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Properties

Label 42.0.431...816.1
Degree $42$
Signature $[0, 21]$
Discriminant $-4.315\times 10^{89}$
Root discriminant \(136.20\)
Ramified primes $2,3,43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729)
 
gp: K = bnfinit(y^42 + 129*y^40 + 7740*y^38 + 286767*y^36 + 7345647*y^34 + 138020841*y^32 + 1969594704*y^30 + 21806227080*y^28 + 189714175596*y^26 + 1305679914396*y^24 + 7121890442160*y^22 + 30713152531815*y^20 + 104028419865825*y^18 + 273674766108555*y^16 + 550045835331480*y^14 + 825068752997220*y^12 + 893824482413655*y^10 + 667335020806575*y^8 + 320320809987156*y^6 + 88509697496451*y^4 + 11544743151711*y^2 + 449795187729, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729)
 

\( x^{42} + 129 x^{40} + 7740 x^{38} + 286767 x^{36} + 7345647 x^{34} + 138020841 x^{32} + \cdots + 449795187729 \) 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:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-431\!\cdots\!816\) \(\medspace = -\,2^{42}\cdot 3^{21}\cdot 43^{41}\) 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:  \(136.20\)
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:  $2\cdot 3^{1/2}43^{41/42}\approx 136.19682156018243$
Ramified primes:   \(2\), \(3\), \(43\) 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:  \(\Q(\sqrt{-129}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(516=2^{2}\cdot 3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{516}(1,·)$, $\chi_{516}(515,·)$, $\chi_{516}(133,·)$, $\chi_{516}(263,·)$, $\chi_{516}(395,·)$, $\chi_{516}(13,·)$, $\chi_{516}(145,·)$, $\chi_{516}(131,·)$, $\chi_{516}(407,·)$, $\chi_{516}(25,·)$, $\chi_{516}(155,·)$, $\chi_{516}(287,·)$, $\chi_{516}(289,·)$, $\chi_{516}(419,·)$, $\chi_{516}(169,·)$, $\chi_{516}(385,·)$, $\chi_{516}(49,·)$, $\chi_{516}(179,·)$, $\chi_{516}(181,·)$, $\chi_{516}(347,·)$, $\chi_{516}(445,·)$, $\chi_{516}(191,·)$, $\chi_{516}(193,·)$, $\chi_{516}(323,·)$, $\chi_{516}(325,·)$, $\chi_{516}(71,·)$, $\chi_{516}(119,·)$, $\chi_{516}(397,·)$, $\chi_{516}(337,·)$, $\chi_{516}(467,·)$, $\chi_{516}(335,·)$, $\chi_{516}(97,·)$, $\chi_{516}(227,·)$, $\chi_{516}(229,·)$, $\chi_{516}(361,·)$, $\chi_{516}(491,·)$, $\chi_{516}(109,·)$, $\chi_{516}(371,·)$, $\chi_{516}(503,·)$, $\chi_{516}(121,·)$, $\chi_{516}(253,·)$, $\chi_{516}(383,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{531441}a^{24}$, $\frac{1}{531441}a^{25}$, $\frac{1}{1594323}a^{26}$, $\frac{1}{1594323}a^{27}$, $\frac{1}{4782969}a^{28}$, $\frac{1}{4782969}a^{29}$, $\frac{1}{14348907}a^{30}$, $\frac{1}{14348907}a^{31}$, $\frac{1}{43046721}a^{32}$, $\frac{1}{43046721}a^{33}$, $\frac{1}{129140163}a^{34}$, $\frac{1}{129140163}a^{35}$, $\frac{1}{387420489}a^{36}$, $\frac{1}{387420489}a^{37}$, $\frac{1}{1162261467}a^{38}$, $\frac{1}{1162261467}a^{39}$, $\frac{1}{3486784401}a^{40}$, $\frac{1}{3486784401}a^{41}$ 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:  $20$
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 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729)
 
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 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729, 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 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729);
 
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 + 129*x^40 + 7740*x^38 + 286767*x^36 + 7345647*x^34 + 138020841*x^32 + 1969594704*x^30 + 21806227080*x^28 + 189714175596*x^26 + 1305679914396*x^24 + 7121890442160*x^22 + 30713152531815*x^20 + 104028419865825*x^18 + 273674766108555*x^16 + 550045835331480*x^14 + 825068752997220*x^12 + 893824482413655*x^10 + 667335020806575*x^8 + 320320809987156*x^6 + 88509697496451*x^4 + 11544743151711*x^2 + 449795187729);
 
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

\(\Q(\sqrt{-129}) \), 3.3.1849.1, 6.0.254030589504.1, 7.7.6321363049.1, 14.0.61568510194571181216996409344.1, \(\Q(\zeta_{43})^+\)

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 R $21^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{14}$ ${\href{/padicField/11.7.0.1}{7} }^{6}$ $21^{2}$ $42$ $21^{2}$ $21^{2}$ $21^{2}$ $42$ ${\href{/padicField/37.6.0.1}{6} }^{7}$ ${\href{/padicField/41.14.0.1}{14} }^{3}$ R ${\href{/padicField/47.7.0.1}{7} }^{6}$ $42$ ${\href{/padicField/59.7.0.1}{7} }^{6}$

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 2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.38$x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$$2$$7$$14$$C_{14}$$[2]^{7}$
\(3\) Copy content Toggle raw display Deg $42$$2$$21$$21$
\(43\) Copy content Toggle raw display Deg $42$$42$$1$$41$