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Properties

Label 16.0.513...056.7
Degree $16$
Signature $[0, 8]$
Discriminant $5.140\times 10^{19}$
Root discriminant \(17.06\)
Ramified primes $2,3,7$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93)
 
gp: K = bnfinit(y^16 - 8*y^15 + 44*y^14 - 168*y^13 + 502*y^12 - 1192*y^11 + 2302*y^10 - 3634*y^9 + 4876*y^8 - 5664*y^7 + 5834*y^6 - 5216*y^5 + 3955*y^4 - 2394*y^3 + 1128*y^2 - 366*y + 93, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93)
 

\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 502 x^{12} - 1192 x^{11} + 2302 x^{10} - 3634 x^{9} + \cdots + 93 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(51399544780206637056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(17.06\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}3^{3/4}7^{1/2}\approx 17.058268835716344$
Ramified primes:   \(2\), \(3\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1407}a^{12}-\frac{2}{469}a^{11}+\frac{305}{1407}a^{10}-\frac{3}{67}a^{9}-\frac{102}{469}a^{8}+\frac{43}{469}a^{7}-\frac{178}{1407}a^{6}+\frac{102}{469}a^{5}-\frac{17}{201}a^{4}+\frac{225}{469}a^{3}-\frac{202}{469}a^{2}-\frac{46}{469}a-\frac{177}{469}$, $\frac{1}{1407}a^{13}+\frac{269}{1407}a^{11}+\frac{120}{469}a^{10}-\frac{228}{469}a^{9}-\frac{100}{469}a^{8}+\frac{596}{1407}a^{7}+\frac{215}{469}a^{6}+\frac{310}{1407}a^{5}-\frac{13}{469}a^{4}+\frac{30}{67}a^{3}+\frac{149}{469}a^{2}+\frac{16}{469}a-\frac{124}{469}$, $\frac{1}{66129}a^{14}-\frac{1}{9447}a^{13}+\frac{5}{66129}a^{12}+\frac{61}{66129}a^{11}-\frac{5396}{22043}a^{10}+\frac{4695}{22043}a^{9}-\frac{29080}{66129}a^{8}+\frac{4681}{9447}a^{7}+\frac{27310}{66129}a^{6}+\frac{7055}{66129}a^{5}-\frac{1141}{3149}a^{4}+\frac{8222}{22043}a^{3}-\frac{2572}{22043}a^{2}-\frac{9666}{22043}a+\frac{7731}{22043}$, $\frac{1}{103756401}a^{15}+\frac{37}{4940781}a^{14}+\frac{15338}{103756401}a^{13}-\frac{2963}{14822343}a^{12}+\frac{656306}{4940781}a^{11}-\frac{4786477}{103756401}a^{10}+\frac{40255268}{103756401}a^{9}-\frac{1333052}{4940781}a^{8}+\frac{35465848}{103756401}a^{7}+\frac{43149767}{103756401}a^{6}+\frac{14294605}{34585467}a^{5}-\frac{335285}{14822343}a^{4}-\frac{7729891}{34585467}a^{3}-\frac{13724555}{34585467}a^{2}-\frac{1592495}{11528489}a+\frac{10786813}{34585467}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{260}{9447} a^{14} + \frac{1820}{9447} a^{13} - \frac{3128}{3149} a^{12} + \frac{32644}{9447} a^{11} - \frac{89458}{9447} a^{10} + \frac{63810}{3149} a^{9} - \frac{328855}{9447} a^{8} + \frac{451564}{9447} a^{7} - \frac{180458}{3149} a^{6} + \frac{566648}{9447} a^{5} - \frac{526343}{9447} a^{4} + \frac{130782}{3149} a^{3} - \frac{78956}{3149} a^{2} + \frac{32030}{3149} a - \frac{9223}{3149} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{121798}{14822343}a^{15}-\frac{262655}{4940781}a^{14}+\frac{3630695}{14822343}a^{13}-\frac{77916092}{103756401}a^{12}+\frac{57927580}{34585467}a^{11}-\frac{5931026}{2207583}a^{10}+\frac{37380326}{14822343}a^{9}-\frac{1899344}{34585467}a^{8}-\frac{335529305}{103756401}a^{7}+\frac{545647010}{103756401}a^{6}-\frac{266673877}{34585467}a^{5}+\frac{131971357}{14822343}a^{4}-\frac{294994492}{34585467}a^{3}+\frac{159498274}{34585467}a^{2}-\frac{12775142}{11528489}a-\frac{3803978}{34585467}$, $\frac{231629}{14822343}a^{15}-\frac{449107}{4940781}a^{14}+\frac{6274108}{14822343}a^{13}-\frac{125994229}{103756401}a^{12}+\frac{91974604}{34585467}a^{11}-\frac{395044001}{103756401}a^{10}+\frac{40462846}{14822343}a^{9}+\frac{104501882}{34585467}a^{8}-\frac{1019251231}{103756401}a^{7}+\frac{1678524811}{103756401}a^{6}-\frac{699457117}{34585467}a^{5}+\frac{370621193}{14822343}a^{4}-\frac{16192006}{735861}a^{3}+\frac{505170494}{34585467}a^{2}-\frac{59547275}{11528489}a+\frac{85021385}{34585467}$, $\frac{231629}{14822343}a^{15}-\frac{236346}{1646927}a^{14}+\frac{11732659}{14822343}a^{13}-\frac{324444487}{103756401}a^{12}+\frac{323299073}{34585467}a^{11}-\frac{2313657995}{103756401}a^{10}+\frac{632222179}{14822343}a^{9}-\frac{2293701020}{34585467}a^{8}+\frac{8977853498}{103756401}a^{7}-\frac{10370643638}{103756401}a^{6}+\frac{3487495741}{34585467}a^{5}-\frac{1306159693}{14822343}a^{4}+\frac{2208242320}{34585467}a^{3}-\frac{1272646147}{34585467}a^{2}+\frac{173447656}{11528489}a-\frac{164855986}{34585467}$, $\frac{2563033}{103756401}a^{15}-\frac{2634367}{11528489}a^{14}+\frac{134802140}{103756401}a^{13}-\frac{540309746}{103756401}a^{12}+\frac{183795456}{11528489}a^{11}-\frac{4004882386}{103756401}a^{10}+\frac{1111842656}{14822343}a^{9}-\frac{4060272178}{34585467}a^{8}+\frac{235212697}{1548603}a^{7}-\frac{17613873583}{103756401}a^{6}+\frac{1945171108}{11528489}a^{5}-\frac{2142489239}{14822343}a^{4}+\frac{3451465907}{34585467}a^{3}-\frac{1738287776}{34585467}a^{2}+\frac{204060201}{11528489}a-\frac{161961320}{34585467}$, $\frac{1845064}{103756401}a^{15}-\frac{2976193}{34585467}a^{14}+\frac{41890649}{103756401}a^{13}-\frac{108595919}{103756401}a^{12}+\frac{79311472}{34585467}a^{11}-\frac{329927179}{103756401}a^{10}+\frac{293989505}{103756401}a^{9}+\frac{2051850}{11528489}a^{8}-\frac{152330711}{103756401}a^{7}+\frac{516855917}{103756401}a^{6}-\frac{196787776}{34585467}a^{5}+\frac{895916767}{103756401}a^{4}-\frac{29420356}{4940781}a^{3}+\frac{126048919}{34585467}a^{2}-\frac{2483266}{11528489}a+\frac{18800}{735861}$, $\frac{846830}{103756401}a^{15}-\frac{2117075}{34585467}a^{14}+\frac{33789151}{103756401}a^{13}-\frac{123302569}{103756401}a^{12}+\frac{117709811}{34585467}a^{11}-\frac{797802830}{103756401}a^{10}+\frac{1460329246}{103756401}a^{9}-\frac{240926687}{11528489}a^{8}+\frac{2784576257}{103756401}a^{7}-\frac{3134816960}{103756401}a^{6}+\frac{1066726222}{34585467}a^{5}-\frac{2786708368}{103756401}a^{4}+\frac{684802975}{34585467}a^{3}-\frac{381078220}{34585467}a^{2}+\frac{50448639}{11528489}a-\frac{30123244}{34585467}$, $\frac{3812}{66129}a^{14}-\frac{3812}{9447}a^{13}+\frac{137171}{66129}a^{12}-\frac{476134}{66129}a^{11}+\frac{1295831}{66129}a^{10}-\frac{916854}{22043}a^{9}+\frac{4638871}{66129}a^{8}-\frac{885622}{9447}a^{7}+\frac{7099741}{66129}a^{6}-\frac{7061114}{66129}a^{5}+\frac{880726}{9447}a^{4}-\frac{1444560}{22043}a^{3}+\frac{658065}{22043}a^{2}-\frac{155725}{22043}a-\frac{75931}{22043}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 8148.71784457 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8148.71784457 \cdot 2}{6\cdot\sqrt{51399544780206637056}}\cr\approx \mathstrut & 0.920295579781 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 502*x^12 - 1192*x^11 + 2302*x^10 - 3634*x^9 + 4876*x^8 - 5664*x^7 + 5834*x^6 - 5216*x^5 + 3955*x^4 - 2394*x^3 + 1128*x^2 - 366*x + 93);
 
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

$C_2^2\wr C_2$ (as 16T39):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-21}) \), 4.0.3024.2, 4.0.189.1, 4.0.1008.1 x2, 4.2.9408.2 x2, \(\Q(\sqrt{-3}, \sqrt{7})\), 4.0.432.1, 4.0.21168.1, 8.0.7169347584.5, 8.0.448084224.7, 8.0.796594176.14, 8.0.146313216.1, 8.0.9144576.2, 8.0.7169347584.8, 8.0.448084224.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 8 siblings: 8.0.9144576.1, 8.0.9144576.2, 8.0.146313216.1, 8.0.7169347584.8, 8.0.146313216.2, 8.0.7169347584.4, 8.0.448084224.2, 8.0.448084224.1
Degree 16 siblings: 16.4.51399544780206637056.3, 16.0.51399544780206637056.2, 16.0.21407557176262656.1, 16.0.200779471797682176.1, 16.0.51399544780206637056.8, 16.0.51399544780206637056.4
Minimal sibling: 8.0.9144576.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$