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Properties

Label 16.0.187...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.873\times 10^{25}$
Root discriminant \(37.98\)
Ramified primes $2,5,29,1109$
Class number $136$ (GRH)
Class group [136] (GRH)
Galois group $C_2\wr D_4$ (as 16T1445)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841)
 
gp: K = bnfinit(y^16 + 27*y^14 + 293*y^12 + 1661*y^10 + 5333*y^8 + 9812*y^6 + 9932*y^4 + 4901*y^2 + 841, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841)
 

\( x^{16} + 27x^{14} + 293x^{12} + 1661x^{10} + 5333x^{8} + 9812x^{6} + 9932x^{4} + 4901x^{2} + 841 \) 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:   \(18727984661882905600000000\) \(\medspace = 2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 1109^{2}\) 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:  \(37.98\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(29\), \(1109\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{2}{7}a^{10}-\frac{1}{7}a^{8}+\frac{1}{7}a^{6}+\frac{1}{7}a^{4}+\frac{2}{7}a^{2}-\frac{1}{7}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{11}-\frac{1}{7}a^{9}+\frac{1}{7}a^{7}+\frac{1}{7}a^{5}+\frac{2}{7}a^{3}-\frac{1}{7}a$, $\frac{1}{120379}a^{14}+\frac{3971}{120379}a^{12}-\frac{56141}{120379}a^{10}-\frac{58659}{120379}a^{8}-\frac{28916}{120379}a^{6}+\frac{15612}{120379}a^{4}+\frac{1203}{120379}a^{2}+\frac{185}{593}$, $\frac{1}{120379}a^{15}+\frac{3971}{120379}a^{13}-\frac{56141}{120379}a^{11}-\frac{58659}{120379}a^{9}-\frac{28916}{120379}a^{7}+\frac{15612}{120379}a^{5}+\frac{1203}{120379}a^{3}+\frac{185}{593}a$ 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

$C_{136}$, which has order $136$ (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:   \( -1 \)  (order $2$) 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:   $\frac{16}{593}a^{14}+\frac{2967}{4151}a^{12}+\frac{30623}{4151}a^{10}+\frac{156591}{4151}a^{8}+\frac{416660}{4151}a^{6}+\frac{555428}{4151}a^{4}+\frac{330426}{4151}a^{2}+\frac{65241}{4151}$, $\frac{735}{17197}a^{14}+\frac{121138}{120379}a^{12}+\frac{1095975}{120379}a^{10}+\frac{4890743}{120379}a^{8}+\frac{11365644}{120379}a^{6}+\frac{13186652}{120379}a^{4}+\frac{6498981}{120379}a^{2}+\frac{28291}{4151}$, $\frac{1934}{120379}a^{14}+\frac{44446}{120379}a^{12}+\frac{383498}{120379}a^{10}+\frac{1567030}{120379}a^{8}+\frac{3130954}{120379}a^{6}+\frac{2936363}{120379}a^{4}+\frac{1380767}{120379}a^{2}+\frac{15709}{4151}$, $\frac{95}{17197}a^{14}+\frac{9574}{120379}a^{12}+\frac{18119}{120379}a^{10}-\frac{263394}{120379}a^{8}-\frac{1395851}{120379}a^{6}-\frac{2240618}{120379}a^{4}-\frac{1091675}{120379}a^{2}-\frac{2826}{4151}$, $\frac{345}{17197}a^{14}+\frac{62827}{120379}a^{12}+\frac{654119}{120379}a^{10}+\frac{3532786}{120379}a^{8}+\frac{10563835}{120379}a^{6}+\frac{16980595}{120379}a^{4}+\frac{12741929}{120379}a^{2}+\frac{97788}{4151}$, $\frac{1269}{120379}a^{14}+\frac{34872}{120379}a^{12}+\frac{52197}{17197}a^{10}+\frac{1830424}{120379}a^{8}+\frac{4526805}{120379}a^{6}+\frac{5176981}{120379}a^{4}+\frac{353206}{17197}a^{2}+\frac{22686}{4151}$, $\frac{7124}{120379}a^{14}+\frac{172309}{120379}a^{12}+\frac{1618663}{120379}a^{10}+\frac{7601458}{120379}a^{8}+\frac{18921979}{120379}a^{6}+\frac{24117083}{120379}a^{4}+\frac{13608893}{120379}a^{2}+\frac{74997}{4151}$ 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:  \( 4572.22444832 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4572.22444832 \cdot 136}{2\cdot\sqrt{18727984661882905600000000}}\cr\approx \mathstrut & 0.174513841524 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841)
 
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 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841, 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 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841);
 
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 + 27*x^14 + 293*x^12 + 1661*x^10 + 5333*x^8 + 9812*x^6 + 9932*x^4 + 4901*x^2 + 841);
 
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

$C_2\wr D_4$ (as 16T1445):

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 2048
The 74 conjugacy class representatives for $C_2\wr D_4$
Character table for $C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.582918125.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.4032060185764742044057600000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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$$2$$8$$16$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(29\) Copy content Toggle raw display 29.4.2.2$x^{4} - 696 x^{2} + 1682$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.4.1$x^{8} + 2784 x^{7} + 2906616 x^{6} + 1348864734 x^{5} + 234834277018 x^{4} + 41857830864 x^{3} + 492109772617 x^{2} + 3561769809750 x + 616658760166$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(1109\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$