Properties

Label 16.0.165...784.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.659\times 10^{18}$
Root discriminant \(13.76\)
Ramified primes $2,17$
Class number $1$
Class group trivial
Galois group $D_8:C_2$ (as 16T45)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^14 - 20*y^13 - 18*y^12 + 36*y^11 + 166*y^10 + 352*y^9 + 518*y^8 + 576*y^7 + 502*y^6 + 348*y^5 + 198*y^4 + 92*y^3 + 34*y^2 + 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*x + 1)
 

\( x^{16} - 6 x^{14} - 20 x^{13} - 18 x^{12} + 36 x^{11} + 166 x^{10} + 352 x^{9} + 518 x^{8} + 576 x^{7} + 502 x^{6} + 348 x^{5} + 198 x^{4} + 92 x^{3} + 34 x^{2} + 8 x + 1 \) 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:   \(1658721111359094784\) \(\medspace = 2^{36}\cdot 17^{6}\) 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:  \(13.76\)
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:  $2^{9/4}17^{1/2}\approx 19.61290618356907$
Ramified primes:   \(2\), \(17\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{34}a^{14}-\frac{3}{34}a^{13}+\frac{3}{17}a^{12}+\frac{2}{17}a^{11}+\frac{5}{34}a^{10}-\frac{1}{34}a^{9}-\frac{3}{34}a^{8}-\frac{8}{17}a^{7}+\frac{13}{34}a^{6}+\frac{13}{34}a^{5}-\frac{4}{17}a^{4}-\frac{7}{17}a^{3}-\frac{5}{34}a^{2}-\frac{3}{34}a+\frac{11}{34}$, $\frac{1}{487934}a^{15}-\frac{736}{243967}a^{14}+\frac{57181}{487934}a^{13}-\frac{8094}{243967}a^{12}-\frac{25673}{243967}a^{11}+\frac{52035}{487934}a^{10}-\frac{23560}{243967}a^{9}-\frac{5871}{243967}a^{8}+\frac{192701}{487934}a^{7}-\frac{39768}{243967}a^{6}-\frac{127123}{487934}a^{5}-\frac{48871}{243967}a^{4}+\frac{111456}{243967}a^{3}-\frac{220373}{487934}a^{2}-\frac{14808}{243967}a-\frac{44842}{243967}$ 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

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:   \( \frac{259881}{487934} a^{15} + \frac{38477}{487934} a^{14} - \frac{1611437}{487934} a^{13} - \frac{5339603}{487934} a^{12} - \frac{5269823}{487934} a^{11} + \frac{4651518}{243967} a^{10} + \frac{44445819}{487934} a^{9} + \frac{2831010}{14351} a^{8} + \frac{142986239}{487934} a^{7} + \frac{160078999}{487934} a^{6} + \frac{140535689}{487934} a^{5} + \frac{97065813}{487934} a^{4} + \frac{54431221}{487934} a^{3} + \frac{12339536}{243967} a^{2} + \frac{8200939}{487934} a + \frac{834596}{243967} \)  (order $8$) 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:   $\frac{145077}{243967}a^{15}-\frac{96570}{243967}a^{14}-\frac{894779}{243967}a^{13}-\frac{4479919}{487934}a^{12}-\frac{624569}{243967}a^{11}+\frac{7014102}{243967}a^{10}+\frac{19946745}{243967}a^{9}+\frac{34081893}{243967}a^{8}+\frac{40544651}{243967}a^{7}+\frac{34519678}{243967}a^{6}+\frac{21344427}{243967}a^{5}+\frac{17506887}{487934}a^{4}+\frac{147717}{14351}a^{3}+\frac{23850}{14351}a^{2}-\frac{298509}{243967}a-\frac{239452}{243967}$, $\frac{9821}{243967}a^{15}+\frac{124104}{243967}a^{14}-\frac{109188}{243967}a^{13}-\frac{992189}{243967}a^{12}-\frac{4585225}{487934}a^{11}-\frac{606117}{243967}a^{10}+\frac{7660260}{243967}a^{9}+\frac{44170935}{487934}a^{8}+\frac{38065950}{243967}a^{7}+\frac{46398657}{243967}a^{6}+\frac{41608162}{243967}a^{5}+\frac{27625532}{243967}a^{4}+\frac{27429569}{487934}a^{3}+\frac{5602318}{243967}a^{2}+\frac{2073876}{243967}a+\frac{801427}{487934}$, $\frac{669}{2159}a^{15}+\frac{625}{2159}a^{14}-\frac{9957}{4318}a^{13}-\frac{16484}{2159}a^{12}-\frac{18818}{2159}a^{11}+\frac{27893}{2159}a^{10}+\frac{277291}{4318}a^{9}+\frac{296478}{2159}a^{8}+\frac{435768}{2159}a^{7}+\frac{474713}{2159}a^{6}+\frac{789323}{4318}a^{5}+\frac{258680}{2159}a^{4}+\frac{133887}{2159}a^{3}+\frac{58103}{2159}a^{2}+\frac{35765}{4318}a+\frac{3316}{2159}$, $\frac{105849}{243967}a^{15}-\frac{43974}{243967}a^{14}-\frac{1472017}{487934}a^{13}-\frac{3464419}{487934}a^{12}-\frac{929095}{487934}a^{11}+\frac{11746107}{487934}a^{10}+\frac{31355063}{487934}a^{9}+\frac{25161051}{243967}a^{8}+\frac{28030193}{243967}a^{7}+\frac{21017036}{243967}a^{6}+\frac{19157779}{487934}a^{5}+\frac{137749}{28702}a^{4}-\frac{2994957}{487934}a^{3}-\frac{3033595}{487934}a^{2}-\frac{3068553}{487934}a-\frac{650627}{243967}$, $\frac{190745}{487934}a^{15}-\frac{193401}{243967}a^{14}-\frac{746023}{487934}a^{13}-\frac{827457}{243967}a^{12}+\frac{1813267}{487934}a^{11}+\frac{3526249}{243967}a^{10}+\frac{7081089}{243967}a^{9}+\frac{10449567}{243967}a^{8}+\frac{21492975}{487934}a^{7}+\frac{9489974}{243967}a^{6}+\frac{16106231}{487934}a^{5}+\frac{6136297}{243967}a^{4}+\frac{517817}{28702}a^{3}+\frac{103617}{14351}a^{2}+\frac{600251}{243967}a+\frac{124432}{243967}$, $\frac{238620}{243967}a^{15}-\frac{352339}{243967}a^{14}-\frac{2250611}{487934}a^{13}-\frac{2792090}{243967}a^{12}+\frac{859141}{243967}a^{11}+\frac{9764723}{243967}a^{10}+\frac{1426922}{14351}a^{9}+\frac{39148377}{243967}a^{8}+\frac{44374143}{243967}a^{7}+\frac{38304277}{243967}a^{6}+\frac{52923971}{487934}a^{5}+\frac{13869173}{243967}a^{4}+\frac{7402606}{243967}a^{3}+\frac{2054716}{243967}a^{2}+\frac{782772}{243967}a+\frac{202597}{243967}$, $\frac{682267}{487934}a^{15}-\frac{57097}{487934}a^{14}-\frac{4416347}{487934}a^{13}-\frac{6447897}{243967}a^{12}-\frac{4721789}{243967}a^{11}+\frac{29564287}{487934}a^{10}+\frac{55265857}{243967}a^{9}+\frac{216636663}{487934}a^{8}+\frac{17538839}{28702}a^{7}+\frac{306195329}{487934}a^{6}+\frac{243584441}{487934}a^{5}+\frac{76461918}{243967}a^{4}+\frac{38721023}{243967}a^{3}+\frac{32309807}{487934}a^{2}+\frac{4557425}{243967}a+\frac{1231013}{487934}$ 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:  \( 1382.13723796 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1382.13723796 \cdot 1}{8\cdot\sqrt{1658721111359094784}}\cr\approx \mathstrut & 0.325846786101 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*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^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*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^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*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^14 - 20*x^13 - 18*x^12 + 36*x^11 + 166*x^10 + 352*x^9 + 518*x^8 + 576*x^7 + 502*x^6 + 348*x^5 + 198*x^4 + 92*x^3 + 34*x^2 + 8*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

$D_8:C_2$ (as 16T45):

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 $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.272.1, 4.0.4352.2, \(\Q(\zeta_{8})\), 8.0.18939904.4

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: 8.2.342102016.2, 8.2.342102016.1
Degree 16 siblings: 16.4.119842600295694598144.2, 16.0.119842600295694598144.2, 16.0.1872540629620228096.4
Minimal sibling: 8.2.342102016.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.16.36.1$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$$8$$2$$36$$D_4\times C_2$$[2, 2, 3]^{2}$
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$