↑

Properties

Label 16.4.703...000.2
Degree $16$
Signature $[4, 6]$
Discriminant $7.037\times 10^{21}$
Root discriminant \(23.20\)
Ramified primes $2,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5.C_2\wr C_2^2$ (as 16T1455)

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 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4)
 
Copy content gp:K = bnfinit(y^16 - 4*y^14 - 2*y^12 - 38*y^8 + 8*y^6 + 64*y^4 - 64*y^2 + 4, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4)
 

\( x^{16} - 4x^{14} - 2x^{12} - 38x^{8} + 8x^{6} + 64x^{4} - 64x^{2} + 4 \) 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:  $[4, 6]$
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:   \(7036874417766400000000\) \(\medspace = 2^{54}\cdot 5^{8}\) 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:  \(23.20\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(5\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{1844686}a^{14}+\frac{265563}{1844686}a^{12}-\frac{171795}{922343}a^{10}+\frac{305117}{1844686}a^{8}+\frac{125704}{922343}a^{6}-\frac{448370}{922343}a^{4}+\frac{360856}{922343}a^{2}+\frac{7620}{922343}$, $\frac{1}{1844686}a^{15}+\frac{265563}{1844686}a^{13}-\frac{171795}{922343}a^{11}+\frac{305117}{1844686}a^{9}+\frac{125704}{922343}a^{7}-\frac{448370}{922343}a^{5}+\frac{360856}{922343}a^{3}+\frac{7620}{922343}a$ 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:  Not computed
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:  $9$
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:   $\frac{23124}{922343}a^{14}-\frac{80882}{922343}a^{12}-\frac{112558}{922343}a^{10}+\frac{125459}{1844686}a^{8}-\frac{891680}{922343}a^{6}-\frac{100434}{922343}a^{4}+\frac{1838732}{922343}a^{2}-\frac{847609}{922343}$, $\frac{40460}{922343}a^{14}-\frac{311597}{1844686}a^{12}-\frac{97704}{922343}a^{10}-\frac{132127}{1844686}a^{8}-\frac{1475610}{922343}a^{6}+\frac{106191}{922343}a^{4}+\frac{2777512}{922343}a^{2}-\frac{2281753}{922343}$, $\frac{10699}{922343}a^{14}-\frac{38149}{1844686}a^{12}-\frac{142767}{1844686}a^{10}-\frac{372531}{1844686}a^{8}-\frac{660339}{922343}a^{6}-\frac{931717}{922343}a^{4}-\frac{258908}{922343}a^{2}-\frac{201951}{922343}$, $\frac{23124}{922343}a^{14}-\frac{80882}{922343}a^{12}-\frac{112558}{922343}a^{10}+\frac{125459}{1844686}a^{8}-\frac{891680}{922343}a^{6}-\frac{100434}{922343}a^{4}+\frac{1838732}{922343}a^{2}-a-\frac{847609}{922343}$, $\frac{172}{922343}a^{14}+\frac{41715}{1844686}a^{12}-\frac{67528}{922343}a^{10}-\frac{93427}{922343}a^{8}-\frac{107945}{922343}a^{6}-\frac{1130342}{922343}a^{4}-\frac{381841}{922343}a^{2}-\frac{145749}{922343}$, $\frac{45265}{922343}a^{15}+\frac{24555}{1844686}a^{14}-\frac{187124}{922343}a^{13}-\frac{65545}{1844686}a^{12}-\frac{53684}{922343}a^{11}-\frac{181029}{1844686}a^{10}-\frac{43077}{922343}a^{9}-\frac{22127}{922343}a^{8}-\frac{1729500}{922343}a^{7}-\frac{420301}{922343}a^{6}+\frac{456987}{922343}a^{5}-\frac{639302}{922343}a^{4}+\frac{2593992}{922343}a^{3}+\frac{792222}{922343}a^{2}-\frac{2840993}{922343}a-\frac{126529}{922343}$, $\frac{73194}{922343}a^{15}-\frac{105475}{1844686}a^{14}-\frac{598663}{1844686}a^{13}+\frac{188571}{922343}a^{12}-\frac{122222}{922343}a^{11}+\frac{376437}{1844686}a^{10}+\frac{42639}{922343}a^{9}+\frac{176381}{1844686}a^{8}-\frac{2875070}{922343}a^{7}+\frac{1895911}{922343}a^{6}+\frac{707349}{922343}a^{5}+\frac{533111}{922343}a^{4}+\frac{5171547}{922343}a^{3}-\frac{2647391}{922343}a^{2}-\frac{5170185}{922343}a+\frac{1485939}{922343}$, $\frac{63584}{922343}a^{15}-\frac{12083}{1844686}a^{14}-\frac{473361}{1844686}a^{13}+\frac{16784}{922343}a^{12}-\frac{210262}{922343}a^{11}+\frac{132127}{1844686}a^{10}-\frac{3334}{922343}a^{9}-\frac{61870}{922343}a^{8}-\frac{2367290}{922343}a^{7}+\frac{217489}{922343}a^{6}+\frac{5757}{922343}a^{5}-\frac{188072}{922343}a^{4}+\frac{4616244}{922343}a^{3}-\frac{1230030}{922343}a^{2}-\frac{2207019}{922343}a-\frac{760503}{922343}$, $\frac{73022}{922343}a^{15}-\frac{23296}{922343}a^{14}-\frac{320189}{922343}a^{13}+\frac{120049}{1844686}a^{12}-\frac{54694}{922343}a^{11}+\frac{180086}{922343}a^{10}+\frac{136066}{922343}a^{9}+\frac{61395}{1844686}a^{8}-\frac{2767125}{922343}a^{7}+\frac{999625}{922343}a^{6}+\frac{1837691}{922343}a^{5}+\frac{1230776}{922343}a^{4}+\frac{5553388}{922343}a^{3}-\frac{1456891}{922343}a^{2}-\frac{6869122}{922343}a-\frac{851328}{922343}$ 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:  \( 49937.7677807 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 49937.7677807 \cdot 1}{2\cdot\sqrt{7036874417766400000000}}\cr\approx \mathstrut & 0.293027528660 \end{aligned}\] (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 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4) 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 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4, 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 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4); 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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 - 2*x^12 - 38*x^8 + 8*x^6 + 64*x^4 - 64*x^2 + 4); 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

$C_2^5.C_2\wr C_2^2$ (as 16T1455):

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); G, transitive_group_identification(G)
 
A solvable group of order 2048
The 44 conjugacy class representatives for $C_2^5.C_2\wr C_2^2$
Character table for $C_2^5.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 8.2.163840000.2

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(b)]
 
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: 16.6.7036874417766400000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.8.54a2.4969$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 268 x^{12} + 528 x^{11} + 892 x^{10} + 1328 x^{9} + 1753 x^{8} + 2040 x^{7} + 2060 x^{6} + 1768 x^{5} + 1268 x^{4} + 736 x^{3} + 340 x^{2} + 112 x + 31$$8$$2$$54$16T1455$$[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4, 4, \frac{9}{2}, \frac{9}{2}]^{2}$$
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
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$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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