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Properties

Label 16.0.176...000.8
Degree $16$
Signature $[0, 8]$
Discriminant $1.761\times 10^{20}$
Root discriminant \(18.42\)
Ramified primes $2,3,5$
Class number $4$
Class group [4]
Galois group $Q_8 : C_2^2$ (as 16T23)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36)
 
gp: K = bnfinit(y^16 - 4*y^14 + 42*y^12 + 4*y^10 - 27*y^8 - 64*y^6 + 148*y^4 - 120*y^2 + 36, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36)
 

\( x^{16} - 4x^{14} + 42x^{12} + 4x^{10} - 27x^{8} - 64x^{6} + 148x^{4} - 120x^{2} + 36 \) 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:   \(176120502681600000000\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 5^{8}\) 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:  \(18.42\)
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:  $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{1111008}a^{14}-\frac{1129}{1111008}a^{12}-\frac{39531}{370336}a^{10}+\frac{96169}{1111008}a^{8}+\frac{5547}{46292}a^{6}-\frac{1}{2}a^{5}+\frac{27127}{138876}a^{4}-\frac{1}{2}a^{3}-\frac{138587}{277752}a^{2}+\frac{30491}{92584}$, $\frac{1}{6666048}a^{15}-\frac{1}{2222016}a^{14}+\frac{137747}{6666048}a^{13}-\frac{137747}{2222016}a^{12}-\frac{13177}{740672}a^{11}+\frac{39531}{740672}a^{10}+\frac{512797}{6666048}a^{9}+\frac{42707}{2222016}a^{8}-\frac{3013}{138876}a^{7}-\frac{2140}{11573}a^{6}-\frac{35668}{104157}a^{5}-\frac{32821}{69438}a^{4}-\frac{555215}{1666512}a^{3}-\frac{289}{555504}a^{2}-\frac{62093}{555504}a+\frac{62093}{185168}$ 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

$C_{4}$, which has order $4$

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:   $\frac{237131}{6666048}a^{15}-\frac{35129}{740672}a^{14}-\frac{801227}{6666048}a^{13}+\frac{127273}{740672}a^{12}+\frac{1053237}{740672}a^{11}-\frac{1430095}{740672}a^{10}+\frac{6766979}{6666048}a^{9}-\frac{671313}{740672}a^{8}-\frac{28121}{138876}a^{7}+\frac{37349}{46292}a^{6}-\frac{1031333}{416628}a^{5}+\frac{72227}{23146}a^{4}+\frac{7425419}{1666512}a^{3}-\frac{986465}{185168}a^{2}-\frac{1236043}{555504}a+\frac{559667}{185168}$, $\frac{237131}{6666048}a^{15}+\frac{35129}{740672}a^{14}-\frac{801227}{6666048}a^{13}-\frac{127273}{740672}a^{12}+\frac{1053237}{740672}a^{11}+\frac{1430095}{740672}a^{10}+\frac{6766979}{6666048}a^{9}+\frac{671313}{740672}a^{8}-\frac{28121}{138876}a^{7}-\frac{37349}{46292}a^{6}-\frac{1031333}{416628}a^{5}-\frac{72227}{23146}a^{4}+\frac{7425419}{1666512}a^{3}+\frac{986465}{185168}a^{2}-\frac{1236043}{555504}a-\frac{559667}{185168}$, $\frac{246731}{2222016}a^{15}+\frac{54693}{740672}a^{14}-\frac{807299}{2222016}a^{13}-\frac{180037}{740672}a^{12}+\frac{3256511}{740672}a^{11}+\frac{2172355}{740672}a^{10}+\frac{8130491}{2222016}a^{9}+\frac{1740589}{740672}a^{8}-\frac{20421}{46292}a^{7}-\frac{3093}{23146}a^{6}-\frac{1017983}{138876}a^{5}-\frac{238041}{46292}a^{4}+\frac{6589043}{555504}a^{3}+\frac{1363173}{185168}a^{2}-\frac{995299}{185168}a-\frac{538031}{185168}$, $\frac{207055}{3333024}a^{15}-\frac{731167}{3333024}a^{13}+\frac{920001}{370336}a^{11}+\frac{4953199}{3333024}a^{9}-\frac{64940}{34719}a^{7}-\frac{2280851}{416628}a^{5}+\frac{5121751}{833256}a^{3}-\frac{637043}{277752}a$, $\frac{27997}{2222016}a^{15}+\frac{126719}{2222016}a^{14}-\frac{83761}{2222016}a^{13}-\frac{440099}{2222016}a^{12}+\frac{370065}{740672}a^{11}+\frac{1687627}{740672}a^{10}+\frac{1165489}{2222016}a^{9}+\frac{3264659}{2222016}a^{8}+\frac{11891}{23146}a^{7}-\frac{57369}{46292}a^{6}-\frac{1753}{138876}a^{5}-\frac{163310}{34719}a^{4}+\frac{591829}{555504}a^{3}+\frac{3569303}{555504}a^{2}-\frac{245705}{185168}a-\frac{389379}{185168}$, $\frac{587245}{6666048}a^{15}+\frac{28061}{2222016}a^{14}-\frac{2088721}{6666048}a^{13}-\frac{156017}{2222016}a^{12}+\frac{2635707}{740672}a^{11}+\frac{432433}{740672}a^{10}+\frac{12921217}{6666048}a^{9}-\frac{1567759}{2222016}a^{8}-\frac{227821}{138876}a^{7}-\frac{64893}{46292}a^{6}-\frac{1326275}{208314}a^{5}-\frac{104989}{69438}a^{4}+\frac{15839197}{1666512}a^{3}+\frac{1165829}{555504}a^{2}-\frac{3576497}{555504}a-\frac{423713}{185168}$, $\frac{27997}{2222016}a^{15}-\frac{126719}{2222016}a^{14}-\frac{83761}{2222016}a^{13}+\frac{440099}{2222016}a^{12}+\frac{370065}{740672}a^{11}-\frac{1687627}{740672}a^{10}+\frac{1165489}{2222016}a^{9}-\frac{3264659}{2222016}a^{8}+\frac{11891}{23146}a^{7}+\frac{57369}{46292}a^{6}-\frac{1753}{138876}a^{5}+\frac{163310}{34719}a^{4}+\frac{591829}{555504}a^{3}-\frac{3569303}{555504}a^{2}-\frac{245705}{185168}a+\frac{389379}{185168}$ 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:  \( 3270.91851736 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3270.91851736 \cdot 4}{2\cdot\sqrt{176120502681600000000}}\cr\approx \mathstrut & 1.19738471462 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36)
 
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^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36, 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^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36);
 
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^14 + 42*x^12 + 4*x^10 - 27*x^8 - 64*x^6 + 148*x^4 - 120*x^2 + 36);
 
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

$D_4:C_2^2$ (as 16T23):

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 32
The 17 conjugacy class representatives for $Q_8 : C_2^2$
Character table for $Q_8 : C_2^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{3}, \sqrt{-10})\), 8.0.3317760000.6, 8.4.530841600.1 x2, 8.0.829440000.1 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

Galois closure: deg 32
Degree 8 siblings: 8.4.368640000.1, 8.4.530841600.1, 8.0.132710400.1, 8.0.51840000.1, 8.0.829440000.1, 8.0.1474560000.1
Degree 16 siblings: 16.0.2174327193600000000.1, 16.0.176120502681600000000.1, 16.0.11007531417600000000.3, 16.0.11007531417600000000.12, 16.0.281792804290560000.2, 16.0.687970713600000000.4, 16.0.176120502681600000000.9, 16.8.176120502681600000000.1
Minimal sibling: 8.0.51840000.1

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.18.56$x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 14$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.56$x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{3} + 14$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$