Properties

Label 16.4.145...000.1
Degree $16$
Signature $(4, 6)$
Discriminant $1.453\times 10^{20}$
Root discriminant \(18.20\)
Ramified primes $2,3,5,61$
Class number $1$
Class group trivial
Galois group $Q_8^2.A_4\wr C_2$ (as 16T1791)

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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 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*x + 1)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 5*y^14 + 5*y^13 + 7*y^12 - 8*y^11 - 6*y^10 - 9*y^9 + 12*y^8 + 28*y^7 - 34*y^6 - 21*y^5 + 27*y^4 + 5*y^3 - 5*y^2 - 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*x + 1)
 

\( x^{16} - x^{15} - 5 x^{14} + 5 x^{13} + 7 x^{12} - 8 x^{11} - 6 x^{10} - 9 x^{9} + 12 x^{8} + 28 x^{7} + \cdots + 1 \) 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:   \(145348100481600000000\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{8}\cdot 61^{4}\) 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:  \(18.20\)
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:  $2^{31/24}3^{1/2}5^{1/2}61^{2/3}\approx 146.92507822159706$
Ramified primes:   \(2\), \(3\), \(5\), \(61\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{6362}a^{15}+\frac{611}{3181}a^{14}-\frac{569}{6362}a^{13}+\frac{757}{6362}a^{12}+\frac{147}{6362}a^{11}-\frac{772}{3181}a^{10}-\frac{1985}{6362}a^{9}-\frac{561}{6362}a^{8}-\frac{1088}{3181}a^{7}+\frac{1277}{6362}a^{6}+\frac{3047}{6362}a^{5}-\frac{836}{3181}a^{4}+\frac{277}{3181}a^{3}+\frac{3175}{6362}a^{2}-\frac{981}{6362}a-\frac{264}{3181}$ 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 $1$
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 $2$
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:   $a$, $\frac{215}{3181}a^{15}-\frac{1293}{3181}a^{14}-\frac{1457}{3181}a^{13}+\frac{6886}{3181}a^{12}+\frac{2976}{3181}a^{11}-\frac{10679}{3181}a^{10}-\frac{521}{3181}a^{9}+\frac{6625}{3181}a^{8}+\frac{18853}{3181}a^{7}+\frac{13713}{3181}a^{6}-\frac{38353}{3181}a^{5}-\frac{27}{3181}a^{4}+\frac{45947}{3181}a^{3}-\frac{1290}{3181}a^{2}-\frac{10512}{3181}a-\frac{2185}{3181}$, $\frac{41}{3181}a^{15}+\frac{1593}{6362}a^{14}-\frac{1062}{3181}a^{13}-\frac{3954}{3181}a^{12}+\frac{8873}{6362}a^{11}+\frac{10175}{6362}a^{10}-\frac{13263}{6362}a^{9}-\frac{3915}{3181}a^{8}-\frac{6510}{3181}a^{7}+\frac{11004}{3181}a^{6}+\frac{43089}{6362}a^{5}-\frac{57579}{6362}a^{4}-\frac{21373}{6362}a^{3}+\frac{47223}{6362}a^{2}+\frac{1132}{3181}a-\frac{2562}{3181}$, $\frac{379}{3181}a^{15}+\frac{605}{6362}a^{14}-\frac{2524}{3181}a^{13}-\frac{1955}{6362}a^{12}+\frac{4817}{3181}a^{11}+\frac{128}{3181}a^{10}-\frac{6379}{6362}a^{9}-\frac{14889}{6362}a^{8}-\frac{7187}{3181}a^{7}+\frac{16376}{3181}a^{6}+\frac{3401}{6362}a^{5}-\frac{36329}{6362}a^{4}+\frac{20}{3181}a^{3}-\frac{14091}{6362}a^{2}+\frac{3937}{6362}a+\frac{3472}{3181}$, $\frac{916}{3181}a^{15}-\frac{3901}{6362}a^{14}-\frac{8583}{6362}a^{13}+\frac{9497}{3181}a^{12}+\frac{11643}{6362}a^{11}-\frac{26147}{6362}a^{10}-\frac{5090}{3181}a^{9}-\frac{4916}{3181}a^{8}+\frac{50257}{6362}a^{7}+\frac{27753}{3181}a^{6}-\frac{108705}{6362}a^{5}-\frac{11034}{3181}a^{4}+\frac{39857}{3181}a^{3}+\frac{866}{3181}a^{2}-\frac{4735}{3181}a-\frac{3453}{6362}$, $\frac{18579}{6362}a^{15}-\frac{11983}{6362}a^{14}-\frac{48209}{3181}a^{13}+\frac{29180}{3181}a^{12}+\frac{148141}{6362}a^{11}-\frac{47574}{3181}a^{10}-\frac{145127}{6362}a^{9}-\frac{110676}{3181}a^{8}+\frac{145751}{6362}a^{7}+\frac{567703}{6362}a^{6}-\frac{214149}{3181}a^{5}-\frac{269606}{3181}a^{4}+\frac{294883}{6362}a^{3}+\frac{203445}{6362}a^{2}-\frac{4206}{3181}a-\frac{40861}{6362}$, $\frac{6743}{3181}a^{15}-\frac{7231}{6362}a^{14}-\frac{35472}{3181}a^{13}+\frac{18032}{3181}a^{12}+\frac{56007}{3181}a^{11}-\frac{31589}{3181}a^{10}-\frac{56465}{3181}a^{9}-\frac{83320}{3181}a^{8}+\frac{88257}{6362}a^{7}+\frac{209790}{3181}a^{6}-\frac{283425}{6362}a^{5}-\frac{431099}{6362}a^{4}+\frac{189935}{6362}a^{3}+\frac{80420}{3181}a^{2}-\frac{1584}{3181}a-\frac{30159}{6362}$, $\frac{3257}{3181}a^{15}-\frac{2558}{3181}a^{14}-\frac{32411}{6362}a^{13}+\frac{12998}{3181}a^{12}+\frac{44611}{6362}a^{11}-\frac{21914}{3181}a^{10}-\frac{20439}{3181}a^{9}-\frac{69367}{6362}a^{8}+\frac{60511}{6362}a^{7}+\frac{90690}{3181}a^{6}-\frac{96071}{3181}a^{5}-\frac{149171}{6362}a^{4}+\frac{131923}{6362}a^{3}+\frac{18630}{3181}a^{2}+\frac{395}{6362}a-\frac{7093}{6362}$, $\frac{32175}{6362}a^{15}-\frac{9198}{3181}a^{14}-\frac{169513}{6362}a^{13}+\frac{44313}{3181}a^{12}+\frac{133391}{3181}a^{11}-\frac{71834}{3181}a^{10}-\frac{260099}{6362}a^{9}-\frac{401987}{6362}a^{8}+\frac{220499}{6362}a^{7}+\frac{1006875}{6362}a^{6}-\frac{662843}{6362}a^{5}-\frac{489638}{3181}a^{4}+\frac{453509}{6362}a^{3}+\frac{376349}{6362}a^{2}+\frac{4569}{6362}a-\frac{68661}{6362}$ Copy content Toggle raw display
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:  \( 8833.53771546 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 3 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 8833.53771546 \cdot 1}{2\cdot\sqrt{145348100481600000000}}\cr\approx \mathstrut & 0.36066100168 \end{aligned}\]

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 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*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 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*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 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*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 - x^15 - 5*x^14 + 5*x^13 + 7*x^12 - 8*x^11 - 6*x^10 - 9*x^9 + 12*x^8 + 28*x^7 - 34*x^6 - 21*x^5 + 27*x^4 + 5*x^3 - 5*x^2 - 2*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

$Q_8^2.A_4\wr C_2$ (as 16T1791):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 18432
The 77 conjugacy class representatives for $Q_8^2.A_4\wr C_2$
Character table for $Q_8^2.A_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 8.4.3014010000.1

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
Minimal sibling: 16.0.28710735897600000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.2.6.12a1.1$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 128 x^{5} + 94 x^{4} + 58 x^{3} + 27 x^{2} + 12 x + 1$$6$$2$$12$12T159$$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{12}$$
\(3\) Copy content Toggle raw display 3.8.2.8a1.2$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(61\) Copy content Toggle raw display 61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.3.4a1.1$x^{6} + 180 x^{5} + 10806 x^{4} + 216720 x^{3} + 21612 x^{2} + 781 x + 8$$3$$2$$4$$C_6$$$[\ ]_{3}^{2}$$
61.6.1.0a1.1$x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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