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Properties

Label 16.0.217678233600000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.177\times 10^{17}$
Root discriminant \(12.12\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $C_4^2:C_2^2$ (as 16T117)

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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 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*x^2 - 2*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 + 29*y^14 - 60*y^13 + 81*y^12 - 106*y^11 + 208*y^10 - 380*y^9 + 427*y^8 - 222*y^7 - 30*y^6 + 82*y^5 + 6*y^4 - 54*y^3 + 28*y^2 - 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*x^2 - 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*x^2 - 2*x + 1)
 

\( x^{16} - 8 x^{15} + 29 x^{14} - 60 x^{13} + 81 x^{12} - 106 x^{11} + 208 x^{10} - 380 x^{9} + 427 x^{8} + \cdots + 1 \) 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:   \(217678233600000000\) \(\medspace = 2^{20}\cdot 3^{12}\cdot 5^{8}\) 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:  \(12.12\)
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:  $2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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.
Maximal CM subfield:  \(\Q(\sqrt{-3}, \sqrt{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19}a^{14}-\frac{7}{19}a^{13}-\frac{1}{19}a^{12}+\frac{5}{19}a^{11}-\frac{5}{19}a^{10}+\frac{2}{19}a^{9}+\frac{2}{19}a^{8}-\frac{6}{19}a^{7}-\frac{5}{19}a^{6}+\frac{6}{19}a^{5}-\frac{4}{19}a^{4}-\frac{3}{19}a^{3}-\frac{4}{19}a+\frac{5}{19}$, $\frac{1}{97881673}a^{15}-\frac{1506353}{97881673}a^{14}+\frac{30427997}{97881673}a^{13}+\frac{31088021}{97881673}a^{12}+\frac{11995211}{97881673}a^{11}+\frac{15266898}{97881673}a^{10}+\frac{1177076}{97881673}a^{9}+\frac{18200460}{97881673}a^{8}-\frac{30477673}{97881673}a^{7}+\frac{37114413}{97881673}a^{6}-\frac{19828104}{97881673}a^{5}-\frac{19299282}{97881673}a^{4}-\frac{15710072}{97881673}a^{3}+\frac{23903915}{97881673}a^{2}-\frac{7557483}{97881673}a+\frac{2386365}{97881673}$ 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:   \( \frac{8690}{312721} a^{15} + \frac{13687}{312721} a^{14} - \frac{15880}{16459} a^{13} + \frac{1139371}{312721} a^{12} - \frac{1993634}{312721} a^{11} + \frac{104505}{16459} a^{10} - \frac{2749402}{312721} a^{9} + \frac{7482859}{312721} a^{8} - \frac{12764514}{312721} a^{7} + \frac{9913544}{312721} a^{6} - \frac{507904}{312721} a^{5} - \frac{3936209}{312721} a^{4} + \frac{1480768}{312721} a^{3} + \frac{1347984}{312721} a^{2} - \frac{1372616}{312721} a + \frac{208767}{312721} \)  (order $6$) 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:   $\frac{33354925}{97881673}a^{15}-\frac{237637535}{97881673}a^{14}+\frac{777224020}{97881673}a^{13}-\frac{1453760719}{97881673}a^{12}+\frac{1863482943}{97881673}a^{11}-\frac{2681753152}{97881673}a^{10}+\frac{5486429294}{97881673}a^{9}-\frac{9139843589}{97881673}a^{8}+\frac{9136289532}{97881673}a^{7}-\frac{4227366435}{97881673}a^{6}-\frac{552850963}{97881673}a^{5}+\frac{1347862875}{97881673}a^{4}+\frac{222464166}{97881673}a^{3}-\frac{962316360}{97881673}a^{2}+\frac{401243449}{97881673}a-\frac{52089289}{97881673}$, $\frac{1208795}{97881673}a^{15}+\frac{181516}{97881673}a^{14}-\frac{32909611}{97881673}a^{13}+\frac{137289857}{97881673}a^{12}-\frac{249345796}{97881673}a^{11}+\frac{238137176}{97881673}a^{10}-\frac{299273663}{97881673}a^{9}+\frac{894772230}{97881673}a^{8}-\frac{1605590363}{97881673}a^{7}+\frac{1223301220}{97881673}a^{6}+\frac{48652822}{97881673}a^{5}-\frac{556150755}{97881673}a^{4}+\frac{11591694}{5151667}a^{3}+\frac{67299479}{97881673}a^{2}-\frac{68695223}{97881673}a+\frac{2527642}{5151667}$, $\frac{5189111}{97881673}a^{15}-\frac{39493085}{97881673}a^{14}+\frac{147492624}{97881673}a^{13}-\frac{335372017}{97881673}a^{12}+\frac{536366657}{97881673}a^{11}-\frac{753143325}{97881673}a^{10}+\frac{1253777640}{97881673}a^{9}-\frac{115534649}{5151667}a^{8}+\frac{2979252888}{97881673}a^{7}-\frac{2531078045}{97881673}a^{6}+\frac{875050253}{97881673}a^{5}+\frac{507484916}{97881673}a^{4}-\frac{432427934}{97881673}a^{3}-\frac{286076339}{97881673}a^{2}+\frac{426263219}{97881673}a-\frac{5764722}{97881673}$, $\frac{6873030}{97881673}a^{15}-\frac{43284699}{97881673}a^{14}+\frac{135250879}{97881673}a^{13}-\frac{258871614}{97881673}a^{12}+\frac{385606603}{97881673}a^{11}-\frac{608866811}{97881673}a^{10}+\frac{1090838557}{97881673}a^{9}-\frac{1691445279}{97881673}a^{8}+\frac{1963569735}{97881673}a^{7}-\frac{1572223950}{97881673}a^{6}+\frac{675416060}{97881673}a^{5}+\frac{93494842}{97881673}a^{4}-\frac{152905040}{97881673}a^{3}-\frac{7702917}{97881673}a^{2}+\frac{54875753}{97881673}a-\frac{51271966}{97881673}$, $\frac{12470196}{97881673}a^{15}-\frac{91654761}{97881673}a^{14}+\frac{309445956}{97881673}a^{13}-\frac{601508620}{97881673}a^{12}+\frac{802307817}{97881673}a^{11}-\frac{1152474821}{97881673}a^{10}+\frac{2289173962}{97881673}a^{9}-\frac{3833775516}{97881673}a^{8}+\frac{4036454578}{97881673}a^{7}-\frac{2304821754}{97881673}a^{6}+\frac{328302031}{97881673}a^{5}+\frac{528222798}{97881673}a^{4}-\frac{305859793}{97881673}a^{3}-\frac{286221727}{97881673}a^{2}+\frac{302397934}{97881673}a-\frac{72417954}{97881673}$, $\frac{28674442}{97881673}a^{15}-\frac{205238560}{97881673}a^{14}+\frac{667234788}{97881673}a^{13}-\frac{1221673302}{97881673}a^{12}+\frac{1498680373}{97881673}a^{11}-\frac{2137521334}{97881673}a^{10}+\frac{4570644666}{97881673}a^{9}-\frac{7628545404}{97881673}a^{8}+\frac{7167643760}{97881673}a^{7}-\frac{2545202859}{97881673}a^{6}-\frac{1138296628}{97881673}a^{5}+\frac{55405159}{5151667}a^{4}+\frac{405789557}{97881673}a^{3}-\frac{847773096}{97881673}a^{2}+\frac{340527315}{97881673}a-\frac{68832551}{97881673}$, $\frac{4378103}{97881673}a^{15}-\frac{36319974}{97881673}a^{14}+\frac{138608370}{97881673}a^{13}-\frac{314130826}{97881673}a^{12}+\frac{490118420}{97881673}a^{11}-\frac{689448701}{97881673}a^{10}+\frac{1177800128}{97881673}a^{9}-\frac{2063888592}{97881673}a^{8}+\frac{2752757928}{97881673}a^{7}-\frac{2338500408}{97881673}a^{6}+\frac{841725261}{97881673}a^{5}+\frac{414896761}{97881673}a^{4}-\frac{206321403}{97881673}a^{3}-\frac{499744971}{97881673}a^{2}+\frac{352954496}{97881673}a-\frac{50432086}{97881673}$ 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:  \( 265.442006674 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 265.442006674 \cdot 1}{6\cdot\sqrt{217678233600000000}}\cr\approx \mathstrut & 0.230329649135 \end{aligned}\]

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 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*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 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*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 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*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 - 8*x^15 + 29*x^14 - 60*x^13 + 81*x^12 - 106*x^11 + 208*x^10 - 380*x^9 + 427*x^8 - 222*x^7 - 30*x^6 + 82*x^5 + 6*x^4 - 54*x^3 + 28*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

$C_4^2:C_2^2$ (as 16T117):

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 64
The 28 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.0.21600.2, 4.0.21600.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.466560000.2, 8.0.7290000.1, 8.0.3240000.1

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: 16.0.13604889600000000.2

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.2.12a1.9$x^{8} + 2 x^{5} + 6 x^{4} + x^{2} + 6 x + 7$$2$$4$$12$$C_4\times C_2$$$[3]^{4}$$
2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(5\) 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$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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