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Properties

Label 16.0.167...744.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.674\times 10^{26}$
Root discriminant \(43.55\)
Ramified primes $2,3,23$
Class number $80$ (GRH)
Class group [2, 2, 2, 10] (GRH)
Galois group $C_2^2:Q_8$ (as 16T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641)
 
gp: K = bnfinit(y^16 + 16*y^14 + 248*y^12 + 1696*y^10 + 8098*y^8 + 17744*y^6 + 32408*y^4 + 27104*y^2 + 14641, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641)
 

\( x^{16} + 16x^{14} + 248x^{12} + 1696x^{10} + 8098x^{8} + 17744x^{6} + 32408x^{4} + 27104x^{2} + 14641 \) 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:   \(167442686719647879731871744\) \(\medspace = 2^{50}\cdot 3^{12}\cdot 23^{4}\) 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:  \(43.55\)
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:  $2^{25/8}3^{3/4}23^{1/2}\approx 95.37259411116924$
Ramified primes:   \(2\), \(3\), \(23\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{40}a^{8}-\frac{3}{10}a^{6}-\frac{1}{5}a^{4}+\frac{3}{10}a^{2}-\frac{19}{40}$, $\frac{1}{40}a^{9}-\frac{3}{10}a^{7}-\frac{1}{5}a^{5}+\frac{3}{10}a^{3}-\frac{19}{40}a$, $\frac{1}{40}a^{10}+\frac{1}{5}a^{6}-\frac{1}{10}a^{4}+\frac{1}{8}a^{2}+\frac{3}{10}$, $\frac{1}{80}a^{11}-\frac{1}{80}a^{10}-\frac{1}{80}a^{9}-\frac{1}{80}a^{8}+\frac{1}{4}a^{7}+\frac{1}{20}a^{6}+\frac{1}{20}a^{5}+\frac{3}{20}a^{4}-\frac{7}{80}a^{3}-\frac{17}{80}a^{2}-\frac{9}{80}a-\frac{33}{80}$, $\frac{1}{80}a^{12}-\frac{1}{80}a^{8}+\frac{3}{10}a^{6}-\frac{3}{80}a^{4}-\frac{1}{5}a^{2}-\frac{29}{80}$, $\frac{1}{880}a^{13}+\frac{1}{176}a^{11}-\frac{1}{80}a^{10}+\frac{3}{440}a^{9}-\frac{1}{80}a^{8}-\frac{1}{44}a^{7}+\frac{1}{20}a^{6}+\frac{321}{880}a^{5}+\frac{3}{20}a^{4}-\frac{307}{880}a^{3}-\frac{17}{80}a^{2}+\frac{89}{440}a-\frac{33}{80}$, $\frac{1}{319000208560}a^{14}+\frac{82437921}{319000208560}a^{12}-\frac{2843387343}{319000208560}a^{10}-\frac{3234540417}{319000208560}a^{8}+\frac{2753540225}{63800041712}a^{6}+\frac{25369280069}{319000208560}a^{4}+\frac{22942212377}{63800041712}a^{2}+\frac{996557947}{2636365360}$, $\frac{1}{3509002294160}a^{15}+\frac{82437921}{3509002294160}a^{13}-\frac{683088995}{350900229416}a^{11}-\frac{1}{80}a^{10}-\frac{451377689}{219312643385}a^{9}-\frac{1}{80}a^{8}+\frac{348717920113}{3509002294160}a^{7}+\frac{1}{20}a^{6}-\frac{564781105767}{3509002294160}a^{5}+\frac{3}{20}a^{4}+\frac{820962280183}{1754501147080}a^{3}-\frac{17}{80}a^{2}-\frac{2066881}{659091340}a-\frac{33}{80}$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{10}$, which has order $80$ (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:   $\frac{4961263}{79750052140}a^{14}+\frac{117211073}{159500104280}a^{12}+\frac{1819347903}{159500104280}a^{10}+\frac{3473811911}{79750052140}a^{8}+\frac{8204095777}{79750052140}a^{6}-\frac{105754126799}{159500104280}a^{4}-\frac{26661649941}{31900020856}a^{2}-\frac{799029739}{659091340}$, $\frac{182741}{31900020856}a^{14}+\frac{3272215}{31900020856}a^{12}+\frac{43265793}{31900020856}a^{10}+\frac{311103041}{31900020856}a^{8}+\frac{726391761}{31900020856}a^{6}+\frac{1503102291}{31900020856}a^{4}+\frac{1304854789}{31900020856}a^{2}+\frac{642638749}{263636536}$, $\frac{3094485}{31900020856}a^{14}+\frac{190906479}{159500104280}a^{12}+\frac{2902503797}{159500104280}a^{10}+\frac{12183335309}{159500104280}a^{8}+\frac{28708262101}{159500104280}a^{6}-\frac{138182994509}{159500104280}a^{4}-\frac{178653454263}{159500104280}a^{2}-\frac{1667374239}{1318182680}$, $\frac{595501869}{1754501147080}a^{15}-\frac{12240623}{79750052140}a^{14}+\frac{451176601}{87725057354}a^{13}-\frac{291756477}{159500104280}a^{12}+\frac{34717419229}{438625286770}a^{11}-\frac{901104547}{31900020856}a^{10}+\frac{437563775677}{877250573540}a^{9}-\frac{8787721963}{79750052140}a^{8}+\frac{3814828776617}{1754501147080}a^{7}-\frac{4148449485}{15950010428}a^{6}+\frac{716402827039}{219312643385}a^{5}+\frac{251452632763}{159500104280}a^{4}+\frac{377846539911}{87725057354}a^{3}+\frac{318485977913}{159500104280}a^{2}+\frac{9121809899}{7250004740}a+\frac{1921131051}{659091340}$, $\frac{113167173}{3509002294160}a^{15}-\frac{4961263}{79750052140}a^{14}+\frac{1693162351}{3509002294160}a^{13}-\frac{117211073}{159500104280}a^{12}+\frac{26799896067}{3509002294160}a^{11}-\frac{1819347903}{159500104280}a^{10}+\frac{171870733459}{3509002294160}a^{9}-\frac{3473811911}{79750052140}a^{8}+\frac{853482862129}{3509002294160}a^{7}-\frac{8204095777}{79750052140}a^{6}+\frac{371752244615}{701800458832}a^{5}+\frac{105754126799}{159500104280}a^{4}+\frac{4314704317183}{3509002294160}a^{3}+\frac{26661649941}{31900020856}a^{2}+\frac{11786755559}{29000018960}a+\frac{1458121079}{659091340}$, $\frac{265269981}{3509002294160}a^{15}+\frac{64}{701161}a^{14}+\frac{4065014031}{3509002294160}a^{13}+\frac{7673}{7011610}a^{12}+\frac{63979924469}{3509002294160}a^{11}+\frac{59042}{3505805}a^{10}+\frac{420285377849}{3509002294160}a^{9}+\frac{233599}{3505805}a^{8}+\frac{2072689752129}{3509002294160}a^{7}+\frac{551176}{3505805}a^{6}+\frac{4526781177323}{3509002294160}a^{5}-\frac{6404893}{7011610}a^{4}+\frac{10453510964497}{3509002294160}a^{3}-\frac{4070198}{3505805}a^{2}+\frac{28661043117}{29000018960}a-\frac{5968624}{3505805}$, $\frac{4936319}{29000018960}a^{15}+\frac{9008821}{159500104280}a^{14}+\frac{75607629}{29000018960}a^{13}+\frac{50424999}{79750052140}a^{12}+\frac{1165482823}{29000018960}a^{11}+\frac{801509469}{79750052140}a^{10}+\frac{7467681169}{29000018960}a^{9}+\frac{5392108617}{159500104280}a^{8}+\frac{33298594851}{29000018960}a^{7}+\frac{12776232749}{159500104280}a^{6}+\frac{11000552317}{5800003792}a^{5}-\frac{56634819127}{79750052140}a^{4}+\frac{7979427929}{2636365360}a^{3}-\frac{13983252365}{15950010428}a^{2}+\frac{5430555033}{5800003792}a-\frac{2174887863}{1318182680}$ 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:  \( 45687.5845647 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 45687.5845647 \cdot 80}{2\cdot\sqrt{167442686719647879731871744}}\cr\approx \mathstrut & 0.343055229767 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641)
 
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 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641, 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 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641);
 
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 + 16*x^14 + 248*x^12 + 1696*x^10 + 8098*x^8 + 17744*x^6 + 32408*x^4 + 27104*x^2 + 14641);
 
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

$C_2^2:Q_8$ (as 16T31):

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 14 conjugacy class representatives for $C_2^2:Q_8$
Character table for $C_2^2:Q_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.0.158976.2, 4.0.39744.5, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.718886928384.34, 8.0.101093474304.27, 8.8.12230590464.1

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

Galois closure: deg 32
Degree 16 sibling: deg 16
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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.50.3$x^{16} + 8 x^{15} + 4 x^{14} + 10 x^{12} + 8 x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{3} + 8 x^{2} + 30$$16$$1$$50$16T31$[2, 2, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.16.12.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 36 x^{12} + 120 x^{11} + 312 x^{10} + 352 x^{9} - 522 x^{8} - 2664 x^{7} - 3672 x^{6} - 1440 x^{5} + 4292 x^{4} + 7720 x^{3} + 6408 x^{2} + 2592 x + 433$$4$$4$$12$$C_4:C_4$$[\ ]_{4}^{4}$
\(23\) Copy content Toggle raw display 23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$