↑

Properties

Label 16.0.239625547119140625.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.396\times 10^{17}$
Root discriminant \(12.20\)
Ramified primes $3,5,59$
Class number $1$
Class group trivial
Galois group $C_2^5:C_4$ (as 16T227)

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 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*x^2 - 2*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 3*y^14 + 3*y^13 + 13*y^12 - 17*y^11 + 5*y^10 + 10*y^9 - 19*y^8 + 10*y^7 + 5*y^6 - 17*y^5 + 13*y^4 + 3*y^3 - 3*y^2 - 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*x^2 - 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*x^2 - 2*x + 1)
 

\( x^{16} - 2 x^{15} - 3 x^{14} + 3 x^{13} + 13 x^{12} - 17 x^{11} + 5 x^{10} + 10 x^{9} - 19 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
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:   \(239625547119140625\) \(\medspace = 3^{4}\cdot 5^{12}\cdot 59^{4}\) 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:  \(12.20\)
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:  $3^{1/2}5^{3/4}59^{1/2}\approx 44.48505546908213$
Ramified primes:   \(3\), \(5\), \(59\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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.
Maximal CM subfield:  \(\Q(\zeta_{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{1}{5}a^{8}-\frac{1}{5}a^{6}+\frac{1}{5}a^{4}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{11}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{695}a^{14}-\frac{44}{695}a^{13}+\frac{37}{695}a^{12}+\frac{161}{695}a^{11}-\frac{114}{695}a^{10}+\frac{162}{695}a^{9}-\frac{152}{695}a^{8}-\frac{162}{695}a^{7}-\frac{13}{695}a^{6}+\frac{162}{695}a^{5}+\frac{164}{695}a^{4}-\frac{256}{695}a^{3}-\frac{241}{695}a^{2}-\frac{322}{695}a+\frac{279}{695}$, $\frac{1}{695}a^{15}+\frac{47}{695}a^{13}-\frac{18}{695}a^{12}-\frac{119}{695}a^{11}+\frac{11}{695}a^{10}-\frac{113}{695}a^{9}+\frac{239}{695}a^{8}-\frac{52}{695}a^{7}+\frac{146}{695}a^{6}+\frac{203}{695}a^{5}+\frac{149}{695}a^{4}+\frac{62}{139}a^{3}+\frac{333}{695}a^{2}-\frac{267}{695}a-\frac{19}{139}$ 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:  Trivial group, which has order $1$
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:  $7$
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:   \( -\frac{8171}{695} a^{15} + \frac{11984}{695} a^{14} + \frac{30809}{695} a^{13} - \frac{7909}{695} a^{12} - \frac{110079}{695} a^{11} + \frac{79893}{695} a^{10} + \frac{358}{695} a^{9} - \frac{16019}{139} a^{8} + \frac{112842}{695} a^{7} - \frac{4623}{139} a^{6} - \frac{51183}{695} a^{5} + \frac{110999}{695} a^{4} - \frac{47864}{695} a^{3} - \frac{48114}{695} a^{2} - \frac{2241}{695} a + \frac{14833}{695} \)  (order $10$) 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:   $\frac{2789}{695}a^{15}+\frac{790}{139}a^{14}+\frac{10647}{695}a^{13}-\frac{428}{139}a^{12}-\frac{37407}{695}a^{11}+\frac{25676}{695}a^{10}+\frac{822}{695}a^{9}-\frac{27508}{695}a^{8}+\frac{36803}{695}a^{7}-\frac{6377}{695}a^{6}-\frac{17312}{695}a^{5}+\frac{36527}{695}a^{4}-\frac{2916}{139}a^{3}-\frac{16697}{695}a^{2}-\frac{280}{139}a+\frac{5083}{695}$, $\frac{8171}{695}a^{15}-\frac{11984}{695}a^{14}-\frac{30809}{695}a^{13}+\frac{7909}{695}a^{12}+\frac{110079}{695}a^{11}-\frac{79893}{695}a^{10}-\frac{358}{695}a^{9}+\frac{16019}{139}a^{8}-\frac{112842}{695}a^{7}+\frac{4623}{139}a^{6}+\frac{51183}{695}a^{5}-\frac{110999}{695}a^{4}+\frac{47864}{695}a^{3}+\frac{48114}{695}a^{2}+\frac{2241}{695}a-\frac{14138}{695}$, $\frac{6984}{695}a^{15}-\frac{2005}{139}a^{14}-\frac{5341}{139}a^{13}+\frac{5987}{695}a^{12}+\frac{94548}{695}a^{11}-\frac{65381}{695}a^{10}-\frac{652}{139}a^{9}+\frac{68812}{695}a^{8}-\frac{18679}{139}a^{7}+\frac{15888}{695}a^{6}+\frac{44319}{695}a^{5}-\frac{93493}{695}a^{4}+\frac{7482}{139}a^{3}+\frac{8616}{139}a^{2}+\frac{3068}{695}a-\frac{12699}{695}$, $\frac{290}{139}a^{15}+\frac{355}{139}a^{14}+\frac{1191}{139}a^{13}-\frac{104}{695}a^{12}-\frac{19696}{695}a^{11}+\frac{1932}{139}a^{10}+\frac{625}{139}a^{9}-\frac{14619}{695}a^{8}+\frac{3440}{139}a^{7}+\frac{274}{695}a^{6}-\frac{2194}{139}a^{5}+\frac{18616}{695}a^{4}-\frac{775}{139}a^{3}-\frac{2120}{139}a^{2}-\frac{1476}{695}a+\frac{1776}{695}$, $\frac{5537}{695}a^{15}+\frac{7834}{695}a^{14}+\frac{4252}{139}a^{13}-\frac{4402}{695}a^{12}-\frac{74612}{695}a^{11}+\frac{50987}{695}a^{10}+\frac{2999}{695}a^{9}-\frac{55757}{695}a^{8}+\frac{73826}{695}a^{7}-\frac{12443}{695}a^{6}-\frac{36993}{695}a^{5}+\frac{74872}{695}a^{4}-\frac{28744}{695}a^{3}-\frac{6883}{139}a^{2}-\frac{1113}{695}a+\frac{1936}{139}$, $\frac{13464}{695}a^{15}+\frac{19648}{695}a^{14}+\frac{51001}{695}a^{13}-\frac{12846}{695}a^{12}-\frac{181883}{695}a^{11}+\frac{130704}{695}a^{10}+\frac{3289}{695}a^{9}-\frac{26630}{139}a^{8}+\frac{184006}{695}a^{7}-\frac{7162}{139}a^{6}-\frac{17379}{139}a^{5}+\frac{182393}{695}a^{4}-\frac{76293}{695}a^{3}-\frac{81376}{695}a^{2}-\frac{3059}{695}a+\frac{24654}{695}$, $\frac{11023}{695}a^{15}+\frac{16246}{695}a^{14}+\frac{41586}{695}a^{13}-\frac{11132}{695}a^{12}-\frac{29822}{139}a^{11}+\frac{108923}{695}a^{10}+\frac{1992}{695}a^{9}-\frac{108646}{695}a^{8}+\frac{151583}{695}a^{7}-\frac{30514}{695}a^{6}-\frac{70911}{695}a^{5}+\frac{29855}{139}a^{4}-\frac{64546}{695}a^{3}-\frac{66601}{695}a^{2}-\frac{1938}{695}a+\frac{20792}{695}$ 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:  \( 628.8983375986766 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 628.8983375986766 \cdot 1}{10\cdot\sqrt{239625547119140625}}\cr\approx \mathstrut & 0.312070584847752 \end{aligned}\]

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 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*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 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*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 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 3*x^14 + 3*x^13 + 13*x^12 - 17*x^11 + 5*x^10 + 10*x^9 - 19*x^8 + 10*x^7 + 5*x^6 - 17*x^5 + 13*x^4 + 3*x^3 - 3*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

$C_2^5:C_4$ (as 16T227):

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 128
The 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, \(\Q(\zeta_{5})\), 4.2.7375.1, 8.0.97903125.1, 8.4.97903125.1, 8.0.54390625.1

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(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: 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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ R

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 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(59\) Copy content Toggle raw display 59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.2.2a1.2$x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
59.2.2.2a1.2$x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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