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Properties

Label 42.42.865...744.1
Degree $42$
Signature $[42, 0]$
Discriminant $8.652\times 10^{85}$
Root discriminant \(111.20\)
Ramified primes $2,43$
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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536)
 
gp: K = bnfinit(y^42 - 86*y^40 + 3440*y^38 - 84968*y^36 + 1450992*y^34 - 18175584*y^32 + 172913664*y^30 - 1276267520*y^28 + 7402351616*y^26 - 33963730944*y^24 + 123504476160*y^22 - 355075368960*y^20 + 801783091200*y^18 - 1406204190720*y^16 + 1884175073280*y^14 - 1884175073280*y^12 + 1360793108480*y^10 - 677317836800*y^8 + 216741707776*y^6 - 39926104064*y^4 + 3471835136*y^2 - 90177536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536)
 

\( x^{42} - 86 x^{40} + 3440 x^{38} - 84968 x^{36} + 1450992 x^{34} - 18175584 x^{32} + 172913664 x^{30} + \cdots - 90177536 \) 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:   \(865\!\cdots\!744\) \(\medspace = 2^{63}\cdot 43^{41}\) 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:  \(111.20\)
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:  $2^{3/2}43^{41/42}\approx 111.20423913711257$
Ramified primes:   \(2\), \(43\) 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:  \(\Q(\sqrt{86}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(344=2^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{344}(1,·)$, $\chi_{344}(3,·)$, $\chi_{344}(17,·)$, $\chi_{344}(51,·)$, $\chi_{344}(9,·)$, $\chi_{344}(337,·)$, $\chi_{344}(115,·)$, $\chi_{344}(145,·)$, $\chi_{344}(19,·)$, $\chi_{344}(25,·)$, $\chi_{344}(153,·)$, $\chi_{344}(27,·)$, $\chi_{344}(289,·)$, $\chi_{344}(291,·)$, $\chi_{344}(163,·)$, $\chi_{344}(49,·)$, $\chi_{344}(41,·)$, $\chi_{344}(171,·)$, $\chi_{344}(305,·)$, $\chi_{344}(179,·)$, $\chi_{344}(155,·)$, $\chi_{344}(57,·)$, $\chi_{344}(193,·)$, $\chi_{344}(323,·)$, $\chi_{344}(147,·)$, $\chi_{344}(97,·)$, $\chi_{344}(75,·)$, $\chi_{344}(81,·)$, $\chi_{344}(211,·)$, $\chi_{344}(185,·)$, $\chi_{344}(331,·)$, $\chi_{344}(91,·)$, $\chi_{344}(225,·)$, $\chi_{344}(227,·)$, $\chi_{344}(131,·)$, $\chi_{344}(273,·)$, $\chi_{344}(235,·)$, $\chi_{344}(243,·)$, $\chi_{344}(169,·)$, $\chi_{344}(121,·)$, $\chi_{344}(123,·)$, $\chi_{344}(281,·)$$\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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536)
 
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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536, 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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536);
 
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 - 86*x^40 + 3440*x^38 - 84968*x^36 + 1450992*x^34 - 18175584*x^32 + 172913664*x^30 - 1276267520*x^28 + 7402351616*x^26 - 33963730944*x^24 + 123504476160*x^22 - 355075368960*x^20 + 801783091200*x^18 - 1406204190720*x^16 + 1884175073280*x^14 - 1884175073280*x^12 + 1360793108480*x^10 - 677317836800*x^8 + 216741707776*x^6 - 39926104064*x^4 + 3471835136*x^2 - 90177536);
 
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 $C_{42}$

Intermediate fields

\(\Q(\sqrt{86}) \), 3.3.1849.1, 6.6.75268322816.1, 7.7.6321363049.1, 14.14.3603461044766854684853927936.1, \(\Q(\zeta_{43})^+\)

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 $42$ $21^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{14}$ ${\href{/padicField/11.7.0.1}{7} }^{6}$ $42$ $21^{2}$ $42$ $42$ $21^{2}$ $42$ ${\href{/padicField/37.3.0.1}{3} }^{14}$ ${\href{/padicField/41.7.0.1}{7} }^{6}$ R ${\href{/padicField/47.14.0.1}{14} }^{3}$ $42$ ${\href{/padicField/59.7.0.1}{7} }^{6}$

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 2.14.21.40$x^{14} + 28 x^{13} + 530 x^{12} + 6592 x^{11} + 70004 x^{10} + 568560 x^{9} + 3483880 x^{8} + 16019456 x^{7} + 59065136 x^{6} + 169683776 x^{5} + 367736672 x^{4} + 550228992 x^{3} + 585563072 x^{2} + 456203520 x + 295014272$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.40$x^{14} + 28 x^{13} + 530 x^{12} + 6592 x^{11} + 70004 x^{10} + 568560 x^{9} + 3483880 x^{8} + 16019456 x^{7} + 59065136 x^{6} + 169683776 x^{5} + 367736672 x^{4} + 550228992 x^{3} + 585563072 x^{2} + 456203520 x + 295014272$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.40$x^{14} + 28 x^{13} + 530 x^{12} + 6592 x^{11} + 70004 x^{10} + 568560 x^{9} + 3483880 x^{8} + 16019456 x^{7} + 59065136 x^{6} + 169683776 x^{5} + 367736672 x^{4} + 550228992 x^{3} + 585563072 x^{2} + 456203520 x + 295014272$$2$$7$$21$$C_{14}$$[3]^{7}$
\(43\) Copy content Toggle raw display Deg $42$$42$$1$$41$