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Properties

Label 16.4.321...808.59
Degree $16$
Signature $[4, 6]$
Discriminant $3.212\times 10^{29}$
Root discriminant \(69.85\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.C_8$ (as 16T1155)

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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 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818)
 
gp: K = bnfinit(y^16 - 72*y^12 + 1764*y^8 - 3024*y^6 - 42768*y^4 - 54432*y^2 + 46818, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 72*x^12 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 72*x^12 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818)
 

\( x^{16} - 72x^{12} + 1764x^{8} - 3024x^{6} - 42768x^{4} - 54432x^{2} + 46818 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(321236373250909071617512439808\) \(\medspace = 2^{79}\cdot 3^{12}\) 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:  \(69.85\)
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:  $2^{2849/512}3^{3/4}\approx 107.8718783637264$
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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$, $\frac{1}{9}a^{11}$, $\frac{1}{189}a^{12}-\frac{2}{63}a^{8}-\frac{2}{21}a^{4}+\frac{3}{7}$, $\frac{1}{189}a^{13}-\frac{2}{63}a^{9}-\frac{2}{21}a^{5}+\frac{3}{7}a$, $\frac{1}{5753907117}a^{14}-\frac{2595043}{5753907117}a^{12}-\frac{45631847}{1917969039}a^{10}-\frac{81979465}{1917969039}a^{8}-\frac{32490173}{639323013}a^{6}-\frac{96726995}{639323013}a^{4}+\frac{3100560}{6874441}a^{2}-\frac{5341611}{213107671}$, $\frac{1}{97816420989}a^{15}+\frac{10592275}{5753907117}a^{13}-\frac{1324277873}{32605473663}a^{11}-\frac{38847916}{1917969039}a^{9}-\frac{1737351541}{10868491221}a^{7}-\frac{225054034}{3622830407}a^{5}+\frac{51221647}{116865497}a^{3}+\frac{1181972556}{3622830407}a$ 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 $1$ (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:  $9$
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:   $\frac{20960}{213107671}a^{14}-\frac{885793}{5753907117}a^{12}-\frac{13497802}{1917969039}a^{10}+\frac{8527810}{639323013}a^{8}+\frac{30383240}{213107671}a^{6}-\frac{387090139}{639323013}a^{4}-\frac{18401168}{6874441}a^{2}+\frac{190479733}{213107671}$, $\frac{126748}{5753907117}a^{14}+\frac{13984}{1917969039}a^{12}+\frac{1050872}{639323013}a^{10}-\frac{1036751}{1917969039}a^{8}-\frac{28187000}{639323013}a^{6}+\frac{17171116}{213107671}a^{4}+\frac{8264208}{6874441}a^{2}-\frac{65447645}{213107671}$, $\frac{349312}{1917969039}a^{14}-\frac{2435570}{5753907117}a^{12}-\frac{24922102}{1917969039}a^{10}+\frac{23744404}{639323013}a^{8}+\frac{54815800}{213107671}a^{6}-\frac{893602303}{639323013}a^{4}-\frac{29843372}{6874441}a^{2}+\frac{1006935757}{213107671}$, $\frac{979715}{213107671}a^{14}+\frac{12464477}{821986731}a^{12}+\frac{539308157}{1917969039}a^{10}-\frac{28128490}{30443953}a^{8}-\frac{1079774387}{213107671}a^{6}+\frac{930243371}{30443953}a^{4}+\frac{663138646}{6874441}a^{2}-\frac{2011102325}{30443953}$, $\frac{7092142}{213107671}a^{14}-\frac{200698648}{1917969039}a^{12}-\frac{1311716018}{639323013}a^{10}+\frac{4097843000}{639323013}a^{8}+\frac{24033054026}{639323013}a^{6}-\frac{46100946466}{213107671}a^{4}-\frac{4983537910}{6874441}a^{2}+\frac{80318509153}{213107671}$, $\frac{8453933}{32605473663}a^{15}-\frac{714865}{1917969039}a^{14}+\frac{124791}{213107671}a^{13}+\frac{39858373}{5753907117}a^{12}-\frac{443080430}{32605473663}a^{11}+\frac{1999338}{213107671}a^{10}-\frac{44146693}{1917969039}a^{9}-\frac{79167321}{213107671}a^{8}-\frac{936290168}{10868491221}a^{7}-\frac{91325303}{213107671}a^{6}-\frac{4833124264}{3622830407}a^{5}+\frac{1097693143}{213107671}a^{4}-\frac{126158640}{116865497}a^{3}+\frac{63817280}{6874441}a^{2}+\frac{20292492692}{3622830407}a-\frac{771710885}{213107671}$, $\frac{354569660}{97816420989}a^{15}-\frac{163612}{185609907}a^{14}+\frac{34035182}{1917969039}a^{13}+\frac{1932337}{185609907}a^{12}+\frac{691771217}{3622830407}a^{11}-\frac{152276}{61869969}a^{10}-\frac{641173657}{639323013}a^{9}-\frac{2590176}{6874441}a^{8}-\frac{25544671060}{10868491221}a^{7}+\frac{8146484}{6874441}a^{6}+\frac{282425063624}{10868491221}a^{5}+\frac{26295973}{6874441}a^{4}+\frac{4679012768}{116865497}a^{3}-\frac{92263776}{6874441}a^{2}-\frac{327252505796}{3622830407}a-\frac{268056849}{6874441}$, $\frac{219118643}{97816420989}a^{15}+\frac{12650122}{5753907117}a^{14}-\frac{97315699}{5753907117}a^{13}-\frac{139576970}{5753907117}a^{12}-\frac{3428573236}{32605473663}a^{11}-\frac{20122409}{639323013}a^{10}+\frac{1978671292}{1917969039}a^{9}+\frac{713549504}{639323013}a^{8}+\frac{5004909794}{10868491221}a^{7}-\frac{454007476}{213107671}a^{6}-\frac{258912090596}{10868491221}a^{5}-\frac{11047701943}{639323013}a^{4}+\frac{1415990774}{116865497}a^{3}+\frac{242900646}{6874441}a^{2}+\frac{663240826770}{3622830407}a+\frac{26605289295}{213107671}$, $\frac{196561741}{97816420989}a^{15}-\frac{13789238}{5753907117}a^{14}-\frac{7728731}{5753907117}a^{13}-\frac{4589201}{5753907117}a^{12}+\frac{3362152636}{32605473663}a^{11}+\frac{9214465}{213107671}a^{10}-\frac{29834278}{639323013}a^{9}+\frac{18417787}{639323013}a^{8}-\frac{8356172234}{10868491221}a^{7}+\frac{2733568294}{639323013}a^{6}+\frac{172042060852}{10868491221}a^{5}+\frac{4584201793}{213107671}a^{4}+\frac{7883459362}{116865497}a^{3}-\frac{25122042}{6874441}a^{2}+\frac{222220005876}{3622830407}a-\frac{18564640125}{213107671}$ 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:  \( 516372003.4504026 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 516372003.4504026 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.448455678777396 \end{aligned}\] (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 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818)
 
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 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818, 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 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818);
 
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 + 1764*x^8 - 3024*x^6 - 42768*x^4 - 54432*x^2 + 46818);
 
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

$C_2^7.C_8$ (as 16T1155):

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 40 conjugacy class representatives for $C_2^7.C_8$
Character table for $C_2^7.C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.173946175488.2

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: 16.8.321236373250909071617512439808.2, 16.12.321236373250909071617512439808.2, 16.8.321236373250909071617512439808.5, 16.8.321236373250909071617512439808.6, 16.4.321236373250909071617512439808.3, 16.12.321236373250909071617512439808.4, 16.8.321236373250909071617512439808.11, 16.4.321236373250909071617512439808.22, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.321236373250909071617512439808.20

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 $16$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ $16$ $16$

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 2.16.79.2532$x^{16} + 40 x^{12} + 32 x^{10} + 32 x^{9} + 36 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 2$$16$$1$$79$16T1155$[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$
\(3\) Copy content Toggle raw display 3.16.12.3$x^{16} - 6 x^{12} + 162$$4$$4$$12$$C_{16} : C_2$$[\ ]_{4}^{8}$