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Properties

Label 16.4.283...625.2
Degree $16$
Signature $[4, 6]$
Discriminant $2.838\times 10^{18}$
Root discriminant \(14.23\)
Ramified primes $3,5,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 3*y^14 - 9*y^13 + 7*y^12 - 15*y^11 + 27*y^10 - 3*y^9 - 15*y^8 - 3*y^7 + 27*y^6 - 15*y^5 + 7*y^4 - 9*y^3 + 3*y^2 - 3*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*x + 1)
 

\( x^{16} - 3 x^{15} + 3 x^{14} - 9 x^{13} + 7 x^{12} - 15 x^{11} + 27 x^{10} - 3 x^{9} - 15 x^{8} + \cdots + 1 \) 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:  $[4, 6]$
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:   \(2837698174072265625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 11^{6}\) 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:  \(14.23\)
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:  $3^{1/2}5^{3/4}11^{1/2}\approx 19.208102881010017$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{12}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{5841}a^{14}-\frac{71}{5841}a^{13}+\frac{287}{5841}a^{12}-\frac{211}{1947}a^{11}+\frac{193}{1947}a^{10}+\frac{62}{1947}a^{9}+\frac{13}{177}a^{8}+\frac{199}{649}a^{7}+\frac{24}{59}a^{6}-\frac{587}{1947}a^{5}+\frac{842}{1947}a^{4}-\frac{860}{1947}a^{3}-\frac{362}{5841}a^{2}-\frac{2018}{5841}a-\frac{1297}{5841}$, $\frac{1}{5841}a^{15}-\frac{211}{5841}a^{13}+\frac{274}{5841}a^{12}+\frac{139}{1947}a^{11}+\frac{136}{1947}a^{10}+\frac{2}{1947}a^{9}-\frac{283}{1947}a^{8}-\frac{953}{1947}a^{7}-\frac{169}{1947}a^{6}+\frac{701}{1947}a^{5}+\frac{512}{1947}a^{4}+\frac{1423}{5841}a^{3}-\frac{73}{177}a^{2}-\frac{1795}{5841}a-\frac{2525}{5841}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

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:  $C_{2}$, which has order $2$ (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:   $\frac{2110}{5841}a^{15}-\frac{5200}{5841}a^{14}+\frac{347}{531}a^{13}-\frac{1999}{649}a^{12}+\frac{2279}{1947}a^{11}-\frac{10528}{1947}a^{10}+\frac{4919}{649}a^{9}+\frac{3349}{1947}a^{8}-\frac{1668}{649}a^{7}-\frac{2641}{649}a^{6}+\frac{15119}{1947}a^{5}-\frac{511}{1947}a^{4}+\frac{20896}{5841}a^{3}-\frac{19192}{5841}a^{2}+\frac{62}{531}a-\frac{1337}{1947}$, $\frac{1204}{5841}a^{15}-\frac{1108}{1947}a^{14}+\frac{248}{531}a^{13}-\frac{3163}{1947}a^{12}+\frac{552}{649}a^{11}-\frac{1772}{649}a^{10}+\frac{8543}{1947}a^{9}+\frac{1027}{1947}a^{8}-\frac{2518}{649}a^{7}-\frac{1252}{1947}a^{6}+\frac{14227}{1947}a^{5}-\frac{5614}{1947}a^{4}-\frac{2630}{5841}a^{3}+\frac{1510}{1947}a^{2}+\frac{332}{531}a-\frac{534}{649}$, $\frac{1498}{5841}a^{15}-\frac{370}{649}a^{14}+\frac{2126}{5841}a^{13}-\frac{13078}{5841}a^{12}+\frac{318}{649}a^{11}-\frac{8026}{1947}a^{10}+\frac{3136}{649}a^{9}+\frac{3284}{1947}a^{8}-\frac{566}{1947}a^{7}-\frac{7664}{1947}a^{6}+\frac{6428}{1947}a^{5}+\frac{1625}{1947}a^{4}+\frac{18448}{5841}a^{3}-\frac{7996}{1947}a^{2}+\frac{4634}{5841}a-\frac{6004}{5841}$, $a$, $\frac{398}{5841}a^{15}-\frac{131}{5841}a^{14}-\frac{691}{5841}a^{13}-\frac{3829}{5841}a^{12}-\frac{469}{649}a^{11}-\frac{2956}{1947}a^{10}-\frac{17}{177}a^{9}+\frac{3625}{1947}a^{8}+\frac{535}{177}a^{7}-\frac{2923}{1947}a^{6}-\frac{4949}{1947}a^{5}+\frac{5210}{1947}a^{4}+\frac{18449}{5841}a^{3}-\frac{2111}{5841}a^{2}-\frac{137}{99}a-\frac{275}{531}$, $\frac{530}{5841}a^{15}-\frac{2228}{5841}a^{14}+\frac{3524}{5841}a^{13}-\frac{6817}{5841}a^{12}+\frac{3161}{1947}a^{11}-\frac{4219}{1947}a^{10}+\frac{8300}{1947}a^{9}-\frac{5857}{1947}a^{8}-\frac{161}{649}a^{7}+\frac{3941}{1947}a^{6}+\frac{6244}{1947}a^{5}-\frac{9374}{1947}a^{4}+\frac{658}{531}a^{3}-\frac{10742}{5841}a^{2}+\frac{3161}{5841}a-\frac{1591}{5841}$, $\frac{1280}{5841}a^{15}-\frac{1390}{5841}a^{14}-\frac{2650}{5841}a^{13}-\frac{6025}{5841}a^{12}-\frac{1286}{649}a^{11}-\frac{1110}{649}a^{10}-\frac{548}{1947}a^{9}+\frac{15302}{1947}a^{8}-\frac{1557}{649}a^{7}-\frac{12710}{1947}a^{6}+\frac{3195}{649}a^{5}+\frac{1442}{177}a^{4}-\frac{16702}{5841}a^{3}-\frac{2512}{5841}a^{2}-\frac{5290}{5841}a+\frac{7067}{5841}$, $\frac{740}{5841}a^{15}-\frac{2173}{5841}a^{14}+\frac{739}{5841}a^{13}-\frac{2935}{5841}a^{12}+\frac{19}{59}a^{11}-\frac{30}{649}a^{10}+\frac{4991}{1947}a^{9}+\frac{5531}{1947}a^{8}-\frac{4006}{649}a^{7}-\frac{972}{649}a^{6}+\frac{16031}{1947}a^{5}-\frac{1172}{649}a^{4}-\frac{32479}{5841}a^{3}+\frac{8618}{5841}a^{2}+\frac{2620}{5841}a-\frac{2851}{5841}$, $\frac{5222}{5841}a^{15}-\frac{14122}{5841}a^{14}+\frac{1311}{649}a^{13}-\frac{43060}{5841}a^{12}+\frac{6949}{1947}a^{11}-\frac{2143}{177}a^{10}+\frac{36341}{1947}a^{9}+\frac{6679}{1947}a^{8}-\frac{29563}{1947}a^{7}-\frac{3530}{1947}a^{6}+\frac{41735}{1947}a^{5}-\frac{6261}{649}a^{4}+\frac{32855}{5841}a^{3}-\frac{14518}{5841}a^{2}-\frac{724}{649}a-\frac{2257}{5841}$ 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:  \( 795.712721841 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 795.712721841 \cdot 1}{2\cdot\sqrt{2837698174072265625}}\cr\approx \mathstrut & 0.232510408075 \end{aligned}\] (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 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*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 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*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 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*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 - 3*x^15 + 3*x^14 - 9*x^13 + 7*x^12 - 15*x^11 + 27*x^10 - 3*x^9 - 15*x^8 - 3*x^7 + 27*x^6 - 15*x^5 + 7*x^4 - 9*x^3 + 3*x^2 - 3*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

$C_4\wr C_2$ (as 16T28):

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 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, \(\Q(\zeta_{15})^+\), 4.2.12375.1, 8.4.153140625.2

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.16471125.2, 8.0.411778125.2
Degree 16 sibling: 16.0.169561224228515625.2
Minimal sibling: 8.0.16471125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(3\) Copy content Toggle raw display 3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
3.2.2.2a1.1$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.4.4.12a1.4$x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$
\(11\) Copy content Toggle raw display 11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.2.2.2a1.1$x^{4} + 14 x^{3} + 53 x^{2} + 39 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
11.4.2.4a1.2$x^{8} + 16 x^{6} + 20 x^{5} + 68 x^{4} + 160 x^{3} + 132 x^{2} + 40 x + 15$$2$$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)