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Properties

Label 16.0.897...896.2
Degree $16$
Signature $[0, 8]$
Discriminant $8.979\times 10^{19}$
Root discriminant \(17.66\)
Ramified primes $2,3,13$
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 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^15 + 18*y^14 - 36*y^13 + 59*y^12 - 114*y^11 + 270*y^10 - 492*y^9 + 609*y^8 - 492*y^7 + 270*y^6 - 114*y^5 + 59*y^4 - 36*y^3 + 18*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
 

\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 59 x^{12} - 114 x^{11} + 270 x^{10} - 492 x^{9} + 609 x^{8} + \cdots + 1 \) 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:   \(89791815397090000896\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 13^{8}\) 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:  \(17.66\)
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:  $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$
Ramified primes:   \(2\), \(3\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a$, $\frac{1}{2061}a^{14}-\frac{3}{229}a^{13}-\frac{332}{2061}a^{12}+\frac{31}{687}a^{11}+\frac{30}{229}a^{10}+\frac{34}{229}a^{9}-\frac{27}{229}a^{8}+\frac{290}{687}a^{7}-\frac{310}{687}a^{6}+\frac{331}{687}a^{5}-\frac{139}{687}a^{4}-\frac{66}{229}a^{3}-\frac{790}{2061}a^{2}-\frac{238}{687}a-\frac{457}{2061}$, $\frac{1}{35037}a^{15}-\frac{7}{35037}a^{14}-\frac{2933}{35037}a^{13}+\frac{1010}{35037}a^{12}+\frac{23}{11679}a^{11}+\frac{1673}{11679}a^{10}+\frac{814}{11679}a^{9}+\frac{320}{3893}a^{8}+\frac{456}{3893}a^{7}+\frac{4894}{11679}a^{6}+\frac{1214}{11679}a^{5}+\frac{686}{11679}a^{4}-\frac{6487}{35037}a^{3}+\frac{11653}{35037}a^{2}+\frac{12743}{35037}a+\frac{8722}{35037}$ 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 $2$

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:   \( \frac{40849}{35037} a^{15} - \frac{76807}{11679} a^{14} + \frac{654586}{35037} a^{13} - \frac{415411}{11679} a^{12} + \frac{221200}{3893} a^{11} - \frac{443461}{3893} a^{10} + \frac{3223306}{11679} a^{9} - \frac{5594257}{11679} a^{8} + \frac{6416906}{11679} a^{7} - \frac{4608509}{11679} a^{6} + \frac{742459}{3893} a^{5} - \frac{280640}{3893} a^{4} + \frac{1493708}{35037} a^{3} - \frac{94212}{3893} a^{2} + \frac{433718}{35037} a - \frac{35491}{11679} \)  (order $4$) 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:   $\frac{15866}{35037}a^{15}-\frac{65009}{35037}a^{14}+\frac{128849}{35037}a^{13}-\frac{164219}{35037}a^{12}+\frac{75748}{11679}a^{11}-\frac{241297}{11679}a^{10}+\frac{642743}{11679}a^{9}-\frac{2732}{51}a^{8}+\frac{34937}{3893}a^{7}+\frac{135464}{3893}a^{6}-\frac{181763}{11679}a^{5}-\frac{17599}{11679}a^{4}+\frac{129490}{35037}a^{3}+\frac{111254}{35037}a^{2}-\frac{11867}{35037}a+\frac{9797}{35037}$, $\frac{4891}{11679}a^{15}-\frac{80696}{35037}a^{14}+\frac{74777}{11679}a^{13}-\frac{419291}{35037}a^{12}+\frac{221879}{11679}a^{11}-\frac{453104}{11679}a^{10}+\frac{1104979}{11679}a^{9}-\frac{625147}{3893}a^{8}+\frac{696909}{3893}a^{7}-\frac{483011}{3893}a^{6}+\frac{710342}{11679}a^{5}-\frac{305461}{11679}a^{4}+\frac{69184}{3893}a^{3}-\frac{301564}{35037}a^{2}+\frac{46207}{11679}a-\frac{40849}{35037}$, $\frac{1442}{35037}a^{15}-\frac{13324}{35037}a^{14}+\frac{38906}{35037}a^{13}-\frac{63958}{35037}a^{12}+\frac{26468}{11679}a^{11}-\frac{50528}{11679}a^{10}+\frac{159160}{11679}a^{9}-\frac{312229}{11679}a^{8}+\frac{195068}{11679}a^{7}+\frac{46654}{11679}a^{6}-\frac{190574}{11679}a^{5}+\frac{87311}{11679}a^{4}-\frac{147938}{35037}a^{3}+\frac{14902}{35037}a^{2}+\frac{9796}{35037}a+\frac{15076}{35037}$, $\frac{1663}{35037}a^{15}+\frac{449}{11679}a^{14}-\frac{42779}{35037}a^{13}+\frac{17655}{3893}a^{12}-\frac{36012}{3893}a^{11}+\frac{155444}{11679}a^{10}-\frac{268543}{11679}a^{9}+\frac{255451}{3893}a^{8}-\frac{1596562}{11679}a^{7}+\frac{638930}{3893}a^{6}-\frac{1277333}{11679}a^{5}+\frac{421633}{11679}a^{4}-\frac{423661}{35037}a^{3}+\frac{154760}{11679}a^{2}-\frac{321415}{35037}a+\frac{22286}{11679}$, $\frac{67705}{35037}a^{15}-\frac{354391}{35037}a^{14}+\frac{929308}{35037}a^{13}-\frac{1636921}{35037}a^{12}+\frac{840631}{11679}a^{11}-\frac{1807907}{11679}a^{10}+\frac{4526840}{11679}a^{9}-\frac{7219523}{11679}a^{8}+\frac{7150220}{11679}a^{7}-\frac{1352737}{3893}a^{6}+\frac{542949}{3893}a^{5}-\frac{254060}{3893}a^{4}+\frac{1760852}{35037}a^{3}-\frac{806018}{35037}a^{2}+\frac{301418}{35037}a-\frac{57908}{35037}$, $\frac{2020}{11679}a^{15}-\frac{58400}{35037}a^{14}+\frac{78125}{11679}a^{13}-\frac{568373}{35037}a^{12}+\frac{333061}{11679}a^{11}-\frac{557374}{11679}a^{10}+\frac{1208446}{11679}a^{9}-\frac{2659450}{11679}a^{8}+\frac{4099682}{11679}a^{7}-\frac{4007120}{11679}a^{6}+\frac{2238752}{11679}a^{5}-\frac{671843}{11679}a^{4}+\frac{232709}{11679}a^{3}-\frac{835741}{35037}a^{2}+\frac{148057}{11679}a-\frac{82102}{35037}$, $\frac{10418}{11679}a^{15}-\frac{174527}{35037}a^{14}+\frac{158603}{11679}a^{13}-\frac{853514}{35037}a^{12}+\frac{432205}{11679}a^{11}-\frac{894280}{11679}a^{10}+\frac{754483}{3893}a^{9}-\frac{3801850}{11679}a^{8}+\frac{3815255}{11679}a^{7}-\frac{2037704}{11679}a^{6}+\frac{454381}{11679}a^{5}-\frac{52958}{11679}a^{4}+\frac{154186}{11679}a^{3}-\frac{406882}{35037}a^{2}+\frac{20832}{3893}a-\frac{20188}{35037}$ 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:  \( 3239.8094211 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3239.8094211 \cdot 2}{4\cdot\sqrt{89791815397090000896}}\cr\approx \mathstrut & 0.41525038052 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*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 - 6*x^15 + 18*x^14 - 36*x^13 + 59*x^12 - 114*x^11 + 270*x^10 - 492*x^9 + 609*x^8 - 492*x^7 + 270*x^6 - 114*x^5 + 59*x^4 - 36*x^3 + 18*x^2 - 6*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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), 4.0.7488.4 x2, 4.2.24336.2 x2, 4.4.8112.1, 4.0.8112.1, \(\Q(i, \sqrt{13})\), 4.2.507.1, 4.2.8112.1, 8.0.65804544.2, 8.0.9475854336.4, 8.0.1052872704.3, 8.4.592240896.1, 8.0.592240896.2, 8.0.9475854336.5, 8.4.9475854336.1

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: 8.4.1052872704.1, 8.4.9475854336.1, 8.4.65804544.1, 8.0.65804544.1, 8.0.9475854336.5, 8.0.1052872704.1, 8.4.592240896.1, 8.0.592240896.2
Degree 16 siblings: 16.0.350749278894882816.3, 16.8.89791815397090000896.1, 16.0.89791815397090000896.9, 16.0.531312517142544384.3, 16.0.1108540930828271616.1, 16.0.89791815397090000896.3
Minimal sibling: 8.4.65804544.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(13\) Copy content Toggle raw display 13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$