Properties

Label 42.0.109...256.1
Degree $42$
Signature $[0, 21]$
Discriminant $-1.090\times 10^{84}$
Root discriminant $100.21$
Ramified primes $2, 7$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 84*x^40 + 3276*x^38 + 78736*x^36 + 1305360*x^34 + 15833664*x^32 + 145435136*x^30 + 1032886400*x^28 + 5741639680*x^26 + 25131904000*x^24 + 86707088384*x^22 + 234966480896*x^20 + 496154537984*x^18 + 805934137344*x^16 + 988445753344*x^14 + 891877195776*x^12 + 571258175488*x^10 + 247113187328*x^8 + 67166797824*x^6 + 10250354688*x^4 + 719323136*x^2 + 14680064)
 
gp: K = bnfinit(x^42 + 84*x^40 + 3276*x^38 + 78736*x^36 + 1305360*x^34 + 15833664*x^32 + 145435136*x^30 + 1032886400*x^28 + 5741639680*x^26 + 25131904000*x^24 + 86707088384*x^22 + 234966480896*x^20 + 496154537984*x^18 + 805934137344*x^16 + 988445753344*x^14 + 891877195776*x^12 + 571258175488*x^10 + 247113187328*x^8 + 67166797824*x^6 + 10250354688*x^4 + 719323136*x^2 + 14680064, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14680064, 0, 719323136, 0, 10250354688, 0, 67166797824, 0, 247113187328, 0, 571258175488, 0, 891877195776, 0, 988445753344, 0, 805934137344, 0, 496154537984, 0, 234966480896, 0, 86707088384, 0, 25131904000, 0, 5741639680, 0, 1032886400, 0, 145435136, 0, 15833664, 0, 1305360, 0, 78736, 0, 3276, 0, 84, 0, 1]);
 

\( x^{42} + 84 x^{40} + 3276 x^{38} + 78736 x^{36} + 1305360 x^{34} + 15833664 x^{32} + 145435136 x^{30} + 1032886400 x^{28} + 5741639680 x^{26} + 25131904000 x^{24} + 86707088384 x^{22} + 234966480896 x^{20} + 496154537984 x^{18} + 805934137344 x^{16} + 988445753344 x^{14} + 891877195776 x^{12} + 571258175488 x^{10} + 247113187328 x^{8} + 67166797824 x^{6} + 10250354688 x^{4} + 719323136 x^{2} + 14680064 \)

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:  \(-10\!\cdots\!256\)\(\medspace = -\,2^{63}\cdot 7^{77}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $100.21$
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:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(5,·)$, $\chi_{392}(65,·)$, $\chi_{392}(9,·)$, $\chi_{392}(13,·)$, $\chi_{392}(45,·)$, $\chi_{392}(25,·)$, $\chi_{392}(281,·)$, $\chi_{392}(285,·)$, $\chi_{392}(325,·)$, $\chi_{392}(289,·)$, $\chi_{392}(293,·)$, $\chi_{392}(305,·)$, $\chi_{392}(169,·)$, $\chi_{392}(173,·)$, $\chi_{392}(157,·)$, $\chi_{392}(177,·)$, $\chi_{392}(181,·)$, $\chi_{392}(137,·)$, $\chi_{392}(57,·)$, $\chi_{392}(61,·)$, $\chi_{392}(193,·)$, $\chi_{392}(69,·)$, $\chi_{392}(269,·)$, $\chi_{392}(81,·)$, $\chi_{392}(341,·)$, $\chi_{392}(121,·)$, $\chi_{392}(345,·)$, $\chi_{392}(349,·)$, $\chi_{392}(101,·)$, $\chi_{392}(225,·)$, $\chi_{392}(229,·)$, $\chi_{392}(337,·)$, $\chi_{392}(361,·)$, $\chi_{392}(237,·)$, $\chi_{392}(381,·)$, $\chi_{392}(113,·)$, $\chi_{392}(117,·)$, $\chi_{392}(233,·)$, $\chi_{392}(249,·)$, $\chi_{392}(125,·)$$\chi_{392}(213,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$, $\frac{1}{262144} a^{36}$, $\frac{1}{262144} a^{37}$, $\frac{1}{524288} a^{38}$, $\frac{1}{524288} a^{39}$, $\frac{1}{1048576} a^{40}$, $\frac{1}{1048576} a^{41}$

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

Class group and class number

not computed

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:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{-14}) \), \(\Q(\zeta_{7})^+\), 6.0.8605184.1, 7.7.13841287201.1, 14.0.2812424737865523319657201664.1, \(\Q(\zeta_{49})^+\)

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 $21^{2}$ $21^{2}$ R $42$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $42$ $21^{2}$

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
2Data not computed
7Data not computed