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Properties

Label 16.0.142...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.427\times 10^{18}$
Root discriminant \(13.63\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 20*y^13 + 35*y^12 + 16*y^11 - 144*y^10 + 132*y^9 + 136*y^8 - 288*y^7 + 54*y^6 + 172*y^5 - 97*y^4 - 20*y^3 + 26*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*x + 1)
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 20 x^{13} + 35 x^{12} + 16 x^{11} - 144 x^{10} + 132 x^{9} + 136 x^{8} + \cdots + 1 \) 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:   \(1426576071720960000\) \(\medspace = 2^{32}\cdot 3^{12}\cdot 5^{4}\) 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:  \(13.63\)
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:  $2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{6}a^{13}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{6}a^{14}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{1159747650}a^{15}-\frac{6222521}{1159747650}a^{14}+\frac{247999}{13644090}a^{13}+\frac{159319}{5154434}a^{12}+\frac{57793481}{115974765}a^{11}-\frac{7038383}{30519675}a^{10}+\frac{230265887}{579873825}a^{9}-\frac{160523288}{579873825}a^{8}-\frac{62699936}{579873825}a^{7}-\frac{62442832}{579873825}a^{6}-\frac{24570254}{579873825}a^{5}+\frac{13524541}{30519675}a^{4}-\frac{63288361}{386582550}a^{3}-\frac{90206609}{1159747650}a^{2}+\frac{94153129}{1159747650}a-\frac{498620297}{1159747650}$ 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:  $2$

Class group and class number

Trivial group, which has order $1$

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:   \( \frac{670679381}{1159747650} a^{15} - \frac{1346197838}{579873825} a^{14} + \frac{33254557}{6822045} a^{13} - \frac{32236662}{2577217} a^{12} + \frac{2584066906}{115974765} a^{11} + \frac{125019077}{30519675} a^{10} - \frac{43276764053}{579873825} a^{9} + \frac{45252364247}{579873825} a^{8} + \frac{29578868384}{579873825} a^{7} - \frac{79845683042}{579873825} a^{6} + \frac{27635705201}{579873825} a^{5} + \frac{1590831596}{30519675} a^{4} - \frac{14019505291}{386582550} a^{3} + \frac{207326623}{579873825} a^{2} + \frac{2525346037}{579873825} a - \frac{119108966}{579873825} \)  (order $24$) 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:   $\frac{338454629}{1159747650}a^{15}-\frac{675299492}{579873825}a^{14}+\frac{32662991}{13644090}a^{13}-\frac{95477293}{15463302}a^{12}+\frac{1268368324}{115974765}a^{11}+\frac{86750993}{30519675}a^{10}-\frac{22051833977}{579873825}a^{9}+\frac{21566866673}{579873825}a^{8}+\frac{17000877431}{579873825}a^{7}-\frac{40166789078}{579873825}a^{6}+\frac{11025073634}{579873825}a^{5}+\frac{887280614}{30519675}a^{4}-\frac{2154818973}{128860850}a^{3}-\frac{877609493}{579873825}a^{2}+\frac{2590269191}{1159747650}a-\frac{31082713}{1159747650}$, $\frac{196990633}{579873825}a^{15}-\frac{1129677361}{1159747650}a^{14}+\frac{19220099}{13644090}a^{13}-\frac{67934989}{15463302}a^{12}+\frac{604244656}{115974765}a^{11}+\frac{480378872}{30519675}a^{10}-\frac{22523314658}{579873825}a^{9}-\frac{350702308}{579873825}a^{8}+\frac{41211359399}{579873825}a^{7}-\frac{25414089287}{579873825}a^{6}-\frac{27847403989}{579873825}a^{5}+\frac{1485324431}{30519675}a^{4}+\frac{431981629}{64430425}a^{3}-\frac{16387120219}{1159747650}a^{2}+\frac{2684232689}{1159747650}a+\frac{153534173}{1159747650}$, $\frac{31082713}{1159747650}a^{15}+\frac{214123777}{1159747650}a^{14}-\frac{6481984}{6822045}a^{13}+\frac{28729333}{15463302}a^{12}-\frac{607290202}{115974765}a^{11}+\frac{346868596}{30519675}a^{10}-\frac{589686469}{579873825}a^{9}-\frac{20000374919}{579873825}a^{8}+\frac{23680491157}{579873825}a^{7}+\frac{12524966759}{579873825}a^{6}-\frac{39327555827}{579873825}a^{5}+\frac{720957208}{30519675}a^{4}+\frac{10233880057}{386582550}a^{3}-\frac{20015025017}{1159747650}a^{2}-\frac{473534224}{579873825}a+\frac{2465938339}{1159747650}$, $\frac{12996}{33575}a^{15}-\frac{166898}{100725}a^{14}+\frac{2783}{790}a^{13}-\frac{70721}{8058}a^{12}+\frac{109182}{6715}a^{11}+\frac{142123}{100725}a^{10}-\frac{5483138}{100725}a^{9}+\frac{2080279}{33575}a^{8}+\frac{3452914}{100725}a^{7}-\frac{10809632}{100725}a^{6}+\frac{1376032}{33575}a^{5}+\frac{4174054}{100725}a^{4}-\frac{3068654}{100725}a^{3}+\frac{1958}{100725}a^{2}+\frac{733379}{201450}a-\frac{36497}{201450}$, $\frac{232699903}{1159747650}a^{15}-\frac{439767394}{579873825}a^{14}+\frac{10238126}{6822045}a^{13}-\frac{29382875}{7731651}a^{12}+\frac{726399278}{115974765}a^{11}+\frac{125028076}{30519675}a^{10}-\frac{16244942539}{579873825}a^{9}+\frac{14203778536}{579873825}a^{8}+\frac{15921524092}{579873825}a^{7}-\frac{33580493221}{579873825}a^{6}+\frac{7616787838}{579873825}a^{5}+\frac{1114226773}{30519675}a^{4}-\frac{3389776011}{128860850}a^{3}-\frac{1686403801}{579873825}a^{2}+\frac{4655632406}{579873825}a-\frac{1098350833}{579873825}$, $\frac{90417722}{579873825}a^{15}-\frac{811809749}{1159747650}a^{14}+\frac{11227028}{6822045}a^{13}-\frac{10899238}{2577217}a^{12}+\frac{938302514}{115974765}a^{11}-\frac{88236452}{30519675}a^{10}-\frac{10440595897}{579873825}a^{9}+\frac{17197892353}{579873825}a^{8}-\frac{1992634184}{579873825}a^{7}-\frac{19382978158}{579873825}a^{6}+\frac{19352589199}{579873825}a^{5}-\frac{201281696}{30519675}a^{4}-\frac{690651389}{64430425}a^{3}+\frac{11171931529}{1159747650}a^{2}-\frac{1104753412}{579873825}a-\frac{262559209}{579873825}$, $\frac{106869977}{579873825}a^{15}-\frac{600476917}{579873825}a^{14}+\frac{18519128}{6822045}a^{13}-\frac{33001137}{5154434}a^{12}+\frac{1558710539}{115974765}a^{11}-\frac{300194657}{30519675}a^{10}-\frac{15121897402}{579873825}a^{9}+\frac{35432357698}{579873825}a^{8}-\frac{11668129694}{579873825}a^{7}-\frac{39780878953}{579873825}a^{6}+\frac{45156048559}{579873825}a^{5}-\frac{124093661}{30519675}a^{4}-\frac{2084822699}{64430425}a^{3}+\frac{7875079907}{579873825}a^{2}+\frac{1443112208}{579873825}a-\frac{1588841363}{1159747650}$ Copy content Toggle raw display
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:  \( 4466.59489456 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4466.59489456 \cdot 1}{24\cdot\sqrt{1426576071720960000}}\cr\approx \mathstrut & 0.378492085631 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 8*x^14 - 20*x^13 + 35*x^12 + 16*x^11 - 144*x^10 + 132*x^9 + 136*x^8 - 288*x^7 + 54*x^6 + 172*x^5 - 97*x^4 - 20*x^3 + 26*x^2 - 4*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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.34560.1, 4.2.8640.2, 4.2.8640.1, 4.2.34560.2, \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{24})\), 8.0.1194393600.7, 8.0.1194393600.8, 8.4.1194393600.1, 8.4.1194393600.2, 8.0.74649600.1, 8.0.1194393600.6

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 16 siblings: 16.0.3482851737600000000.5, 16.0.55725627801600000000.1, 16.0.891610044825600000000.13, 16.0.891610044825600000000.7, 16.0.891610044825600000000.14, 16.0.891610044825600000000.8, 16.8.891610044825600000000.2
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 R R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# 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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$