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Properties

Label 42.0.201...808.1
Degree $42$
Signature $[0, 21]$
Discriminant $-2.012\times 10^{84}$
Root discriminant \(101.68\)
Ramified primes $2,43$
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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152)
 
gp: K = bnfinit(y^42 + 82*y^40 + 3120*y^38 + 73112*y^36 + 1181040*y^34 + 13948704*y^32 + 124658688*y^30 + 860738560*y^28 + 4647988224*y^26 + 19746355200*y^24 + 66060533760*y^22 + 173408901120*y^20 + 354276249600*y^18 + 555941191680*y^16 + 657270374400*y^14 + 569634324480*y^12 + 348109864960*y^10 + 141764198400*y^8 + 35283533824*y^6 + 4642570240*y^4 + 242221056*y^2 + 2097152, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152)
 

\( x^{42} + 82 x^{40} + 3120 x^{38} + 73112 x^{36} + 1181040 x^{34} + 13948704 x^{32} + 124658688 x^{30} + \cdots + 2097152 \) 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:   \(-201\!\cdots\!808\) \(\medspace = -\,2^{63}\cdot 43^{40}\) 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:  \(101.68\)
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:  $2^{3/2}43^{20/21}\approx 101.67852486149111$
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:  \(\Q(\sqrt{-2}) \)
$\card{ \Gal(K/\Q) }$:  $42$
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$$\chi_{344}(1,·)$, $\chi_{344}(259,·)$, $\chi_{344}(17,·)$, $\chi_{344}(9,·)$, $\chi_{344}(11,·)$, $\chi_{344}(337,·)$, $\chi_{344}(145,·)$, $\chi_{344}(275,·)$, $\chi_{344}(25,·)$, $\chi_{344}(281,·)$, $\chi_{344}(283,·)$, $\chi_{344}(289,·)$, $\chi_{344}(35,·)$, $\chi_{344}(49,·)$, $\chi_{344}(41,·)$, $\chi_{344}(299,·)$, $\chi_{344}(305,·)$, $\chi_{344}(307,·)$, $\chi_{344}(57,·)$, $\chi_{344}(315,·)$, $\chi_{344}(193,·)$, $\chi_{344}(67,·)$, $\chi_{344}(97,·)$, $\chi_{344}(203,·)$, $\chi_{344}(81,·)$, $\chi_{344}(339,·)$, $\chi_{344}(139,·)$, $\chi_{344}(185,·)$, $\chi_{344}(267,·)$, $\chi_{344}(219,·)$, $\chi_{344}(225,·)$, $\chi_{344}(99,·)$, $\chi_{344}(195,·)$, $\chi_{344}(273,·)$, $\chi_{344}(107,·)$, $\chi_{344}(59,·)$, $\chi_{344}(83,·)$, $\chi_{344}(187,·)$, $\chi_{344}(169,·)$, $\chi_{344}(153,·)$, $\chi_{344}(121,·)$, $\chi_{344}(251,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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:  $20$
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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152)
 
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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152, 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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152);
 
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 + 82*x^40 + 3120*x^38 + 73112*x^36 + 1181040*x^34 + 13948704*x^32 + 124658688*x^30 + 860738560*x^28 + 4647988224*x^26 + 19746355200*x^24 + 66060533760*x^22 + 173408901120*x^20 + 354276249600*x^18 + 555941191680*x^16 + 657270374400*x^14 + 569634324480*x^12 + 348109864960*x^10 + 141764198400*x^8 + 35283533824*x^6 + 4642570240*x^4 + 242221056*x^2 + 2097152);
 
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

\(\Q(\sqrt{-2}) \), 3.3.1849.1, 6.0.1750426112.2, 7.7.6321363049.1, 14.0.83801419645740806624509952.1, \(\Q(\zeta_{43})^+\)

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 $21^{2}$ $42$ ${\href{/padicField/7.6.0.1}{6} }^{7}$ ${\href{/padicField/11.7.0.1}{7} }^{6}$ $42$ $21^{2}$ $21^{2}$ $42$ $42$ $42$ ${\href{/padicField/37.6.0.1}{6} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{6}$ 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 2.14.21.6$x^{14} + 158 x^{12} + 1568 x^{11} + 1332 x^{10} - 32384 x^{9} + 95128 x^{8} + 683008 x^{7} + 3694128 x^{6} + 18752000 x^{5} + 49924512 x^{4} + 51764736 x^{3} - 10775616 x^{2} + 27525120 x + 167639168$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.6$x^{14} + 158 x^{12} + 1568 x^{11} + 1332 x^{10} - 32384 x^{9} + 95128 x^{8} + 683008 x^{7} + 3694128 x^{6} + 18752000 x^{5} + 49924512 x^{4} + 51764736 x^{3} - 10775616 x^{2} + 27525120 x + 167639168$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.6$x^{14} + 158 x^{12} + 1568 x^{11} + 1332 x^{10} - 32384 x^{9} + 95128 x^{8} + 683008 x^{7} + 3694128 x^{6} + 18752000 x^{5} + 49924512 x^{4} + 51764736 x^{3} - 10775616 x^{2} + 27525120 x + 167639168$$2$$7$$21$$C_{14}$$[3]^{7}$
\(43\) Copy content Toggle raw display 43.21.20.1$x^{21} + 43$$21$$1$$20$$C_{21}$$[\ ]_{21}$
43.21.20.1$x^{21} + 43$$21$$1$$20$$C_{21}$$[\ ]_{21}$