Properties

Label 42.42.1236219045...0021.1
Degree $42$
Signature $[42, 0]$
Discriminant $3^{21}\cdot 7^{77}$
Root discriminant $61.36$
Ramified primes $3, 7$
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 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*x + 1)
 
gp: K = bnfinit(x^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 56, 392, -2170, -15974, 29211, 235249, -198948, -1816836, 798357, 8580418, -2063243, -27041546, 3643150, 60174899, -4577216, -98285670, 4206314, 121090515, -2877896, -114717330, 1480051, 84672315, -573300, -49085400, 166257, 22428252, -35525, -8069424, 5425, 2272424, -560, -494802, 35, 81585, -1, -9842, 0, 819, 0, -42, 0, 1]);
 

Normalized defining polynomial

\( x^{42} - 42 x^{40} + 819 x^{38} - 9842 x^{36} - x^{35} + 81585 x^{34} + 35 x^{33} - 494802 x^{32} - 560 x^{31} + 2272424 x^{30} + 5425 x^{29} - 8069424 x^{28} - 35525 x^{27} + 22428252 x^{26} + 166257 x^{25} - 49085400 x^{24} - 573300 x^{23} + 84672315 x^{22} + 1480051 x^{21} - 114717330 x^{20} - 2877896 x^{19} + 121090515 x^{18} + 4206314 x^{17} - 98285670 x^{16} - 4577216 x^{15} + 60174899 x^{14} + 3643150 x^{13} - 27041546 x^{12} - 2063243 x^{11} + 8580418 x^{10} + 798357 x^{9} - 1816836 x^{8} - 198948 x^{7} + 235249 x^{6} + 29211 x^{5} - 15974 x^{4} - 2170 x^{3} + 392 x^{2} + 56 x + 1 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1236219045653317330764639083558080790009887216550178193020957667462329030021=3^{21}\cdot 7^{77}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $61.36$
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:  $3, 7$
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:  \(147=3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{147}(1,·)$, $\chi_{147}(130,·)$, $\chi_{147}(131,·)$, $\chi_{147}(4,·)$, $\chi_{147}(5,·)$, $\chi_{147}(142,·)$, $\chi_{147}(143,·)$, $\chi_{147}(16,·)$, $\chi_{147}(17,·)$, $\chi_{147}(146,·)$, $\chi_{147}(20,·)$, $\chi_{147}(22,·)$, $\chi_{147}(25,·)$, $\chi_{147}(26,·)$, $\chi_{147}(37,·)$, $\chi_{147}(38,·)$, $\chi_{147}(41,·)$, $\chi_{147}(43,·)$, $\chi_{147}(46,·)$, $\chi_{147}(47,·)$, $\chi_{147}(58,·)$, $\chi_{147}(59,·)$, $\chi_{147}(62,·)$, $\chi_{147}(64,·)$, $\chi_{147}(67,·)$, $\chi_{147}(68,·)$, $\chi_{147}(79,·)$, $\chi_{147}(80,·)$, $\chi_{147}(83,·)$, $\chi_{147}(85,·)$, $\chi_{147}(88,·)$, $\chi_{147}(89,·)$, $\chi_{147}(100,·)$, $\chi_{147}(101,·)$, $\chi_{147}(104,·)$, $\chi_{147}(106,·)$, $\chi_{147}(109,·)$, $\chi_{147}(110,·)$, $\chi_{147}(121,·)$, $\chi_{147}(122,·)$, $\chi_{147}(125,·)$, $\chi_{147}(127,·)$$\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:  \( 1966641158581684500000000 \) (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

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 7.7.13841287201.1, 14.14.2932917071205091238064909.1, \(\Q(\zeta_{49})^+\)

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 $42$ R $21^{2}$ R $42$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$ $21^{2}$ $42$ $21^{2}$

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
3Data not computed
7Data not computed