↑

Properties

Label 16.0.134...464.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.346\times 10^{26}$
Root discriminant \(42.96\)
Ramified primes $2,3,7$
Class number $72$ (GRH)
Class group [2, 6, 6] (GRH)
Galois group $Q_{16}:C_2$ (as 16T50)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81)
 
gp: K = bnfinit(y^16 + 40*y^14 + 608*y^12 + 4520*y^10 + 17782*y^8 + 36792*y^6 + 36288*y^4 + 12312*y^2 + 81, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81)
 

\( x^{16} + 40x^{14} + 608x^{12} + 4520x^{10} + 17782x^{8} + 36792x^{6} + 36288x^{4} + 12312x^{2} + 81 \) 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:   \(134588851614250885095358464\) \(\medspace = 2^{58}\cdot 3^{4}\cdot 7^{8}\) 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:  \(42.96\)
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:  $2^{29/8}3^{1/2}7^{1/2}\approx 56.53838276009771$
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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{2}$, $\frac{1}{12}a^{11}-\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{12}a^{8}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{12}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{36}a^{12}-\frac{1}{18}a^{10}-\frac{1}{36}a^{8}+\frac{1}{18}a^{6}+\frac{1}{36}a^{4}-\frac{1}{4}$, $\frac{1}{36}a^{13}+\frac{1}{36}a^{11}-\frac{1}{12}a^{10}+\frac{1}{18}a^{9}-\frac{1}{12}a^{8}+\frac{1}{18}a^{7}+\frac{13}{36}a^{5}-\frac{1}{3}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{19968876}a^{14}-\frac{82793}{19968876}a^{12}-\frac{1315729}{19968876}a^{10}-\frac{1012285}{19968876}a^{8}+\frac{1310863}{19968876}a^{6}+\frac{471103}{6656292}a^{4}+\frac{622637}{2218764}a^{2}-\frac{276485}{739588}$, $\frac{1}{19968876}a^{15}-\frac{82793}{19968876}a^{13}+\frac{87086}{4992219}a^{11}-\frac{1}{12}a^{10}+\frac{162947}{4992219}a^{9}-\frac{1}{12}a^{8}+\frac{1310863}{19968876}a^{7}+\frac{2689867}{6656292}a^{5}-\frac{1}{3}a^{4}-\frac{260359}{554691}a^{3}-\frac{1}{4}a^{2}-\frac{22897}{184897}a-\frac{1}{4}$ 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

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:   $\frac{56}{72351}a^{14}+\frac{2095}{72351}a^{12}+\frac{9589}{24117}a^{10}+\frac{367891}{144702}a^{8}+\frac{65674}{8039}a^{6}+\frac{966029}{72351}a^{4}+\frac{82919}{8039}a^{2}+\frac{39919}{16078}$, $\frac{41171}{4992219}a^{14}+\frac{1569665}{4992219}a^{12}+\frac{44090627}{9984438}a^{10}+\frac{286620347}{9984438}a^{8}+\frac{447405983}{4992219}a^{6}+\frac{22505392}{184897}a^{4}+\frac{18616701}{369794}a^{2}+\frac{224637}{369794}$, $\frac{5938}{4992219}a^{14}+\frac{989641}{19968876}a^{12}+\frac{3906770}{4992219}a^{10}+\frac{117991949}{19968876}a^{8}+\frac{222282473}{9984438}a^{6}+\frac{270176327}{6656292}a^{4}+\frac{12276897}{369794}a^{2}+\frac{6941115}{739588}$, $\frac{14309}{1109382}a^{14}+\frac{3213295}{6656292}a^{12}+\frac{22005653}{3328146}a^{10}+\frac{277430093}{6656292}a^{8}+\frac{210436745}{1664073}a^{6}+\frac{1119125971}{6656292}a^{4}+\frac{36653705}{554691}a^{2}-\frac{244299}{739588}$, $\frac{85243}{19968876}a^{14}+\frac{784510}{4992219}a^{12}+\frac{41770805}{19968876}a^{10}+\frac{124953055}{9984438}a^{8}+\frac{684990835}{19968876}a^{6}+\frac{118838225}{3328146}a^{4}+\frac{5199469}{2218764}a^{2}-\frac{960724}{184897}$, $\frac{120293}{19968876}a^{14}+\frac{1124021}{4992219}a^{12}+\frac{61338589}{19968876}a^{10}+\frac{95674456}{4992219}a^{8}+\frac{1133873783}{19968876}a^{6}+\frac{120279841}{1664073}a^{4}+\frac{20110905}{739588}a^{2}+\frac{61788}{184897}$, $\frac{52798}{4992219}a^{14}+\frac{1879030}{4992219}a^{12}+\frac{23828738}{4992219}a^{10}+\frac{268062121}{9984438}a^{8}+\frac{704871941}{9984438}a^{6}+\frac{268709879}{3328146}a^{4}+\frac{10813201}{369794}a^{2}+\frac{132914}{184897}$ 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:  \( 44651.6325002 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 44651.6325002 \cdot 72}{2\cdot\sqrt{134588851614250885095358464}}\cr\approx \mathstrut & 0.336569103205 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81)
 
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 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81, 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 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81);
 
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 + 40*x^14 + 608*x^12 + 4520*x^10 + 17782*x^8 + 36792*x^6 + 36288*x^4 + 12312*x^2 + 81);
 
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

$Q_{16}:C_2$ (as 16T50):

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 11 conjugacy class representatives for $Q_{16}:C_2$
Character table for $Q_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), 4.4.25088.1 x2, 4.4.7168.1 x2, \(\Q(\sqrt{2}, \sqrt{7})\), 8.8.10070523904.1

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 16 sibling: 16.0.222483611852129014137225216.1
Minimal sibling: 16.0.222483611852129014137225216.1

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.58.70$x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{10} + 30 x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$$16$$1$$58$16T50$[2, 3, 7/2, 9/2]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$