·)$, $\chi_{49}(6,·)$, $\chi_{49}(8,·)$, $\chi_{49}(9,·)$, $\chi_{49}(10,·)$, $\chi_{49}(11,·)$, $\chi_{49}(12,·)$, $\chi_{49}(13,·)$, $\chi_{49}(15,·)$, $\chi_{49}(16,·)$, $\chi_{49}(17,·)$, $\chi_{49}(18,·)$, $\chi_{49}(19,·)$, $\chi_{49}(20,·)$, $\chi_{49}(22,·)$, $\chi_{49}(23,·)$, $\chi_{49}(24,·)$, $\chi_{49}(25,·)$, $\chi_{49}(26,·)$, $\chi_{49}(27,·)$, $\chi_{49}(29,·)$, $\chi_{49}(30,·)$, $\chi_{49}(31,·)$, $\chi_{49}(32,·)$, $\chi_{49}(33,·)$, $\chi_{49}(34,·)$, $\chi_{49}(36,·)$, $\chi_{49}(37,·)$, $\chi_{49}(38,·)$, $\chi_{49}(39,·)$, $\chi_{49}(40,·)$, $\chi_{49}(41,·)$, $\chi_{49}(43,·)$, $\chi_{49}(44,·)$, $\chi_{49}(45,·)$, $\chi_{49}(46,·)$, $\chi_{49}(47,·)$$\chi_{49}(48,·)$$\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

$C_{43}$, which has order $43$ (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:  \( a \) (order $98$)
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);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{21}\cdot 1776855897760068.5 \cdot 43}{98\sqrt{118181386580595879976868414312001964434038548836769923458287039207}}\approx 0.131038008164406$ (assuming GRH)

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

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, \(\Q(\zeta_{49})^+\)

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 $21^{2}$ $42$ $42$ R $21^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $42$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$ $42$ $21^{2}$ $42$

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
7Data not computed