↑

Properties

Label 16.12.835...000.1
Degree $16$
Signature $[12, 2]$
Discriminant $8.353\times 10^{22}$
Root discriminant \(27.08\)
Ramified primes $2,5,151,559001$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^4.C_2\wr C_2^2$ (as 16T1769)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 + 18*y^14 + 14*y^13 - 101*y^12 + 60*y^11 + 179*y^10 - 158*y^9 - 204*y^8 + 136*y^7 + 221*y^6 - 42*y^5 - 168*y^4 - 14*y^3 + 54*y^2 + 12*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1)
 

\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 101 x^{12} + 60 x^{11} + 179 x^{10} - 158 x^{9} - 204 x^{8} + \cdots - 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[12, 2]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(83530755611033600000000\) \(\medspace = 2^{24}\cdot 5^{8}\cdot 151^{2}\cdot 559001\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(27.08\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\), \(151\), \(559001\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{559001}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $16a^{14}-112a^{13}+205a^{12}+226a^{11}-1019a^{10}+354a^{9}+1367a^{8}-530a^{7}-1293a^{6}-70a^{5}+1091a^{4}+278a^{3}-409a^{2}-104a+9$, $a^{2}-a-1$, $17a^{14}-119a^{13}+217a^{12}+245a^{11}-1089a^{10}+363a^{9}+1485a^{8}-561a^{7}-1410a^{6}-82a^{5}+1183a^{4}+316a^{3}-447a^{2}-118a+12$, $a^{14}-7a^{13}+12a^{12}+19a^{11}-70a^{10}+9a^{9}+118a^{8}-31a^{7}-117a^{6}-12a^{5}+92a^{4}+38a^{3}-38a^{2}-14a+3$, $17a^{14}-119a^{13}+217a^{12}+245a^{11}-1089a^{10}+363a^{9}+1485a^{8}-561a^{7}-1410a^{6}-82a^{5}+1183a^{4}+316a^{3}-447a^{2}-118a+11$, $17a^{15}-119a^{14}+216a^{13}+251a^{12}-1096a^{11}+343a^{10}+1527a^{9}-543a^{8}-1471a^{7}-112a^{6}+1230a^{5}+366a^{4}-461a^{3}-148a^{2}+5a$, $17a^{15}-119a^{14}+217a^{13}+245a^{12}-1089a^{11}+363a^{10}+1485a^{9}-561a^{8}-1410a^{7}-82a^{6}+1183a^{5}+316a^{4}-447a^{3}-118a^{2}+12a$, $16a^{14}-112a^{13}+205a^{12}+226a^{11}-1019a^{10}+354a^{9}+1367a^{8}-530a^{7}-1293a^{6}-70a^{5}+1091a^{4}+278a^{3}-409a^{2}-103a+9$, $5a^{15}-10a^{14}-111a^{13}+391a^{12}+36a^{11}-1485a^{10}+979a^{9}+1982a^{8}-1238a^{7}-2054a^{6}+235a^{5}+1803a^{4}+310a^{3}-681a^{2}-163a+16$, $4a^{15}-23a^{14}+16a^{13}+122a^{12}-185a^{11}-239a^{10}+469a^{9}+306a^{8}-532a^{7}-416a^{6}+285a^{5}+418a^{4}-39a^{3}-170a^{2}-16a+5$, $a^{11}-5a^{10}+3a^{9}+18a^{8}-21a^{7}-21a^{6}+19a^{5}+28a^{4}-22a^{2}-8a+1$, $a^{15}-8a^{14}+19a^{13}+7a^{12}-89a^{11}+79a^{10}+109a^{9}-149a^{8}-86a^{7}+105a^{6}+104a^{5}-54a^{4}-76a^{3}+24a^{2}+16a-1$, $4a^{15}-24a^{14}+24a^{13}+101a^{12}-180a^{11}-165a^{10}+354a^{9}+284a^{8}-373a^{7}-445a^{6}+176a^{5}+390a^{4}+60a^{3}-141a^{2}-64a+5$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 723941.623378 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 723941.623378 \cdot 1}{2\cdot\sqrt{83530755611033600000000}}\cr\approx \mathstrut & 0.202520985159 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 101*x^12 + 60*x^11 + 179*x^10 - 158*x^9 - 204*x^8 + 136*x^7 + 221*x^6 - 42*x^5 - 168*x^4 - 14*x^3 + 54*x^2 + 12*x - 1); 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

$C_4^4.C_2\wr C_2^2$ (as 16T1769):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 16384
The 220 conjugacy class representatives for $C_4^4.C_2\wr C_2^2$
Character table for $C_4^4.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.6.386560000.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.8.2.24a1.41$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 7 x^{8} + 2 x^{7} + 3 x^{6} + 2 x^{5} + 7 x^{4} + 6 x^{3} + 6 x^{2} + 7$$2$$8$$24$$C_8\times C_2$$$[3]^{8}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(151\) Copy content Toggle raw display 151.2.1.0a1.1$x^{2} + 149 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
151.2.1.0a1.1$x^{2} + 149 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
151.2.1.0a1.1$x^{2} + 149 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
151.2.1.0a1.1$x^{2} + 149 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
151.2.2.2a1.2$x^{4} + 298 x^{3} + 22213 x^{2} + 1788 x + 187$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
151.4.1.0a1.1$x^{4} + 13 x^{2} + 89 x + 6$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(559001\) Copy content Toggle raw display $\Q_{559001}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{559001}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{559001}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{559001}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)