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Properties

Label 16.0.513...056.2
Degree $16$
Signature $[0, 8]$
Discriminant $5.140\times 10^{19}$
Root discriminant \(17.06\)
Ramified primes $2,3,7$
Class number $2$
Class group [2]
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 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 4*y^13 - 25*y^12 - 34*y^11 + 126*y^10 + 408*y^9 + 577*y^8 + 408*y^7 + 126*y^6 - 34*y^5 - 25*y^4 - 4*y^3 + 2*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*x^2 - 2*x + 1)
 

\( x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 25 x^{12} - 34 x^{11} + 126 x^{10} + 408 x^{9} + 577 x^{8} + \cdots + 1 \) 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:   \(51399544780206637056\) \(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{8}\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{18627}a^{14}+\frac{2768}{18627}a^{13}-\frac{754}{18627}a^{12}+\frac{2430}{6209}a^{11}+\frac{8570}{18627}a^{10}+\frac{2329}{6209}a^{9}-\frac{566}{6209}a^{8}-\frac{3617}{18627}a^{7}-\frac{566}{6209}a^{6}+\frac{2329}{6209}a^{5}+\frac{8570}{18627}a^{4}+\frac{2430}{6209}a^{3}-\frac{754}{18627}a^{2}+\frac{2768}{18627}a+\frac{1}{18627}$, $\frac{1}{5085171}a^{15}+\frac{116}{5085171}a^{14}-\frac{126632}{5085171}a^{13}+\frac{23230}{1695057}a^{12}+\frac{2152421}{5085171}a^{11}-\frac{1100915}{5085171}a^{10}-\frac{1847639}{5085171}a^{9}-\frac{45509}{5085171}a^{8}-\frac{1020595}{5085171}a^{7}+\frac{2336933}{5085171}a^{6}+\frac{2117762}{5085171}a^{5}+\frac{1730692}{5085171}a^{4}+\frac{193969}{565019}a^{3}+\frac{65168}{5085171}a^{2}+\frac{1618852}{5085171}a+\frac{934907}{5085171}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}$, which has order $2$

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:   \( -\frac{661534}{5085171} a^{15} + \frac{4620697}{5085171} a^{14} - \frac{9678412}{5085171} a^{13} + \frac{4371266}{1695057} a^{12} - \frac{283628}{5085171} a^{11} - \frac{54141616}{5085171} a^{10} - \frac{149729317}{5085171} a^{9} + \frac{191557952}{5085171} a^{8} + \frac{720094387}{5085171} a^{7} + \frac{953641231}{5085171} a^{6} + \frac{483119269}{5085171} a^{5} + \frac{57527594}{5085171} a^{4} - \frac{26766925}{1695057} a^{3} - \frac{5850686}{5085171} a^{2} - \frac{5046499}{5085171} a - \frac{713957}{5085171} \)  (order $12$) 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:   $\frac{529016}{5085171}a^{15}+\frac{1374379}{5085171}a^{14}-\frac{5240917}{5085171}a^{13}+\frac{1899358}{1695057}a^{12}-\frac{24589085}{5085171}a^{11}-\frac{77596966}{5085171}a^{10}+\frac{27441329}{5085171}a^{9}+\frac{556961654}{5085171}a^{8}+\frac{1088233339}{5085171}a^{7}+\frac{1043918077}{5085171}a^{6}+\frac{463479619}{5085171}a^{5}+\frac{45608525}{5085171}a^{4}-\frac{6705406}{1695057}a^{3}+\frac{18773605}{5085171}a^{2}+\frac{9897362}{5085171}a-\frac{3634430}{5085171}$, $\frac{9236}{1695057}a^{15}-\frac{506746}{1695057}a^{14}+\frac{1631428}{1695057}a^{13}-\frac{916974}{565019}a^{12}+\frac{1309824}{565019}a^{11}+\frac{8860532}{1695057}a^{10}+\frac{725827}{565019}a^{9}-\frac{21719378}{565019}a^{8}-\frac{102026546}{1695057}a^{7}-\frac{28454814}{565019}a^{6}-\frac{6460395}{565019}a^{5}-\frac{3317068}{1695057}a^{4}+\frac{2764546}{1695057}a^{3}-\frac{444632}{1695057}a^{2}+\frac{1164867}{565019}a-\frac{89130}{565019}$, $\frac{3283426}{5085171}a^{15}-\frac{10036471}{5085171}a^{14}+\frac{14238058}{5085171}a^{13}-\frac{2245907}{565019}a^{12}-\frac{71038714}{5085171}a^{11}-\frac{20826914}{5085171}a^{10}+\frac{502285885}{5085171}a^{9}+\frac{854253997}{5085171}a^{8}+\frac{561269333}{5085171}a^{7}-\frac{163565035}{5085171}a^{6}-\frac{275645821}{5085171}a^{5}-\frac{19475762}{5085171}a^{4}+\frac{11875865}{565019}a^{3}+\frac{16429982}{5085171}a^{2}-\frac{11032655}{5085171}a+\frac{1220372}{5085171}$, $\frac{2343053}{5085171}a^{15}-\frac{6789248}{5085171}a^{14}+\frac{10730666}{5085171}a^{13}-\frac{1933420}{565019}a^{12}-\frac{47704445}{5085171}a^{11}-\frac{29598250}{5085171}a^{10}+\frac{312620843}{5085171}a^{9}+\frac{645207971}{5085171}a^{8}+\frac{758991319}{5085171}a^{7}+\frac{471354271}{5085171}a^{6}+\frac{226132120}{5085171}a^{5}+\frac{9011102}{5085171}a^{4}-\frac{14535133}{1695057}a^{3}-\frac{50786279}{5085171}a^{2}-\frac{3644344}{5085171}a+\frac{5420638}{5085171}$, $\frac{1694146}{5085171}a^{15}-\frac{6458113}{5085171}a^{14}+\frac{11243920}{5085171}a^{13}-\frac{5506526}{1695057}a^{12}-\frac{27103984}{5085171}a^{11}+\frac{14475913}{5085171}a^{10}+\frac{271806874}{5085171}a^{9}+\frac{255826216}{5085171}a^{8}-\frac{36153451}{5085171}a^{7}-\frac{397502431}{5085171}a^{6}-\frac{242109241}{5085171}a^{5}-\frac{32702906}{5085171}a^{4}+\frac{23498513}{1695057}a^{3}+\frac{11330762}{5085171}a^{2}+\frac{7681348}{5085171}a+\frac{6426356}{5085171}$, $\frac{964813}{5085171}a^{15}+\frac{650060}{5085171}a^{14}-\frac{7010219}{5085171}a^{13}+\frac{3756709}{1695057}a^{12}-\frac{47258728}{5085171}a^{11}-\frac{77432990}{5085171}a^{10}+\frac{121121320}{5085171}a^{9}+\frac{783239815}{5085171}a^{8}+\frac{1070163893}{5085171}a^{7}+\frac{676603751}{5085171}a^{6}-\frac{82628782}{5085171}a^{5}-\frac{175330301}{5085171}a^{4}-\frac{19377080}{1695057}a^{3}+\frac{47058656}{5085171}a^{2}-\frac{26251766}{5085171}a+\frac{6791039}{5085171}$, $\frac{4166770}{5085171}a^{15}-\frac{12665650}{5085171}a^{14}+\frac{20254075}{5085171}a^{13}-\frac{3548271}{565019}a^{12}-\frac{81965917}{5085171}a^{11}-\frac{41324279}{5085171}a^{10}+\frac{584340760}{5085171}a^{9}+\frac{1055081470}{5085171}a^{8}+\frac{1092729962}{5085171}a^{7}+\frac{534791267}{5085171}a^{6}+\frac{321632567}{5085171}a^{5}+\frac{86468818}{5085171}a^{4}+\frac{2126552}{565019}a^{3}-\frac{58935679}{5085171}a^{2}+\frac{6833746}{5085171}a-\frac{8100310}{5085171}$ 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:  \( 6495.33719206 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6495.33719206 \cdot 2}{12\cdot\sqrt{51399544780206637056}}\cr\approx \mathstrut & 0.366783475699 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - 4*x^13 - 25*x^12 - 34*x^11 + 126*x^10 + 408*x^9 + 577*x^8 + 408*x^7 + 126*x^6 - 34*x^5 - 25*x^4 - 4*x^3 + 2*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

$C_2^2\wr C_2$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.3024.2, 4.0.189.1, 4.0.1008.1, 4.0.1008.2, 4.2.84672.5 x2, \(\Q(\zeta_{12})\), 4.0.21168.1 x2, 8.0.7169347584.11, 8.0.9144576.3, 8.0.16257024.2, 8.0.7169347584.8, 8.0.448084224.2, 8.0.7169347584.4, 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, 16.0.21407557176262656.1, 16.0.200779471797682176.1, 16.0.51399544780206637056.8, 16.0.51399544780206637056.4, 16.0.51399544780206637056.7
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 ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 3.8.6.2$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}$
\(7\) 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}$