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Properties

Label 16.0.176...000.7
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 $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34)
 
gp: K = bnfinit(y^16 - 8*y^15 + 44*y^14 - 168*y^13 + 532*y^12 - 1372*y^11 + 2998*y^10 - 5464*y^9 + 8367*y^8 - 10628*y^7 + 11082*y^6 - 9292*y^5 + 6080*y^4 - 2972*y^3 + 992*y^2 - 192*y + 34, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34)
 

\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 532 x^{12} - 1372 x^{11} + 2998 x^{10} - 5464 x^{9} + \cdots + 34 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5}a^{10}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}$, $\frac{1}{65}a^{13}-\frac{1}{65}a^{11}-\frac{1}{65}a^{10}-\frac{3}{65}a^{9}+\frac{8}{65}a^{8}-\frac{7}{65}a^{7}+\frac{6}{13}a^{6}+\frac{1}{5}a^{5}+\frac{18}{65}a^{4}+\frac{1}{13}a^{3}+\frac{2}{5}a^{2}-\frac{4}{65}a+\frac{29}{65}$, $\frac{1}{2258815}a^{14}-\frac{7}{2258815}a^{13}-\frac{11240}{451763}a^{12}-\frac{114472}{2258815}a^{11}-\frac{128488}{2258815}a^{10}-\frac{190746}{2258815}a^{9}+\frac{546249}{2258815}a^{8}-\frac{492374}{2258815}a^{7}-\frac{1004}{451763}a^{6}-\frac{4891}{38285}a^{5}-\frac{154277}{451763}a^{4}+\frac{147216}{2258815}a^{3}+\frac{441229}{2258815}a^{2}+\frac{460803}{2258815}a+\frac{120212}{451763}$, $\frac{1}{191999275}a^{15}+\frac{7}{38399855}a^{14}+\frac{395269}{191999275}a^{13}+\frac{14240359}{191999275}a^{12}-\frac{10809231}{191999275}a^{11}+\frac{1502777}{38399855}a^{10}+\frac{38614743}{191999275}a^{9}+\frac{19127138}{38399855}a^{8}-\frac{81672733}{191999275}a^{7}+\frac{82624983}{191999275}a^{6}-\frac{36382959}{191999275}a^{5}+\frac{19701791}{191999275}a^{4}-\frac{79662432}{191999275}a^{3}+\frac{47001727}{191999275}a^{2}-\frac{14379202}{191999275}a-\frac{933289}{11294075}$ 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

$C_{4}$, 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:   \( -\frac{4823098}{191999275} a^{15} + \frac{7234647}{38399855} a^{14} - \frac{192552307}{191999275} a^{13} + \frac{54074046}{14769175} a^{12} - \frac{2159919287}{191999275} a^{11} + \frac{11271469}{404209} a^{10} - \frac{11265206284}{191999275} a^{9} + \frac{126137526}{1238705} a^{8} - \frac{1495663829}{10105225} a^{7} + \frac{33855565581}{191999275} a^{6} - \frac{2530855236}{14769175} a^{5} + \frac{25319626792}{191999275} a^{4} - \frac{15106279619}{191999275} a^{3} + \frac{346455961}{10105225} a^{2} - \frac{1731843354}{191999275} a + \frac{11091147}{11294075} \)  (order $8$) 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:   $\frac{4664138}{191999275}a^{15}-\frac{1294195}{7679971}a^{14}+\frac{170250157}{191999275}a^{13}-\frac{599920273}{191999275}a^{12}+\frac{1828818722}{191999275}a^{11}-\frac{881883744}{38399855}a^{10}+\frac{9141681464}{191999275}a^{9}-\frac{3082689071}{38399855}a^{8}+\frac{21831097801}{191999275}a^{7}-\frac{24956631956}{191999275}a^{6}+\frac{22912318683}{191999275}a^{5}-\frac{16239280432}{191999275}a^{4}+\frac{8411997249}{191999275}a^{3}-\frac{3088152604}{191999275}a^{2}+\frac{666990029}{191999275}a-\frac{12600977}{11294075}$, $\frac{342422}{191999275}a^{15}+\frac{11599}{38399855}a^{14}-\frac{1045807}{191999275}a^{13}+\frac{21908298}{191999275}a^{12}-\frac{61284302}{191999275}a^{11}+\frac{45508612}{38399855}a^{10}-\frac{437788679}{191999275}a^{9}+\frac{171921631}{38399855}a^{8}-\frac{924367491}{191999275}a^{7}+\frac{769612371}{191999275}a^{6}+\frac{195500247}{191999275}a^{5}-\frac{1175000423}{191999275}a^{4}+\frac{1738130026}{191999275}a^{3}-\frac{1534335701}{191999275}a^{2}+\frac{831228731}{191999275}a-\frac{9586163}{11294075}$, $\frac{24243}{650845}a^{15}-\frac{10062853}{38399855}a^{14}+\frac{51955212}{38399855}a^{13}-\frac{181538777}{38399855}a^{12}+\frac{541480227}{38399855}a^{11}-\frac{1287650552}{38399855}a^{10}+\frac{27296937}{404209}a^{9}-\frac{4251421258}{38399855}a^{8}+\frac{5746444817}{38399855}a^{7}-\frac{328181643}{2021045}a^{6}+\frac{89703576}{650845}a^{5}-\frac{3394141629}{38399855}a^{4}+\frac{1552766004}{38399855}a^{3}-\frac{90158967}{7679971}a^{2}+\frac{62772767}{38399855}a+\frac{713489}{2258815}$, $\frac{630818}{191999275}a^{15}-\frac{87357}{2953835}a^{14}+\frac{32563107}{191999275}a^{13}-\frac{132077848}{191999275}a^{12}+\frac{432260787}{191999275}a^{11}-\frac{233326031}{38399855}a^{10}+\frac{2648592274}{191999275}a^{9}-\frac{1014302446}{38399855}a^{8}+\frac{8169959916}{191999275}a^{7}-\frac{11027489061}{191999275}a^{6}+\frac{12408222753}{191999275}a^{5}-\frac{11469000137}{191999275}a^{4}+\frac{8580346104}{191999275}a^{3}-\frac{5007396129}{191999275}a^{2}+\frac{2139865164}{191999275}a-\frac{34115947}{11294075}$, $\frac{421748}{191999275}a^{15}-\frac{632622}{38399855}a^{14}+\frac{14502452}{191999275}a^{13}-\frac{3560931}{14769175}a^{12}+\frac{113410002}{191999275}a^{11}-\frac{2316677}{2021045}a^{10}+\frac{270179124}{191999275}a^{9}-\frac{663426}{1238705}a^{8}-\frac{30779936}{10105225}a^{7}+\frac{1687952634}{191999275}a^{6}-\frac{221611589}{14769175}a^{5}+\frac{3319027038}{191999275}a^{4}-\frac{2600806386}{191999275}a^{3}+\frac{74694189}{10105225}a^{2}-\frac{25842002}{14769175}a-\frac{1464417}{11294075}$, $\frac{2116189}{191999275}a^{15}-\frac{4199086}{38399855}a^{14}+\frac{115889381}{191999275}a^{13}-\frac{24338491}{10105225}a^{12}+\frac{1427600476}{191999275}a^{11}-\frac{737356831}{38399855}a^{10}+\frac{130746808}{3254225}a^{9}-\frac{2686675441}{38399855}a^{8}+\frac{18516516043}{191999275}a^{7}-\frac{49748696}{476425}a^{6}+\frac{494685044}{6193525}a^{5}-\frac{6250146431}{191999275}a^{4}-\frac{1598414238}{191999275}a^{3}+\frac{4335788998}{191999275}a^{2}-\frac{2256064933}{191999275}a+\frac{8110689}{11294075}$, $\frac{7286816}{191999275}a^{15}-\frac{19542}{95285}a^{14}+\frac{9617261}{10105225}a^{13}-\frac{504904391}{191999275}a^{12}+\frac{1309242089}{191999275}a^{11}-\frac{452208881}{38399855}a^{10}+\frac{3081495903}{191999275}a^{9}-\frac{279582203}{38399855}a^{8}-\frac{3657329963}{191999275}a^{7}+\frac{12465329398}{191999275}a^{6}-\frac{20893123764}{191999275}a^{5}+\frac{23452012321}{191999275}a^{4}-\frac{18655547527}{191999275}a^{3}+\frac{10208154592}{191999275}a^{2}-\frac{3310171977}{191999275}a+\frac{24380871}{11294075}$ 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:  \( 10910.5936005 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10910.5936005 \cdot 4}{8\cdot\sqrt{176120502681600000000}}\cr\approx \mathstrut & 0.998509893732 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34)
 
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 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34, 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 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34);
 
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 - 8*x^15 + 44*x^14 - 168*x^13 + 532*x^12 - 1372*x^11 + 2998*x^10 - 5464*x^9 + 8367*x^8 - 10628*x^7 + 11082*x^6 - 9292*x^5 + 6080*x^4 - 2972*x^3 + 992*x^2 - 192*x + 34);
 
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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{15}) \), 4.0.1280.1, 4.0.320.1, 4.0.2880.1, 4.0.11520.1, \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{15})\), 8.0.3317760000.4, 8.0.6553600.1, 8.0.530841600.3, 8.0.13271040000.3, 8.0.13271040000.4, 8.0.3317760000.11, 8.0.207360000.3

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 siblings: 16.0.281792804290560000.1, 16.0.11007531417600000000.6, 16.0.11007531417600000000.9, 16.0.176120502681600000000.4, 16.8.176120502681600000000.2, 16.0.176120502681600000000.5, 16.0.176120502681600000000.10
Minimal sibling: 16.0.281792804290560000.1

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.18.53$x^{8} + 2 x^{6} + 4 x^{3} + 2$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
\(3\) 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}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$