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Properties

Label 16.0.189...384.4
Degree $16$
Signature $[0, 8]$
Discriminant $1.891\times 10^{21}$
Root discriminant \(21.37\)
Ramified primes $2,3$
Class number $2$ (GRH)
Class group [2] (GRH)
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 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9)
 
gp: K = bnfinit(y^16 + 72*y^12 - 80*y^10 + 366*y^8 - 288*y^6 + 88*y^4 + 48*y^2 + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9)
 

\( x^{16} + 72x^{12} - 80x^{10} + 366x^{8} - 288x^{6} + 88x^{4} + 48x^{2} + 9 \) 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:   \(1891079497931380752384\) \(\medspace = 2^{58}\cdot 3^{8}\) 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:  \(21.37\)
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:  $2^{29/8}3^{1/2}\approx 21.36950004471431$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{6}a^{8}+\frac{1}{3}a^{2}-\frac{1}{2}$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{5}+\frac{4}{9}a^{3}+\frac{1}{6}a$, $\frac{1}{18}a^{10}+\frac{1}{9}a^{4}-\frac{1}{6}a^{2}$, $\frac{1}{36}a^{11}-\frac{1}{36}a^{10}-\frac{1}{36}a^{9}-\frac{1}{12}a^{8}-\frac{1}{9}a^{5}+\frac{4}{9}a^{4}+\frac{7}{36}a^{3}-\frac{1}{12}a^{2}-\frac{1}{12}a-\frac{1}{4}$, $\frac{1}{108}a^{12}+\frac{1}{54}a^{10}-\frac{1}{36}a^{8}+\frac{2}{27}a^{6}+\frac{43}{108}a^{4}+\frac{5}{18}a^{2}-\frac{5}{12}$, $\frac{1}{108}a^{13}-\frac{1}{108}a^{11}-\frac{1}{36}a^{10}-\frac{1}{12}a^{8}+\frac{2}{27}a^{7}-\frac{53}{108}a^{5}+\frac{4}{9}a^{4}+\frac{1}{12}a^{3}-\frac{1}{12}a^{2}-\frac{1}{3}a-\frac{1}{4}$, $\frac{1}{534276}a^{14}+\frac{1159}{534276}a^{12}-\frac{125}{5508}a^{10}+\frac{36101}{534276}a^{8}-\frac{39877}{534276}a^{6}-\frac{250207}{534276}a^{4}+\frac{3751}{10476}a^{2}-\frac{7333}{59364}$, $\frac{1}{534276}a^{15}+\frac{1159}{534276}a^{13}+\frac{7}{1377}a^{11}-\frac{1}{36}a^{10}-\frac{4211}{267138}a^{9}-\frac{1}{12}a^{8}-\frac{39877}{534276}a^{7}+\frac{46613}{534276}a^{5}+\frac{4}{9}a^{4}+\frac{283}{2619}a^{3}-\frac{1}{12}a^{2}-\frac{11087}{29682}a-\frac{1}{4}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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:   $\frac{781}{44523}a^{14}+\frac{43}{6596}a^{12}+\frac{584}{459}a^{10}-\frac{165433}{178092}a^{8}+\frac{10837}{1649}a^{6}-\frac{543401}{178092}a^{4}+\frac{2168}{873}a^{2}+\frac{7861}{19788}$, $\frac{910}{133569}a^{14}-\frac{1031}{534276}a^{12}+\frac{1337}{2754}a^{10}-\frac{365599}{534276}a^{8}+\frac{614725}{267138}a^{6}-\frac{1242583}{534276}a^{4}+\frac{562}{2619}a^{2}+\frac{16733}{59364}$, $\frac{683}{59364}a^{14}+\frac{77}{59364}a^{12}+\frac{509}{612}a^{10}-\frac{48335}{59364}a^{8}+\frac{259435}{59364}a^{6}-\frac{130835}{59364}a^{4}+\frac{2879}{1164}a^{2}+\frac{4521}{6596}$, $\frac{4501}{89046}a^{15}+\frac{797}{9894}a^{14}+\frac{377}{19788}a^{13}+\frac{6743}{178092}a^{12}+\frac{6763}{1836}a^{11}+\frac{10705}{1836}a^{10}-\frac{236017}{89046}a^{9}-\frac{109897}{29682}a^{8}+\frac{199421}{9894}a^{7}+\frac{1276060}{44523}a^{6}-\frac{1707625}{178092}a^{5}-\frac{1903573}{178092}a^{4}+\frac{5187}{388}a^{3}+\frac{20011}{3492}a^{2}-\frac{19237}{9894}a+\frac{12464}{4947}$, $\frac{3155}{267138}a^{15}-\frac{3877}{534276}a^{14}+\frac{406}{133569}a^{13}-\frac{1567}{534276}a^{12}+\frac{4657}{5508}a^{11}-\frac{1391}{2754}a^{10}-\frac{383065}{534276}a^{9}+\frac{52417}{133569}a^{8}+\frac{497258}{133569}a^{7}-\frac{643625}{534276}a^{6}-\frac{356585}{267138}a^{5}+\frac{648313}{534276}a^{4}-\frac{30521}{10476}a^{3}+\frac{5990}{2619}a^{2}+\frac{27743}{59364}a-\frac{5131}{29682}$, $\frac{689}{267138}a^{15}-\frac{7349}{534276}a^{14}-\frac{779}{534276}a^{13}+\frac{1243}{534276}a^{12}+\frac{262}{1377}a^{11}-\frac{5495}{5508}a^{10}-\frac{163105}{534276}a^{9}+\frac{674153}{534276}a^{8}+\frac{185705}{133569}a^{7}-\frac{3081241}{534276}a^{6}-\frac{647191}{534276}a^{5}+\frac{2702087}{534276}a^{4}+\frac{12581}{5238}a^{3}-\frac{47393}{10476}a^{2}+\frac{41423}{59364}a-\frac{12295}{59364}$, $\frac{65351}{534276}a^{15}+\frac{12245}{534276}a^{14}-\frac{5801}{267138}a^{13}-\frac{5935}{534276}a^{12}+\frac{12125}{1377}a^{11}+\frac{2279}{1377}a^{10}-\frac{6062765}{534276}a^{9}-\frac{351794}{133569}a^{8}+\frac{24792985}{534276}a^{7}+\frac{5149597}{534276}a^{6}-\frac{11473633}{267138}a^{5}-\frac{5925287}{534276}a^{4}+\frac{42896}{2619}a^{3}+\frac{35425}{5238}a^{2}+\frac{264145}{59364}a-\frac{14635}{29682}$ Copy content Toggle raw display (assuming GRH)
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:  \( 114037.664959 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 114037.664959 \cdot 2}{2\cdot\sqrt{1891079497931380752384}}\cr\approx \mathstrut & 6.36989641168 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9)
 
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 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9, 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 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9);
 
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 + 72*x^12 - 80*x^10 + 366*x^8 - 288*x^6 + 88*x^4 + 48*x^2 + 9);
 
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}, \sqrt{3})\), 4.0.3072.2 x2, 4.2.4608.1 x2, 8.0.339738624.6, 8.0.7247757312.3 x4, 8.2.10871635968.3 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: 8.2.10871635968.3, 8.0.7247757312.3
Minimal sibling: 8.0.7247757312.3

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.16.58.11$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 26 x^{8} + 16 x^{6} + 4 x^{4} + 16 x^{2} + 6$$16$$1$$58$$D_{8}$$[2, 3, 7/2, 9/2]$
\(3\) Copy content Toggle raw display 3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$