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Properties

Label 16.0.444...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.450\times 10^{19}$
Root discriminant \(16.91\)
Ramified primes $3,5,11$
Class number $2$
Class group [2]
Galois group $D_{8}$ (as 16T13)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1)
 
gp: K = bnfinit(y^16 - 7*y^15 + 26*y^14 - 62*y^13 + 86*y^12 - 71*y^11 + 78*y^10 - 126*y^9 + 175*y^8 - 126*y^7 + 78*y^6 - 71*y^5 + 86*y^4 - 62*y^3 + 26*y^2 - 7*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*x + 1)
 

\( x^{16} - 7 x^{15} + 26 x^{14} - 62 x^{13} + 86 x^{12} - 71 x^{11} + 78 x^{10} - 126 x^{9} + 175 x^{8} + \cdots + 1 \) 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:   \(44499647686531640625\) \(\medspace = 3^{12}\cdot 5^{8}\cdot 11^{8}\) 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:  \(16.91\)
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:  $3^{3/4}5^{1/2}11^{1/2}\approx 16.90527678711191$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}+\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{4}{11}a^{7}+\frac{1}{11}a^{6}+\frac{1}{11}a^{4}+\frac{4}{11}a^{3}-\frac{4}{11}a^{2}+\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{11}a^{11}+\frac{4}{11}a^{9}+\frac{2}{11}a^{8}+\frac{3}{11}a^{7}-\frac{5}{11}a^{6}+\frac{1}{11}a^{5}-\frac{1}{11}a^{4}-\frac{2}{11}a^{3}+\frac{3}{11}a^{2}-\frac{2}{11}a-\frac{5}{11}$, $\frac{1}{33}a^{12}+\frac{1}{33}a^{11}+\frac{1}{33}a^{10}+\frac{2}{33}a^{9}-\frac{5}{33}a^{8}-\frac{14}{33}a^{7}-\frac{7}{33}a^{6}-\frac{1}{3}a^{5}+\frac{16}{33}a^{4}+\frac{1}{3}a^{3}+\frac{13}{33}a^{2}-\frac{1}{3}a-\frac{8}{33}$, $\frac{1}{33}a^{13}+\frac{1}{33}a^{10}-\frac{7}{33}a^{9}-\frac{3}{11}a^{8}+\frac{7}{33}a^{7}-\frac{4}{33}a^{6}-\frac{2}{11}a^{5}-\frac{5}{33}a^{4}+\frac{2}{33}a^{3}+\frac{3}{11}a^{2}+\frac{1}{11}a+\frac{8}{33}$, $\frac{1}{165}a^{14}-\frac{2}{165}a^{11}+\frac{2}{165}a^{10}-\frac{5}{11}a^{9}+\frac{31}{165}a^{8}+\frac{56}{165}a^{7}+\frac{17}{55}a^{6}+\frac{5}{33}a^{5}+\frac{47}{165}a^{4}+\frac{6}{55}a^{3}-\frac{5}{11}a^{2}-\frac{8}{33}a-\frac{3}{55}$, $\frac{1}{5445}a^{15}-\frac{8}{5445}a^{14}-\frac{13}{1089}a^{13}+\frac{1}{1815}a^{12}-\frac{82}{5445}a^{11}+\frac{4}{495}a^{10}-\frac{1319}{5445}a^{9}-\frac{2107}{5445}a^{8}-\frac{1447}{5445}a^{7}-\frac{428}{5445}a^{6}+\frac{97}{495}a^{5}-\frac{1798}{5445}a^{4}-\frac{106}{605}a^{3}-\frac{244}{1089}a^{2}-\frac{404}{5445}a+\frac{1222}{5445}$ 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:   \( -1 \)  (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:   $\frac{4517}{5445}a^{15}-\frac{31747}{5445}a^{14}+\frac{23548}{1089}a^{13}-\frac{92998}{1815}a^{12}+\frac{379663}{5445}a^{11}-\frac{26449}{495}a^{10}+\frac{302837}{5445}a^{9}-\frac{106231}{1089}a^{8}+\frac{151994}{1089}a^{7}-\frac{511042}{5445}a^{6}+\frac{23819}{495}a^{5}-\frac{264068}{5445}a^{4}+\frac{40266}{605}a^{3}-\frac{50471}{1089}a^{2}+\frac{77582}{5445}a-\frac{11752}{5445}$, $a$, $\frac{54}{121}a^{15}-\frac{1599}{605}a^{14}+\frac{3121}{363}a^{13}-\frac{2115}{121}a^{12}+\frac{10013}{605}a^{11}-\frac{1289}{165}a^{10}+\frac{7532}{363}a^{9}-\frac{18714}{605}a^{8}+\frac{65243}{1815}a^{7}-\frac{4712}{1815}a^{6}+\frac{149}{11}a^{5}-\frac{23344}{1815}a^{4}+\frac{31129}{1815}a^{3}+\frac{76}{121}a^{2}-\frac{201}{121}a+\frac{403}{1815}$, $\frac{94}{605}a^{15}-\frac{383}{363}a^{14}+\frac{1405}{363}a^{13}-\frac{16424}{1815}a^{12}+\frac{7263}{605}a^{11}-\frac{302}{33}a^{10}+\frac{5814}{605}a^{9}-\frac{10376}{605}a^{8}+\frac{49612}{1815}a^{7}-\frac{1959}{121}a^{6}+\frac{548}{55}a^{5}-\frac{12839}{1815}a^{4}+\frac{1801}{121}a^{3}-\frac{2698}{363}a^{2}+\frac{5312}{1815}a-\frac{344}{363}$, $\frac{4498}{5445}a^{15}-\frac{5989}{1089}a^{14}+\frac{21452}{1089}a^{13}-\frac{81962}{1815}a^{12}+\frac{315551}{5445}a^{11}-\frac{4342}{99}a^{10}+\frac{296548}{5445}a^{9}-\frac{466702}{5445}a^{8}+\frac{622628}{5445}a^{7}-\frac{74083}{1089}a^{6}+\frac{23926}{495}a^{5}-\frac{234361}{5445}a^{4}+\frac{19186}{363}a^{3}-\frac{33064}{1089}a^{2}+\frac{65788}{5445}a-\frac{3463}{1089}$, $\frac{3904}{5445}a^{15}-\frac{5428}{1089}a^{14}+\frac{19802}{1089}a^{13}-\frac{76561}{1815}a^{12}+\frac{299018}{5445}a^{11}-\frac{3793}{99}a^{10}+\frac{238699}{5445}a^{9}-\frac{459871}{5445}a^{8}+\frac{616094}{5445}a^{7}-\frac{72862}{1089}a^{6}+\frac{15988}{495}a^{5}-\frac{236638}{5445}a^{4}+\frac{6905}{121}a^{3}-\frac{35308}{1089}a^{2}+\frac{44239}{5445}a-\frac{1285}{1089}$, $\frac{2522}{5445}a^{15}-\frac{3032}{1089}a^{14}+\frac{9982}{1089}a^{13}-\frac{34273}{1815}a^{12}+\frac{100459}{5445}a^{11}-\frac{833}{99}a^{10}+\frac{106307}{5445}a^{9}-\frac{167768}{5445}a^{8}+\frac{213382}{5445}a^{7}-\frac{7409}{1089}a^{6}+\frac{6509}{495}a^{5}-\frac{56654}{5445}a^{4}+\frac{2253}{121}a^{3}-\frac{4439}{1089}a^{2}+\frac{8567}{5445}a-\frac{314}{1089}$ 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:  \( 1120.36474159 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1120.36474159 \cdot 2}{2\cdot\sqrt{44499647686531640625}}\cr\approx \mathstrut & 0.407962358155 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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 - 7*x^15 + 26*x^14 - 62*x^13 + 86*x^12 - 71*x^11 + 78*x^10 - 126*x^9 + 175*x^8 - 126*x^7 + 78*x^6 - 71*x^5 + 86*x^4 - 62*x^3 + 26*x^2 - 7*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

$D_8$ (as 16T13):

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 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.2.2475.1 x2, 4.0.5445.1 x2, 8.0.741200625.3, 8.2.606436875.2 x4, 8.0.1334161125.1 x4

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 8 siblings: 8.2.606436875.2, 8.0.1334161125.1
Minimal sibling: 8.2.606436875.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) 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}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(11\) Copy content Toggle raw display 11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 11$$2$$1$$1$$C_2$$[\ ]_{2}$