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Properties

Label 16.0.10749542400000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.075\times 10^{16}$
Root discriminant \(10.05\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $D_8:C_4$ (as 16T114)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*x^2 - 7*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + y^14 + 6*y^13 - 15*y^12 + 13*y^11 + 15*y^10 - 45*y^9 + 42*y^8 + 14*y^7 - 65*y^6 + 50*y^5 + 3*y^4 - 29*y^3 + 21*y^2 - 7*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*x^2 - 7*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*x^2 - 7*x + 1)
 

\( x^{16} - 2 x^{15} + x^{14} + 6 x^{13} - 15 x^{12} + 13 x^{11} + 15 x^{10} - 45 x^{9} + 42 x^{8} + \cdots + 1 \) 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:   \(10749542400000000\) \(\medspace = 2^{22}\cdot 3^{8}\cdot 5^{8}\) 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:  \(10.05\)
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:  $2^{11/4}3^{1/2}5^{1/2}\approx 26.054222497305222$
Ramified primes:   \(2\), \(3\), \(5\) 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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{158}a^{15}-\frac{6}{79}a^{14}-\frac{37}{158}a^{13}-\frac{19}{158}a^{12}-\frac{31}{79}a^{11}-\frac{39}{79}a^{10}-\frac{37}{79}a^{9}+\frac{63}{158}a^{8}-\frac{35}{158}a^{7}-\frac{31}{158}a^{6}+\frac{4}{79}a^{5}+\frac{49}{158}a^{4}+\frac{33}{79}a^{3}-\frac{57}{158}a^{2}-\frac{41}{158}a-\frac{71}{158}$ 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:   \( \frac{305}{158} a^{15} - \frac{250}{79} a^{14} + \frac{85}{79} a^{13} + \frac{1789}{158} a^{12} - \frac{3821}{158} a^{11} + \frac{1377}{79} a^{10} + \frac{2461}{79} a^{9} - \frac{5521}{79} a^{8} + \frac{8759}{158} a^{7} + \frac{3054}{79} a^{6} - \frac{7707}{79} a^{5} + \frac{4352}{79} a^{4} + \frac{3145}{158} a^{3} - \frac{3044}{79} a^{2} + \frac{1687}{79} a - \frac{641}{158} \)  (order $6$) 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:   $\frac{490}{79}a^{15}-\frac{1727}{158}a^{14}+\frac{633}{158}a^{13}+\frac{2935}{79}a^{12}-\frac{13123}{158}a^{11}+\frac{9749}{158}a^{10}+\frac{16197}{158}a^{9}-\frac{38985}{158}a^{8}+\frac{15714}{79}a^{7}+\frac{19943}{158}a^{6}-\frac{28233}{79}a^{5}+\frac{16900}{79}a^{4}+\frac{10565}{158}a^{3}-\frac{24813}{158}a^{2}+\frac{6691}{79}a-\frac{2825}{158}$, $\frac{1175}{79}a^{15}-\frac{3631}{158}a^{14}+\frac{503}{158}a^{13}+\frac{7300}{79}a^{12}-\frac{28701}{158}a^{11}+\frac{16491}{158}a^{10}+\frac{44693}{158}a^{9}-\frac{86659}{158}a^{8}+\frac{28237}{79}a^{7}+\frac{63583}{158}a^{6}-\frac{63596}{79}a^{5}+\frac{27792}{79}a^{4}+\frac{38733}{158}a^{3}-\frac{52343}{158}a^{2}+\frac{11549}{79}a-\frac{4031}{158}$, $\frac{906}{79}a^{15}-\frac{3021}{158}a^{14}+\frac{290}{79}a^{13}+\frac{11313}{158}a^{12}-\frac{11695}{79}a^{11}+\frac{7226}{79}a^{10}+\frac{34103}{158}a^{9}-\frac{35510}{79}a^{8}+\frac{48365}{158}a^{7}+\frac{23659}{79}a^{6}-\frac{105031}{158}a^{5}+\frac{49525}{158}a^{4}+\frac{29611}{158}a^{3}-\frac{44113}{158}a^{2}+\frac{10096}{79}a-\frac{1837}{79}$, $\frac{233}{158}a^{15}-\frac{347}{158}a^{14}-\frac{5}{79}a^{13}+\frac{1577}{158}a^{12}-\frac{2991}{158}a^{11}+\frac{1497}{158}a^{10}+\frac{5115}{158}a^{9}-\frac{4866}{79}a^{8}+\frac{6223}{158}a^{7}+\frac{3617}{79}a^{6}-\frac{7600}{79}a^{5}+\frac{7625}{158}a^{4}+\frac{3607}{158}a^{3}-\frac{3599}{79}a^{2}+\frac{4035}{158}a-\frac{411}{79}$, $\frac{31}{79}a^{15}-\frac{349}{158}a^{14}+\frac{313}{158}a^{13}+\frac{201}{79}a^{12}-\frac{2185}{158}a^{11}+\frac{2827}{158}a^{10}+\frac{231}{158}a^{9}-\frac{6759}{158}a^{8}+\frac{3971}{79}a^{7}-\frac{1369}{158}a^{6}-\frac{5045}{79}a^{5}+\frac{4995}{79}a^{4}+\frac{221}{158}a^{3}-\frac{5193}{158}a^{2}+\frac{1336}{79}a-\frac{531}{158}$, $\frac{799}{158}a^{15}-\frac{1293}{158}a^{14}+\frac{299}{158}a^{13}+\frac{2482}{79}a^{12}-\frac{5098}{79}a^{11}+\frac{3283}{79}a^{10}+\frac{14739}{158}a^{9}-\frac{31033}{158}a^{8}+\frac{11258}{79}a^{7}+\frac{19945}{158}a^{6}-\frac{45827}{158}a^{5}+\frac{12268}{79}a^{4}+\frac{11101}{158}a^{3}-\frac{10013}{79}a^{2}+\frac{10691}{158}a-\frac{2219}{158}$, $\frac{1171}{158}a^{15}-\frac{1807}{158}a^{14}+\frac{281}{158}a^{13}+\frac{3609}{79}a^{12}-\frac{7150}{79}a^{11}+\frac{4180}{79}a^{10}+\frac{21971}{158}a^{9}-\frac{43147}{158}a^{8}+\frac{14386}{79}a^{7}+\frac{30849}{158}a^{6}-\frac{63549}{158}a^{5}+\frac{14509}{79}a^{4}+\frac{17799}{158}a^{3}-\frac{13426}{79}a^{2}+\frac{12661}{158}a-\frac{2403}{158}$ 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:  \( 52.8621079118 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 52.8621079118 \cdot 1}{6\cdot\sqrt{10749542400000000}}\cr\approx \mathstrut & 0.206413066048 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*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 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*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 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*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 - 2*x^15 + x^14 + 6*x^13 - 15*x^12 + 13*x^11 + 15*x^10 - 45*x^9 + 42*x^8 + 14*x^7 - 65*x^6 + 50*x^5 + 3*x^4 - 29*x^3 + 21*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

$D_8:C_4$ (as 16T114):

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 64
The 28 conjugacy class representatives for $D_8:C_4$
Character table for $D_8:C_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.3240000.1

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 sibling: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.8.22.32$x^{8} + 8 x^{7} + 88 x^{6} + 96 x^{5} + 188 x^{4} + 272 x^{3} + 528 x^{2} + 532$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
\(3\) Copy content Toggle raw display 3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(5\) Copy content Toggle raw display 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$