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Properties

Label 16.0.4897760256000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.898\times 10^{15}$
Root discriminant \(9.56\)
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 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 6*y^13 - 10*y^12 + 28*y^11 - 8*y^10 - 90*y^9 + 253*y^8 - 396*y^7 + 433*y^6 - 352*y^5 + 218*y^4 - 102*y^3 + 35*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*x^2 - 8*x + 1)
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 6 x^{13} - 10 x^{12} + 28 x^{11} - 8 x^{10} - 90 x^{9} + 253 x^{8} + \cdots + 1 \) 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:   \(4897760256000000\) \(\medspace = 2^{16}\cdot 3^{14}\cdot 5^{6}\) 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:  \(9.56\)
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:  $2\cdot 3^{7/8}5^{3/4}\approx 17.487937673507474$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31}a^{15}-\frac{1}{31}a^{14}+\frac{5}{31}a^{13}+\frac{9}{31}a^{12}-\frac{14}{31}a^{11}-\frac{14}{31}a^{10}+\frac{12}{31}a^{9}+\frac{8}{31}a^{8}-\frac{2}{31}a^{7}+\frac{1}{31}a^{6}+\frac{2}{31}a^{5}-\frac{5}{31}a^{4}-\frac{14}{31}a^{3}+\frac{11}{31}a^{2}+\frac{6}{31}a+\frac{10}{31}$ 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:   \( \frac{628}{31} a^{15} - \frac{2705}{31} a^{14} + \frac{5248}{31} a^{13} - \frac{3555}{31} a^{12} - \frac{7893}{31} a^{11} + \frac{19945}{31} a^{10} - \frac{3624}{31} a^{9} - \frac{64912}{31} a^{8} + \frac{170050}{31} a^{7} - \frac{249666}{31} a^{6} + \frac{250775}{31} a^{5} - \frac{183529}{31} a^{4} + \frac{99119}{31} a^{3} - \frac{38848}{31} a^{2} + \frac{10030}{31} a - \frac{1408}{31} \)  (order $12$) 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:   $\frac{405}{31}a^{15}-\frac{1738}{31}a^{14}+\frac{3420}{31}a^{13}-\frac{2462}{31}a^{12}-\frac{4802}{31}a^{11}+\frac{12775}{31}a^{10}-\frac{3076}{31}a^{9}-\frac{40625}{31}a^{8}+\frac{110046}{31}a^{7}-\frac{165538}{31}a^{6}+\frac{170938}{31}a^{5}-\frac{128970}{31}a^{4}+\frac{71768}{31}a^{3}-\frac{29025}{31}a^{2}+\frac{7762}{31}a-\frac{1158}{31}$, $\frac{1779}{31}a^{15}-\frac{5375}{31}a^{14}+\frac{8089}{31}a^{13}-\frac{140}{31}a^{12}-\frac{21806}{31}a^{11}+\frac{28352}{31}a^{10}+\frac{24293}{31}a^{9}-\frac{149820}{31}a^{8}+\frac{290787}{31}a^{7}-\frac{346382}{31}a^{6}+\frac{290928}{31}a^{5}-\frac{176636}{31}a^{4}+\frac{78417}{31}a^{3}-\frac{23273}{31}a^{2}+\frac{4102}{31}a-\frac{128}{31}$, $\frac{333}{31}a^{15}-\frac{2100}{31}a^{14}+\frac{4734}{31}a^{13}-\frac{4784}{31}a^{12}-\frac{4290}{31}a^{11}+\frac{18619}{31}a^{10}-\frac{11938}{31}a^{9}-\frac{43898}{31}a^{8}+\frac{145127}{31}a^{7}-\frac{237344}{31}a^{6}+\frac{256292}{31}a^{5}-\frac{199631}{31}a^{4}+\frac{113479}{31}a^{3}-\frac{47053}{31}a^{2}+\frac{12879}{31}a-\frac{2033}{31}$, $\frac{375}{31}a^{15}-\frac{2359}{31}a^{14}+\frac{5161}{31}a^{13}-\frac{4964}{31}a^{12}-\frac{5312}{31}a^{11}+\frac{20604}{31}a^{10}-\frac{11496}{31}a^{9}-\frac{50878}{31}a^{8}+\frac{159706}{31}a^{7}-\frac{254569}{31}a^{6}+\frac{268559}{31}a^{5}-\frac{205266}{31}a^{4}+\frac{114627}{31}a^{3}-\frac{46808}{31}a^{2}+\frac{12511}{31}a-\frac{1954}{31}$, $\frac{640}{31}a^{15}-\frac{1973}{31}a^{14}+\frac{3045}{31}a^{13}-\frac{285}{31}a^{12}-\frac{7751}{31}a^{11}+\frac{10663}{31}a^{10}+\frac{7897}{31}a^{9}-\frac{54090}{31}a^{8}+\frac{108088}{31}a^{7}-\frac{132567}{31}a^{6}+\frac{115484}{31}a^{5}-\frac{73911}{31}a^{4}+\frac{35401}{31}a^{3}-\frac{11994}{31}a^{2}+\frac{2693}{31}a-\frac{265}{31}$, $\frac{12}{31}a^{15}+\frac{577}{31}a^{14}-\frac{1738}{31}a^{13}+\frac{2619}{31}a^{12}+\frac{49}{31}a^{11}-\frac{7298}{31}a^{10}+\frac{9227}{31}a^{9}+\frac{8187}{31}a^{8}-\frac{48787}{31}a^{7}+\frac{93105}{31}a^{6}-\frac{108445}{31}a^{5}+\frac{88724}{31}a^{4}-\frac{51876}{31}a^{3}+\frac{22018}{31}a^{2}-\frac{6066}{31}a+\frac{957}{31}$, $\frac{116}{31}a^{15}+\frac{535}{31}a^{14}-\frac{2117}{31}a^{13}+\frac{3958}{31}a^{12}-\frac{1438}{31}a^{11}-\frac{8909}{31}a^{10}+\frac{15404}{31}a^{9}+\frac{2385}{31}a^{8}-\frac{54885}{31}a^{7}+\frac{120365}{31}a^{6}-\frac{151048}{31}a^{5}+\frac{131232}{31}a^{4}-\frac{81232}{31}a^{3}+\frac{36399}{31}a^{2}-\frac{10588}{31}a+\frac{1749}{31}$ 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:  \( 59.5214550029 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 59.5214550029 \cdot 1}{12\cdot\sqrt{4897760256000000}}\cr\approx \mathstrut & 0.172160088985 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*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 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*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 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*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 - 4*x^15 + 8*x^14 - 6*x^13 - 10*x^12 + 28*x^11 - 8*x^10 - 90*x^9 + 253*x^8 - 396*x^7 + 433*x^6 - 352*x^5 + 218*x^4 - 102*x^3 + 35*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

$D_8:C_4$ (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.4665600.1

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 ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$2$$8$$16$
\(3\) Copy content Toggle raw display 3.16.14.3$x^{16} + 36$$8$$2$$14$16T49$[\ ]_{8}^{4}$
\(5\) Copy content Toggle raw display 5.8.0.1$x^{8} + x^{4} + 3 x^{2} + 4 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.6.4$x^{8} - 20 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$