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Properties

Label 16.0.389...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.900\times 10^{19}$
Root discriminant \(16.77\)
Ramified primes $5,29,109$
Class number $2$
Class group [2]
Galois group $C_2^6:D_4$ (as 16T969)

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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 - 2*x^15 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 + 9*y^14 - 14*y^13 + 53*y^12 - 64*y^11 + 115*y^10 - 135*y^9 + 209*y^8 - 186*y^7 + 204*y^6 - 149*y^5 + 121*y^4 - 63*y^3 + 28*y^2 - 7*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*x + 1)
 

\( x^{16} - 2 x^{15} + 9 x^{14} - 14 x^{13} + 53 x^{12} - 64 x^{11} + 115 x^{10} - 135 x^{9} + 209 x^{8} + \cdots + 1 \) 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:   \(38999408308687890625\) \(\medspace = 5^{8}\cdot 29^{4}\cdot 109^{4}\) 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:  \(16.77\)
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:  $5^{1/2}29^{1/2}109^{1/2}\approx 125.7179382586272$
Ramified primes:   \(5\), \(29\), \(109\) 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:  8.0.6244950625.4

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1069942743617}a^{15}-\frac{465573626959}{1069942743617}a^{14}-\frac{522800500049}{1069942743617}a^{13}-\frac{175958233490}{1069942743617}a^{12}+\frac{212003060382}{1069942743617}a^{11}+\frac{88377629972}{1069942743617}a^{10}-\frac{483601359088}{1069942743617}a^{9}+\frac{359093905859}{1069942743617}a^{8}+\frac{135144920852}{1069942743617}a^{7}+\frac{205184377516}{1069942743617}a^{6}+\frac{46384805888}{1069942743617}a^{5}+\frac{183701034910}{1069942743617}a^{4}+\frac{247178181203}{1069942743617}a^{3}+\frac{11914510016}{97267522147}a^{2}+\frac{507966461668}{1069942743617}a+\frac{8126481065}{1069942743617}$ 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:  $C_{2}$, which has order $2$
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:  $7$
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:   \( -1 \)  (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:   $\frac{749440103047}{1069942743617}a^{15}-\frac{1193678770348}{1069942743617}a^{14}+\frac{6208342111780}{1069942743617}a^{13}-\frac{7885749640460}{1069942743617}a^{12}+\frac{36062394343266}{1069942743617}a^{11}-\frac{32718227160332}{1069942743617}a^{10}+\frac{70221173147413}{1069942743617}a^{9}-\frac{70226755079693}{1069942743617}a^{8}+\frac{122126558994481}{1069942743617}a^{7}-\frac{84237998493863}{1069942743617}a^{6}+\frac{108965365581825}{1069942743617}a^{5}-\frac{60679860879907}{1069942743617}a^{4}+\frac{56263712501255}{1069942743617}a^{3}-\frac{1726582885028}{97267522147}a^{2}+\frac{8613858374626}{1069942743617}a+\frac{51773575988}{1069942743617}$, $\frac{537576283419}{1069942743617}a^{15}-\frac{1838589367537}{1069942743617}a^{14}+\frac{5987887099186}{1069942743617}a^{13}-\frac{13486130617404}{1069942743617}a^{12}+\frac{35689075629006}{1069942743617}a^{11}-\frac{68574030259740}{1069942743617}a^{10}+\frac{90298900713264}{1069942743617}a^{9}-\frac{130544537481849}{1069942743617}a^{8}+\frac{172452130530387}{1069942743617}a^{7}-\frac{202014525633511}{1069942743617}a^{6}+\frac{171125777754742}{1069942743617}a^{5}-\frac{156367440357367}{1069942743617}a^{4}+\frac{108736992797040}{1069942743617}a^{3}-\frac{6025582540506}{97267522147}a^{2}+\frac{23431842535865}{1069942743617}a-\frac{4691283348861}{1069942743617}$, $\frac{552585972408}{1069942743617}a^{15}-\frac{587307242167}{1069942743617}a^{14}+\frac{4108110251375}{1069942743617}a^{13}-\frac{3547121230921}{1069942743617}a^{12}+\frac{23684048580557}{1069942743617}a^{11}-\frac{11202397764298}{1069942743617}a^{10}+\frac{40016265646118}{1069942743617}a^{9}-\frac{31017824552791}{1069942743617}a^{8}+\frac{66300475436042}{1069942743617}a^{7}-\frac{24432385424350}{1069942743617}a^{6}+\frac{56057031775362}{1069942743617}a^{5}-\frac{19155685430114}{1069942743617}a^{4}+\frac{24745937479074}{1069942743617}a^{3}-\frac{420877343956}{97267522147}a^{2}+\frac{3111674796883}{1069942743617}a-\frac{1020425328931}{1069942743617}$, $\frac{690769610952}{1069942743617}a^{15}-\frac{937750044883}{1069942743617}a^{14}+\frac{5276710261759}{1069942743617}a^{13}-\frac{5689588448067}{1069942743617}a^{12}+\frac{30146785250767}{1069942743617}a^{11}-\frac{20973879664701}{1069942743617}a^{10}+\frac{49745816876192}{1069942743617}a^{9}-\frac{44816171923348}{1069942743617}a^{8}+\frac{85018315083910}{1069942743617}a^{7}-\frac{40688660296271}{1069942743617}a^{6}+\frac{60656501237770}{1069942743617}a^{5}-\frac{24484253122261}{1069942743617}a^{4}+\frac{23237882491661}{1069942743617}a^{3}-\frac{53725747960}{97267522147}a^{2}-\frac{1745836726583}{1069942743617}a+\frac{1521561018920}{1069942743617}$, $\frac{763436800699}{1069942743617}a^{15}-\frac{1149700548415}{1069942743617}a^{14}+\frac{5960062649538}{1069942743617}a^{13}-\frac{7197532607799}{1069942743617}a^{12}+\frac{34169148120924}{1069942743617}a^{11}-\frac{28477628120079}{1069942743617}a^{10}+\frac{57971739220284}{1069942743617}a^{9}-\frac{60098687295816}{1069942743617}a^{8}+\frac{102025336917577}{1069942743617}a^{7}-\frac{61460215937266}{1069942743617}a^{6}+\frac{76268574127936}{1069942743617}a^{5}-\frac{43690262503341}{1069942743617}a^{4}+\frac{32414102090169}{1069942743617}a^{3}-\frac{761791509103}{97267522147}a^{2}+\frac{928249364928}{1069942743617}a+\frac{537576283419}{1069942743617}$, $\frac{107651732597}{1069942743617}a^{15}-\frac{92546525040}{1069942743617}a^{14}+\frac{706998353803}{1069942743617}a^{13}-\frac{446429746852}{1069942743617}a^{12}+\frac{3981618491791}{1069942743617}a^{11}-\frac{803915008689}{1069942743617}a^{10}+\frac{4534450953162}{1069942743617}a^{9}-\frac{3041508529144}{1069942743617}a^{8}+\frac{7705923600642}{1069942743617}a^{7}+\frac{1104635495848}{1069942743617}a^{6}+\frac{1541414104181}{1069942743617}a^{5}+\frac{768434036463}{1069942743617}a^{4}-\frac{977182776785}{1069942743617}a^{3}+\frac{252131455758}{97267522147}a^{2}-\frac{2294635780942}{1069942743617}a+\frac{218291773500}{1069942743617}$, $\frac{1251495669432}{1069942743617}a^{15}-\frac{2336669114172}{1069942743617}a^{14}+\frac{10914985774383}{1069942743617}a^{13}-\frac{15882309168224}{1069942743617}a^{12}+\frac{63686679540651}{1069942743617}a^{11}-\frac{70198495259358}{1069942743617}a^{10}+\frac{131373739480360}{1069942743617}a^{9}-\frac{143942108148937}{1069942743617}a^{8}+\frac{233110503225799}{1069942743617}a^{7}-\frac{187964027003139}{1069942743617}a^{6}+\frac{212860413862384}{1069942743617}a^{5}-\frac{136622576297132}{1069942743617}a^{4}+\frac{116003802280089}{1069942743617}a^{3}-\frac{4254143631037}{97267522147}a^{2}+\frac{19413896249686}{1069942743617}a-\frac{1765353109374}{1069942743617}$ 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:  \( 558.777866748 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 558.777866748 \cdot 2}{2\cdot\sqrt{38999408308687890625}}\cr\approx \mathstrut & 0.217344739479 \end{aligned}\]

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 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*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 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*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 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*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 + 9*x^14 - 14*x^13 + 53*x^12 - 64*x^11 + 115*x^10 - 135*x^9 + 209*x^8 - 186*x^7 + 204*x^6 - 149*x^5 + 121*x^4 - 63*x^3 + 28*x^2 - 7*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

$C_2^6:D_4$ (as 16T969):

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 512
The 44 conjugacy class representatives for $C_2^6:D_4$
Character table for $C_2^6:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.57293125.2, 8.4.57293125.1, 8.0.6244950625.4

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: 16.8.38999408308687890625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) 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$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(109\) Copy content Toggle raw display $\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{109}$$x + 103$$1$$1$$0$Trivial$$[\ ]$$
109.1.2.1a1.1$x^{2} + 109$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.1.2.1a1.1$x^{2} + 109$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.1.2.1a1.2$x^{2} + 654$$2$$1$$1$$C_2$$$[\ ]_{2}$$
109.2.1.0a1.1$x^{2} + 108 x + 6$$1$$2$$0$$C_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)