Properties

Label 16.0.3091162921500672.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.091\times 10^{15}$
Root discriminant \(9.29\)
Ramified primes $2,3,193$
Class number $1$
Class group trivial
Galois group $C_4\wr C_2^2$ (as 16T1202)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + 8*y^14 - 18*y^13 + 35*y^12 - 62*y^11 + 92*y^10 - 118*y^9 + 138*y^8 - 142*y^7 + 127*y^6 - 104*y^5 + 77*y^4 - 46*y^3 + 20*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*x + 1)
 

\( x^{16} - 2 x^{15} + 8 x^{14} - 18 x^{13} + 35 x^{12} - 62 x^{11} + 92 x^{10} - 118 x^{9} + 138 x^{8} - 142 x^{7} + 127 x^{6} - 104 x^{5} + 77 x^{4} - 46 x^{3} + 20 x^{2} - 6 x + 1 \) 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:   \(3091162921500672\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 193^{3}\) 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:  \(9.29\)
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:  not computed
Ramified primes:   \(2\), \(3\), \(193\) 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:  \(\Q(\sqrt{193}) \)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}+\frac{8}{19}a^{13}-\frac{2}{19}a^{12}-\frac{7}{19}a^{11}-\frac{3}{19}a^{9}+\frac{9}{19}a^{8}+\frac{4}{19}a^{7}-\frac{1}{19}a^{6}-\frac{8}{19}a^{5}-\frac{6}{19}a^{4}+\frac{8}{19}a^{3}+\frac{8}{19}a^{2}-\frac{2}{19}a+\frac{9}{19}$ 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:   \( -\frac{6292}{19} a^{15} + 420 a^{14} - \frac{44503}{19} a^{13} + \frac{80699}{19} a^{12} - \frac{161213}{19} a^{11} + 14327 a^{10} - \frac{379820}{19} a^{9} + \frac{464770}{19} a^{8} - \frac{528516}{19} a^{7} + \frac{507094}{19} a^{6} - \frac{428407}{19} a^{5} + \frac{341220}{19} a^{4} - \frac{235054}{19} a^{3} + \frac{117624}{19} a^{2} - \frac{39894}{19} a + \frac{8599}{19} \)  (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:   $\frac{11035}{19}a^{15}-736a^{14}+\frac{78039}{19}a^{13}-\frac{141447}{19}a^{12}+\frac{282615}{19}a^{11}-25111a^{10}+\frac{665696}{19}a^{9}-\frac{814433}{19}a^{8}+\frac{926139}{19}a^{7}-\frac{888417}{19}a^{6}+\frac{750494}{19}a^{5}-\frac{597678}{19}a^{4}+\frac{411679}{19}a^{3}-\frac{205878}{19}a^{2}+\frac{69757}{19}a-\frac{15027}{19}$, $418a^{15}-530a^{14}+2956a^{13}-5360a^{12}+10706a^{11}-18078a^{10}+25221a^{9}-30858a^{8}+35091a^{7}-33662a^{6}+28437a^{5}-22648a^{4}+15600a^{3}-7802a^{2}+2645a-570$, $\frac{4694}{19}a^{15}-312a^{14}+\frac{33182}{19}a^{13}-\frac{60023}{19}a^{12}+\frac{120035}{19}a^{11}-10657a^{10}+\frac{282489}{19}a^{9}-\frac{345468}{19}a^{8}+\frac{392791}{19}a^{7}-\frac{376619}{19}a^{6}+\frac{318052}{19}a^{5}-\frac{253238}{19}a^{4}+\frac{174352}{19}a^{3}-\frac{87088}{19}a^{2}+\frac{29448}{19}a-\frac{6337}{19}$, $\frac{9936}{19}a^{15}-662a^{14}+\frac{70254}{19}a^{13}-\frac{127279}{19}a^{12}+\frac{254322}{19}a^{11}-22598a^{10}+\frac{598940}{19}a^{9}-\frac{732839}{19}a^{8}+\frac{833317}{19}a^{7}-\frac{799329}{19}a^{6}+\frac{675306}{19}a^{5}-\frac{537846}{19}a^{4}+\frac{370454}{19}a^{3}-\frac{185277}{19}a^{2}+\frac{62835}{19}a-\frac{13556}{19}$, $\frac{11321}{19}a^{15}-754a^{14}+\frac{80042}{19}a^{13}-\frac{144983}{19}a^{12}+\frac{289714}{19}a^{11}-25741a^{10}+\frac{682204}{19}a^{9}-\frac{834678}{19}a^{8}+\frac{949038}{19}a^{7}-\frac{910268}{19}a^{6}+\frac{768916}{19}a^{5}-\frac{612352}{19}a^{4}+\frac{421719}{19}a^{3}-\frac{210848}{19}a^{2}+\frac{71446}{19}a-\frac{15398}{19}$, $\frac{212}{19}a^{15}-15a^{14}+\frac{1506}{19}a^{13}-\frac{2818}{19}a^{12}+\frac{5546}{19}a^{11}-496a^{10}+\frac{13196}{19}a^{9}-\frac{16104}{19}a^{8}+\frac{18347}{19}a^{7}-\frac{17635}{19}a^{6}+\frac{14872}{19}a^{5}-\frac{11817}{19}a^{4}+\frac{8175}{19}a^{3}-\frac{4061}{19}a^{2}+\frac{1343}{19}a-\frac{277}{19}$, $\frac{1635}{19}a^{15}-108a^{14}+\frac{11541}{19}a^{13}-\frac{20826}{19}a^{12}+\frac{41641}{19}a^{11}-3698a^{10}+\frac{97866}{19}a^{9}-\frac{119710}{19}a^{8}+\frac{136025}{19}a^{7}-\frac{130322}{19}a^{6}+\frac{110002}{19}a^{5}-\frac{87577}{19}a^{4}+\frac{60219}{19}a^{3}-\frac{30031}{19}a^{2}+\frac{10125}{19}a-\frac{2176}{19}$ 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:  \( 43.8698041698 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 43.8698041698 \cdot 1}{12\cdot\sqrt{3091162921500672}}\cr\approx \mathstrut & 0.159721157504 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*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 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*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 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*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 - 2*x^15 + 8*x^14 - 18*x^13 + 35*x^12 - 62*x^11 + 92*x^10 - 118*x^9 + 138*x^8 - 142*x^7 + 127*x^6 - 104*x^5 + 77*x^4 - 46*x^3 + 20*x^2 - 6*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

$C_4\wr C_2^2$ (as 16T1202):

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 1024
The 106 conjugacy class representatives for $C_4\wr C_2^2$ are not computed
Character table for $C_4\wr C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.4002048.1

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 siblings: 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 ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 3.8.4.1$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}$
\(193\) Copy content Toggle raw display $\Q_{193}$$x + 188$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 188$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 188$$1$$1$$0$Trivial$[\ ]$
$\Q_{193}$$x + 188$$1$$1$$0$Trivial$[\ ]$
193.2.0.1$x^{2} + 192 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} + 192 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} + 192 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.2.0.1$x^{2} + 192 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
193.4.3.1$x^{4} + 193$$4$$1$$3$$C_4$$[\ ]_{4}$