↑

Properties

Label 16.0.180355734956640625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.804\times 10^{17}$
Root discriminant \(11.98\)
Ramified primes $5,59,131$
Class number $1$
Class group trivial
Galois group $Q_8^2.A_4^2.D_4$ (as 16T1871)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 1)
 
gp: K = bnfinit(y^16 - 5*y^15 + 14*y^14 - 28*y^13 + 46*y^12 - 63*y^11 + 73*y^10 - 72*y^9 + 57*y^8 - 34*y^7 + 23*y^6 - 21*y^5 + 8*y^4 + 8*y^3 - 7*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 1)
 

\( x^{16} - 5 x^{15} + 14 x^{14} - 28 x^{13} + 46 x^{12} - 63 x^{11} + 73 x^{10} - 72 x^{9} + 57 x^{8} + \cdots + 1 \) 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:   \(180355734956640625\) \(\medspace = 5^{8}\cdot 59^{3}\cdot 131^{3}\) 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:  \(11.98\)
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:  $5^{1/2}59^{1/2}131^{3/4}\approx 665.0655921280412$
Ramified primes:   \(5\), \(59\), \(131\) 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:  \(\Q(\sqrt{7729}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6911}a^{15}-\frac{3300}{6911}a^{14}+\frac{2511}{6911}a^{13}-\frac{1306}{6911}a^{12}-\frac{2237}{6911}a^{11}-\frac{3185}{6911}a^{10}-\frac{3161}{6911}a^{9}+\frac{546}{6911}a^{8}-\frac{2153}{6911}a^{7}+\frac{3415}{6911}a^{6}-\frac{1294}{6911}a^{5}-\frac{378}{6911}a^{4}+\frac{1538}{6911}a^{3}-\frac{1939}{6911}a^{2}+\frac{3234}{6911}a+\frac{732}{6911}$ 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:   \( -1 \)  (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:   $\frac{22955}{6911}a^{15}-\frac{110605}{6911}a^{14}+\frac{306349}{6911}a^{13}-\frac{614391}{6911}a^{12}+\frac{1021223}{6911}a^{11}-\frac{1416961}{6911}a^{10}+\frac{1677207}{6911}a^{9}-\frac{1710141}{6911}a^{8}+\frac{1422112}{6911}a^{7}-\frac{953866}{6911}a^{6}+\frac{711541}{6911}a^{5}-\frac{584209}{6911}a^{4}+\frac{245287}{6911}a^{3}+\frac{73116}{6911}a^{2}-\frac{84424}{6911}a+\frac{9330}{6911}$, $\frac{41408}{6911}a^{15}-\frac{216349}{6911}a^{14}+\frac{635305}{6911}a^{13}-\frac{1334096}{6911}a^{12}+\frac{2292889}{6911}a^{11}-\frac{3298414}{6911}a^{10}+\frac{4053498}{6911}a^{9}-\frac{4288844}{6911}a^{8}+\frac{3794615}{6911}a^{7}-\frac{2734407}{6911}a^{6}+\frac{1961755}{6911}a^{5}-\frac{1574517}{6911}a^{4}+\frac{885247}{6911}a^{3}-\frac{32669}{6911}a^{2}-\frac{208305}{6911}a+\frac{75031}{6911}$, $\frac{19675}{6911}a^{15}-\frac{109231}{6911}a^{14}+\frac{335825}{6911}a^{13}-\frac{733018}{6911}a^{12}+\frac{1295541}{6911}a^{11}-\frac{1917185}{6911}a^{10}+\frac{2425175}{6911}a^{9}-\frac{2644057}{6911}a^{8}+\frac{2436827}{6911}a^{7}-\frac{1850765}{6911}a^{6}+\frac{1320675}{6911}a^{5}-\frac{1023742}{6911}a^{4}+\frac{653426}{6911}a^{3}-\frac{139325}{6911}a^{2}-\frac{118114}{6911}a+\frac{75597}{6911}$, $a$, $\frac{10699}{6911}a^{15}-\frac{60600}{6911}a^{14}+\frac{188729}{6911}a^{13}-\frac{413512}{6911}a^{12}+\frac{731696}{6911}a^{11}-\frac{1083201}{6911}a^{10}+\frac{1371273}{6911}a^{9}-\frac{1490917}{6911}a^{8}+\frac{1374705}{6911}a^{7}-\frac{1038022}{6911}a^{6}+\frac{730793}{6911}a^{5}-\frac{574900}{6911}a^{4}+\frac{380076}{6911}a^{3}-\frac{67649}{6911}a^{2}-\frac{78832}{6911}a+\frac{36060}{6911}$, $\frac{48172}{6911}a^{15}-\frac{256485}{6911}a^{14}+\frac{756869}{6911}a^{13}-\frac{1598240}{6911}a^{12}+\frac{2753037}{6911}a^{11}-\frac{3977445}{6911}a^{10}+\frac{4898270}{6911}a^{9}-\frac{5205337}{6911}a^{8}+\frac{4622520}{6911}a^{7}-\frac{3346988}{6911}a^{6}+\frac{2393858}{6911}a^{5}-\frac{1926700}{6911}a^{4}+\frac{1087643}{6911}a^{3}-\frac{58631}{6911}a^{2}-\frac{262132}{6911}a+\frac{91825}{6911}$, $\frac{50717}{6911}a^{15}-\frac{271942}{6911}a^{14}+\frac{809977}{6911}a^{13}-\frac{1722217}{6911}a^{12}+\frac{2982599}{6911}a^{11}-\frac{4329128}{6911}a^{10}+\frac{5360966}{6911}a^{9}-\frac{5723203}{6911}a^{8}+\frac{5121150}{6911}a^{7}-\frac{3743778}{6911}a^{6}+\frac{2666704}{6911}a^{5}-\frac{2128500}{6911}a^{4}+\frac{1249180}{6911}a^{3}-\frac{114220}{6911}a^{2}-\frac{283336}{6911}a+\frac{116439}{6911}$ 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:  \( 69.5094645498 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 69.5094645498 \cdot 1}{2\cdot\sqrt{180355734956640625}}\cr\approx \mathstrut & 0.198786970866 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 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 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 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 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 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 - 5*x^15 + 14*x^14 - 28*x^13 + 46*x^12 - 63*x^11 + 73*x^10 - 72*x^9 + 57*x^8 - 34*x^7 + 23*x^6 - 21*x^5 + 8*x^4 + 8*x^3 - 7*x^2 + 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

$Q_8^2.A_4^2.D_4$ (as 16T1871):

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 73728
The 104 conjugacy class representatives for $Q_8^2.A_4^2.D_4$
Character table for $Q_8^2.A_4^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 8.0.4830625.1

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 ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ $16$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ R

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
\(5\) Copy content Toggle raw display 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(59\) Copy content Toggle raw display $\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 57$$1$$1$$0$Trivial$[\ ]$
59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.1$x^{2} + 118$$2$$1$$1$$C_2$$[\ ]_{2}$
59.8.0.1$x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
\(131\) Copy content Toggle raw display $\Q_{131}$$x + 129$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 129$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 129$$1$$1$$0$Trivial$[\ ]$
$\Q_{131}$$x + 129$$1$$1$$0$Trivial$[\ ]$
131.2.0.1$x^{2} + 127 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
131.3.0.1$x^{3} + 3 x + 129$$1$$3$$0$$C_3$$[\ ]^{3}$
131.3.0.1$x^{3} + 3 x + 129$$1$$3$$0$$C_3$$[\ ]^{3}$
131.4.3.2$x^{4} + 262$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$