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Properties

Label 42.42.109...256.1
Degree $42$
Signature $[42, 0]$
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(y^42 - 84*y^40 + 3276*y^38 - 78736*y^36 + 1305360*y^34 - 15833664*y^32 + 145435136*y^30 - 1032886400*y^28 + 5741639680*y^26 - 25131904000*y^24 + 86707088384*y^22 - 234966480896*y^20 + 496154537984*y^18 - 805934137344*y^16 + 988445753344*y^14 - 891877195776*y^12 + 571258175488*y^10 - 247113187328*y^8 + 67166797824*y^6 - 10250354688*y^4 + 719323136*y^2 - 14680064, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{42} - 84 x^{40} + 3276 x^{38} - 78736 x^{36} + 1305360 x^{34} - 15833664 x^{32} + 145435136 x^{30} + \cdots - 14680064 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(109\!\cdots\!256\) \(\medspace = 2^{63}\cdot 7^{77}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(100.21\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}7^{11/6}\approx 100.20546326574731$
Ramified primes:   \(2\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{14}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(3,·)$, $\chi_{392}(65,·)$, $\chi_{392}(9,·)$, $\chi_{392}(139,·)$, $\chi_{392}(131,·)$, $\chi_{392}(25,·)$, $\chi_{392}(281,·)$, $\chi_{392}(283,·)$, $\chi_{392}(289,·)$, $\chi_{392}(27,·)$, $\chi_{392}(305,·)$, $\chi_{392}(169,·)$, $\chi_{392}(171,·)$, $\chi_{392}(307,·)$, $\chi_{392}(177,·)$, $\chi_{392}(243,·)$, $\chi_{392}(137,·)$, $\chi_{392}(57,·)$, $\chi_{392}(59,·)$, $\chi_{392}(19,·)$, $\chi_{392}(193,·)$, $\chi_{392}(195,·)$, $\chi_{392}(339,·)$, $\chi_{392}(75,·)$, $\chi_{392}(337,·)$, $\chi_{392}(83,·)$, $\chi_{392}(121,·)$, $\chi_{392}(355,·)$, $\chi_{392}(345,·)$, $\chi_{392}(225,·)$, $\chi_{392}(227,·)$, $\chi_{392}(81,·)$, $\chi_{392}(361,·)$, $\chi_{392}(363,·)$, $\chi_{392}(113,·)$, $\chi_{392}(115,·)$, $\chi_{392}(187,·)$, $\chi_{392}(233,·)$, $\chi_{392}(249,·)$, $\chi_{392}(251,·)$, $\chi_{392}(299,·)$$\rbrace$
This is not a CM field.

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{14}) \), \(\Q(\zeta_{7})^+\), 6.6.8605184.1, 7.7.13841287201.1, 14.14.2812424737865523319657201664.1, \(\Q(\zeta_{49})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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 $21^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{6}$ $42$ ${\href{/padicField/19.6.0.1}{6} }^{7}$ $42$ ${\href{/padicField/29.14.0.1}{14} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{14}$ $42$ ${\href{/padicField/41.14.0.1}{14} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{6}$ $21^{2}$ $42$ $42$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $42$$2$$21$$63$
\(7\) Copy content Toggle raw display Deg $42$$42$$1$$77$