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Properties

Label 16.0.7213895789838336.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.214\times 10^{15}$
Root discriminant \(9.80\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $Q_8 : C_2$ (as 16T11)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^15 + 36*y^14 - 108*y^13 + 244*y^12 - 436*y^11 + 636*y^10 - 780*y^9 + 831*y^8 - 780*y^7 + 636*y^6 - 436*y^5 + 244*y^4 - 108*y^3 + 36*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*x^2 - 8*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*x^2 - 8*x + 1)
 

\( x^{16} - 8 x^{15} + 36 x^{14} - 108 x^{13} + 244 x^{12} - 436 x^{11} + 636 x^{10} - 780 x^{9} + 831 x^{8} + \cdots + 1 \) 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:   \(7213895789838336\) \(\medspace = 2^{40}\cdot 3^{8}\) 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:  \(9.80\)
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:  $2^{5/2}3^{1/2}\approx 9.797958971132712$
Ramified primes:   \(2\), \(3\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}-\frac{2}{13}a^{13}-\frac{3}{13}a^{12}+\frac{6}{13}a^{11}-\frac{3}{13}a^{10}-\frac{5}{13}a^{9}-\frac{2}{13}a^{8}+\frac{6}{13}a^{7}-\frac{2}{13}a^{6}-\frac{5}{13}a^{5}-\frac{3}{13}a^{4}+\frac{6}{13}a^{3}-\frac{3}{13}a^{2}-\frac{2}{13}a+\frac{1}{13}$, $\frac{1}{949}a^{15}-\frac{9}{949}a^{14}-\frac{28}{949}a^{13}+\frac{66}{949}a^{12}+\frac{397}{949}a^{11}-\frac{322}{949}a^{10}+\frac{228}{949}a^{9}+\frac{306}{949}a^{8}-\frac{278}{949}a^{7}+\frac{9}{949}a^{6}-\frac{176}{949}a^{5}+\frac{105}{949}a^{4}+\frac{358}{949}a^{3}+\frac{45}{949}a^{2}+\frac{210}{949}a-\frac{72}{949}$ 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:   \( -\frac{305}{73} a^{15} + \frac{30064}{949} a^{14} - \frac{127325}{949} a^{13} + \frac{355890}{949} a^{12} - \frac{743292}{949} a^{11} + \frac{1224316}{949} a^{10} - \frac{1630370}{949} a^{9} + \frac{1827160}{949} a^{8} - \frac{1803130}{949} a^{7} + \frac{1565132}{949} a^{6} - \frac{1135044}{949} a^{5} + \frac{655826}{949} a^{4} - \frac{296365}{949} a^{3} + \frac{99413}{949} a^{2} - \frac{19503}{949} a + \frac{1802}{949} \)  (order $24$) 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:   $\frac{2718}{949}a^{15}-\frac{20593}{949}a^{14}+\frac{86978}{949}a^{13}-\frac{243136}{949}a^{12}+\frac{508186}{949}a^{11}-\frac{841251}{949}a^{10}+\frac{1126105}{949}a^{9}-\frac{1275218}{949}a^{8}+\frac{1270000}{949}a^{7}-\frac{1112586}{949}a^{6}+\frac{817601}{949}a^{5}-\frac{487315}{949}a^{4}+\frac{229466}{949}a^{3}-\frac{81944}{949}a^{2}+\frac{17367}{949}a-\frac{2027}{949}$, $\frac{3272}{949}a^{15}-\frac{24776}{949}a^{14}+\frac{105922}{949}a^{13}-\frac{300085}{949}a^{12}+\frac{639940}{949}a^{11}-\frac{1079937}{949}a^{10}+\frac{1482805}{949}a^{9}-\frac{1718456}{949}a^{8}+\frac{1748095}{949}a^{7}-\frac{1561879}{949}a^{6}+\frac{1188684}{949}a^{5}-\frac{743775}{949}a^{4}+\frac{371880}{949}a^{3}-\frac{10849}{73}a^{2}+\frac{37201}{949}a-\frac{5999}{949}$, $\frac{2398}{949}a^{15}-\frac{20341}{949}a^{14}+\frac{90755}{949}a^{13}-\frac{267760}{949}a^{12}+\frac{44531}{73}a^{11}-\frac{75816}{73}a^{10}+\frac{1349087}{949}a^{9}-\frac{1540600}{949}a^{8}+\frac{1540584}{949}a^{7}-\frac{1363593}{949}a^{6}+\frac{1015176}{949}a^{5}-\frac{602237}{949}a^{4}+\frac{21423}{73}a^{3}-\frac{97950}{949}a^{2}+\frac{1681}{73}a-\frac{2493}{949}$, $\frac{3871}{949}a^{15}-\frac{30021}{949}a^{14}+\frac{129665}{949}a^{13}-\frac{371073}{949}a^{12}+\frac{794158}{949}a^{11}-\frac{1342530}{949}a^{10}+\frac{1838815}{949}a^{9}-\frac{2118140}{949}a^{8}+\frac{2134767}{949}a^{7}-\frac{1893675}{949}a^{6}+\frac{1423221}{949}a^{5}-\frac{867322}{949}a^{4}+\frac{414480}{949}a^{3}-\frac{146786}{949}a^{2}+\frac{32686}{949}a-\frac{3429}{949}$, $\frac{40}{949}a^{15}-\frac{4010}{949}a^{14}+\frac{24211}{949}a^{13}-\frac{90800}{949}a^{12}+\frac{220791}{949}a^{11}-\frac{420439}{949}a^{10}+\frac{625240}{949}a^{9}-\frac{764334}{949}a^{8}+\frac{61992}{73}a^{7}-\frac{760081}{949}a^{6}+\frac{609080}{949}a^{5}-\frac{401461}{949}a^{4}+\frac{210690}{949}a^{3}-\frac{87844}{949}a^{2}+\frac{24241}{949}a-\frac{5581}{949}$, $\frac{144}{949}a^{15}+\frac{1113}{949}a^{14}-\frac{9799}{949}a^{13}+\frac{44033}{949}a^{12}-\frac{119127}{949}a^{11}+\frac{245340}{949}a^{10}-\frac{30010}{73}a^{9}+\frac{504256}{949}a^{8}-\frac{550375}{949}a^{7}+\frac{534561}{949}a^{6}-\frac{449255}{949}a^{5}+\frac{314420}{949}a^{4}-\frac{172193}{949}a^{3}+\frac{73275}{949}a^{2}-\frac{22028}{949}a+\frac{4378}{949}$, $\frac{343}{949}a^{15}-\frac{580}{73}a^{14}+\frac{42007}{949}a^{13}-\frac{11466}{73}a^{12}+\frac{355244}{949}a^{11}-\frac{661742}{949}a^{10}+\frac{970702}{949}a^{9}-\frac{1168235}{949}a^{8}+\frac{1216018}{949}a^{7}-\frac{1127756}{949}a^{6}+\frac{68390}{73}a^{5}-\frac{43725}{73}a^{4}+\frac{285876}{949}a^{3}-\frac{110709}{949}a^{2}+\frac{26843}{949}a-\frac{4475}{949}$ 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:  \( 168.461389974 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 168.461389974 \cdot 1}{24\cdot\sqrt{7213895789838336}}\cr\approx \mathstrut & 0.200744267691 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*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 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*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 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*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 - 8*x^15 + 36*x^14 - 108*x^13 + 244*x^12 - 436*x^11 + 636*x^10 - 780*x^9 + 831*x^8 - 780*x^7 + 636*x^6 - 436*x^5 + 244*x^4 - 108*x^3 + 36*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

$D_4:C_2$ (as 16T11):

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 10 conjugacy class representatives for $Q_8 : C_2$
Character table for $Q_8 : C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{24})\), 8.0.9437184.1 x2, 8.0.5308416.2 x2, 8.4.84934656.1 x2

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: 8.4.84934656.1, 8.0.5308416.2, 8.0.9437184.1
Minimal sibling: 8.0.5308416.2

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 2.16.40.1$x^{16} - 4 x^{15} + 16 x^{14} + 8 x^{13} - 8 x^{12} + 8 x^{11} + 16 x^{10} + 32 x^{9} + 56 x^{8} - 56 x^{7} + 112 x^{5} + 112 x^{4} + 196$$8$$2$$40$$Q_8 : C_2$$[2, 3, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$