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Properties

Label 16.0.6634204312890625.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.634\times 10^{15}$
Root discriminant \(9.75\)
Ramified primes $5,19$
Class number $1$
Class group trivial
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 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*x^3 + 34*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 7*y^15 + 27*y^14 - 73*y^13 + 160*y^12 - 304*y^11 + 509*y^10 - 730*y^9 + 869*y^8 - 839*y^7 + 654*y^6 - 416*y^5 + 221*y^4 - 98*y^3 + 34*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*x^3 + 34*x^2 - 8*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*x^3 + 34*x^2 - 8*x + 1)
 

\( x^{16} - 7 x^{15} + 27 x^{14} - 73 x^{13} + 160 x^{12} - 304 x^{11} + 509 x^{10} - 730 x^{9} + 869 x^{8} + \cdots + 1 \) 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:   \(6634204312890625\) \(\medspace = 5^{8}\cdot 19^{8}\) 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:  \(9.75\)
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:  $5^{1/2}19^{1/2}\approx 9.746794344808963$
Ramified primes:   \(5\), \(19\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}+\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{85}a^{15}-\frac{1}{17}a^{14}+\frac{1}{5}a^{13}-\frac{39}{85}a^{12}-\frac{3}{85}a^{11}+\frac{6}{17}a^{10}-\frac{26}{85}a^{9}-\frac{1}{5}a^{8}-\frac{3}{17}a^{7}-\frac{19}{85}a^{6}+\frac{21}{85}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{38}{85}a^{2}+\frac{5}{17}a+\frac{42}{85}$ 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:   \( -1 \)  (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:   $a^{15}-\frac{21}{5}a^{14}+\frac{48}{5}a^{13}-\frac{61}{5}a^{12}+\frac{52}{5}a^{11}+\frac{12}{5}a^{10}-\frac{216}{5}a^{9}+\frac{719}{5}a^{8}-278a^{7}+364a^{6}-\frac{1631}{5}a^{5}+\frac{1072}{5}a^{4}-\frac{552}{5}a^{3}+\frac{258}{5}a^{2}-\frac{82}{5}a+\frac{11}{5}$, $\frac{331}{85}a^{15}-\frac{2029}{85}a^{14}+\frac{417}{5}a^{13}-\frac{17533}{85}a^{12}+\frac{7227}{17}a^{11}-\frac{65397}{85}a^{10}+\frac{20763}{17}a^{9}-\frac{8083}{5}a^{8}+\frac{29250}{17}a^{7}-\frac{122059}{85}a^{6}+\frac{80442}{85}a^{5}-\frac{2538}{5}a^{4}+230a^{3}-\frac{1408}{17}a^{2}+\frac{1577}{85}a-\frac{174}{85}$, $\frac{5}{17}a^{15}-\frac{346}{85}a^{14}+\frac{104}{5}a^{13}-\frac{5701}{85}a^{12}+\frac{13627}{85}a^{11}-\frac{27368}{85}a^{10}+\frac{48429}{85}a^{9}-\frac{4408}{5}a^{8}+\frac{19305}{17}a^{7}-\frac{19917}{17}a^{6}+\frac{80119}{85}a^{5}-\frac{2979}{5}a^{4}+\frac{1534}{5}a^{3}-\frac{11392}{85}a^{2}+\frac{3668}{85}a-\frac{599}{85}$, $a^{15}-6a^{14}+20a^{13}-47a^{12}+93a^{11}-164a^{10}+252a^{9}-314a^{8}+303a^{7}-222a^{6}+129a^{5}-65a^{4}+27a^{3}-6a^{2}+a-1$, $\frac{32}{17}a^{15}-\frac{868}{85}a^{14}+\frac{162}{5}a^{13}-\frac{6138}{85}a^{12}+\frac{11726}{85}a^{11}-\frac{19799}{85}a^{10}+\frac{28922}{85}a^{9}-\frac{1929}{5}a^{8}+\frac{5300}{17}a^{7}-\frac{2461}{17}a^{6}+\frac{572}{85}a^{5}+\frac{223}{5}a^{4}-\frac{188}{5}a^{3}+\frac{2204}{85}a^{2}-\frac{1236}{85}a+\frac{328}{85}$, $\frac{67}{17}a^{15}-\frac{2202}{85}a^{14}+\frac{473}{5}a^{13}-\frac{20647}{85}a^{12}+\frac{43484}{85}a^{11}-\frac{80101}{85}a^{10}+\frac{129823}{85}a^{9}-\frac{10441}{5}a^{8}+\frac{39404}{17}a^{7}-\frac{34610}{17}a^{6}+\frac{120323}{85}a^{5}-\frac{3998}{5}a^{4}+\frac{1908}{5}a^{3}-\frac{12549}{85}a^{2}+\frac{3156}{85}a-\frac{363}{85}$, $\frac{462}{85}a^{15}-\frac{3024}{85}a^{14}+\frac{648}{5}a^{13}-\frac{28252}{85}a^{12}+\frac{59542}{85}a^{11}-\frac{109747}{85}a^{10}+\frac{177929}{85}a^{9}-\frac{14314}{5}a^{8}+\frac{54136}{17}a^{7}-\frac{238873}{85}a^{6}+\frac{167343}{85}a^{5}-1120a^{4}+\frac{2674}{5}a^{3}-\frac{17702}{85}a^{2}+\frac{4682}{85}a-\frac{622}{85}$ 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:  \( 11.4915268613 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11.4915268613 \cdot 1}{2\cdot\sqrt{6634204312890625}}\cr\approx \mathstrut & 0.171353218687 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*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 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*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 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*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 - 7*x^15 + 27*x^14 - 73*x^13 + 160*x^12 - 304*x^11 + 509*x^10 - 730*x^9 + 869*x^8 - 839*x^7 + 654*x^6 - 416*x^5 + 221*x^4 - 98*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

$D_8$ (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

\(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-19})\), 4.0.1805.1 x2, 4.2.475.1 x2, 8.0.81450625.1, 8.0.16290125.1 x4, 8.2.4286875.1 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: 8.2.4286875.1, 8.0.16290125.1
Minimal sibling: 8.2.4286875.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ R ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(5\) 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}$
\(19\) Copy content Toggle raw display 19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$