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Properties

Label 16.4.513114342400000000.2
Degree $16$
Signature $[4, 6]$
Discriminant $5.131\times 10^{17}$
Root discriminant \(12.79\)
Ramified primes $2,5,11,37$
Class number $1$
Class group trivial
Galois group $C_2^7.C_2\wr D_4$ (as 16T1772)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*x^2 - 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 6*y^14 - 12*y^13 + 25*y^12 - 10*y^11 - 15*y^10 - 36*y^9 + 83*y^8 + 2*y^7 - 81*y^6 + 32*y^5 + 20*y^4 - 12*y^3 + 3*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*x^2 - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*x^2 - 1)
 

\( x^{16} - 4 x^{15} + 6 x^{14} - 12 x^{13} + 25 x^{12} - 10 x^{11} - 15 x^{10} - 36 x^{9} + 83 x^{8} + \cdots - 1 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(513114342400000000\) \(\medspace = 2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 37^{2}\) 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:  \(12.79\)
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:   \(2\), \(5\), \(11\), \(37\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{714194109}a^{15}+\frac{53675077}{714194109}a^{14}+\frac{64179110}{714194109}a^{13}+\frac{59201113}{714194109}a^{12}-\frac{10977509}{238064703}a^{11}+\frac{76503227}{714194109}a^{10}+\frac{26568043}{714194109}a^{9}-\frac{230315488}{714194109}a^{8}-\frac{26717602}{79354901}a^{7}+\frac{197563781}{714194109}a^{6}-\frac{52833233}{714194109}a^{5}+\frac{347822080}{714194109}a^{4}+\frac{66767962}{714194109}a^{3}-\frac{276819460}{714194109}a^{2}+\frac{135604225}{714194109}a-\frac{90622693}{714194109}$ 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:  $9$
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:   $\frac{93382436}{714194109}a^{15}-\frac{516225034}{714194109}a^{14}+\frac{1057374247}{714194109}a^{13}-\frac{1767220525}{714194109}a^{12}+\frac{1269809591}{238064703}a^{11}-\frac{3833188649}{714194109}a^{10}-\frac{1060768831}{714194109}a^{9}-\frac{1371015518}{714194109}a^{8}+\frac{1498308909}{79354901}a^{7}-\frac{8668485347}{714194109}a^{6}-\frac{11669782792}{714194109}a^{5}+\frac{12051100778}{714194109}a^{4}+\frac{1086069437}{714194109}a^{3}-\frac{2565441464}{714194109}a^{2}-\frac{68549632}{714194109}a-\frac{413550722}{714194109}$, $\frac{111134804}{714194109}a^{15}-\frac{8073664}{714194109}a^{14}-\frac{914451812}{714194109}a^{13}+\frac{833742398}{714194109}a^{12}-\frac{743164684}{238064703}a^{11}+\frac{8800940122}{714194109}a^{10}-\frac{4174617844}{714194109}a^{9}-\frac{7523258741}{714194109}a^{8}-\frac{1071523004}{79354901}a^{7}+\frac{28263590524}{714194109}a^{6}-\frac{4109043196}{714194109}a^{5}-\frac{18253095229}{714194109}a^{4}+\frac{7547060072}{714194109}a^{3}+\frac{1910461186}{714194109}a^{2}-\frac{909107464}{714194109}a+\frac{844301839}{714194109}$, $\frac{91585642}{714194109}a^{15}-\frac{228437030}{714194109}a^{14}+\frac{341022374}{714194109}a^{13}-\frac{1187063678}{714194109}a^{12}+\frac{503083969}{238064703}a^{11}-\frac{575507221}{714194109}a^{10}+\frac{1944193132}{714194109}a^{9}-\frac{3090200788}{714194109}a^{8}+\frac{190818927}{79354901}a^{7}-\frac{2759701447}{714194109}a^{6}+\frac{2369186110}{714194109}a^{5}+\frac{2979769708}{714194109}a^{4}-\frac{3288761552}{714194109}a^{3}-\frac{1223961094}{714194109}a^{2}+\frac{1535822158}{714194109}a+\frac{68390501}{714194109}$, $\frac{162243715}{714194109}a^{15}-\frac{554572019}{714194109}a^{14}+\frac{656352449}{714194109}a^{13}-\frac{1662023756}{714194109}a^{12}+\frac{1144818559}{238064703}a^{11}-\frac{209587471}{714194109}a^{10}-\frac{1265028812}{714194109}a^{9}-\frac{8974391014}{714194109}a^{8}+\frac{1081465303}{79354901}a^{7}+\frac{5224036508}{714194109}a^{6}-\frac{5884957880}{714194109}a^{5}-\frac{4688203085}{714194109}a^{4}+\frac{3067555966}{714194109}a^{3}+\frac{1811157848}{714194109}a^{2}+\frac{158064862}{714194109}a-\frac{376986712}{714194109}$, $a$, $\frac{7952824}{714194109}a^{15}-\frac{379023089}{714194109}a^{14}+\frac{1346951027}{714194109}a^{13}-\frac{2150182958}{714194109}a^{12}+\frac{1605042853}{238064703}a^{11}-\frac{8832664597}{714194109}a^{10}+\frac{4498990981}{714194109}a^{9}+\frac{95302847}{714194109}a^{8}+\frac{1566187968}{79354901}a^{7}-\frac{24919432603}{714194109}a^{6}+\frac{5559971542}{714194109}a^{5}+\frac{15440044930}{714194109}a^{4}-\frac{10183610921}{714194109}a^{3}-\frac{960297739}{714194109}a^{2}+\frac{2002103182}{714194109}a-\frac{911143279}{714194109}$, $\frac{634508387}{714194109}a^{15}-\frac{2294518081}{714194109}a^{14}+\frac{3177383785}{714194109}a^{13}-\frac{7085713360}{714194109}a^{12}+\frac{4595399783}{238064703}a^{11}-\frac{3230969675}{714194109}a^{10}-\frac{7377079237}{714194109}a^{9}-\frac{23611384307}{714194109}a^{8}+\frac{4638812318}{79354901}a^{7}+\frac{7495328113}{714194109}a^{6}-\frac{41394003784}{714194109}a^{5}+\frac{14263941728}{714194109}a^{4}+\frac{7890161945}{714194109}a^{3}-\frac{5007118430}{714194109}a^{2}+\frac{2012477318}{714194109}a-\frac{471031748}{714194109}$, $\frac{135347929}{714194109}a^{15}-\frac{533646827}{714194109}a^{14}+\frac{931463942}{714194109}a^{13}-\frac{2071923524}{714194109}a^{12}+\frac{1318840924}{238064703}a^{11}-\frac{2834213059}{714194109}a^{10}+\frac{825299833}{714194109}a^{9}-\frac{5507481316}{714194109}a^{8}+\frac{1140645669}{79354901}a^{7}-\frac{4496889415}{714194109}a^{6}-\frac{3009338393}{714194109}a^{5}+\frac{4322071615}{714194109}a^{4}-\frac{4796576675}{714194109}a^{3}+\frac{1585008383}{714194109}a^{2}+\frac{2282722729}{714194109}a-\frac{621581200}{714194109}$, $\frac{152059679}{238064703}a^{15}-\frac{536155183}{238064703}a^{14}+\frac{646586335}{238064703}a^{13}-\frac{1449637828}{238064703}a^{12}+\frac{989716479}{79354901}a^{11}+\frac{145389427}{238064703}a^{10}-\frac{2780295661}{238064703}a^{9}-\frac{6191166521}{238064703}a^{8}+\frac{3195778089}{79354901}a^{7}+\frac{5261499802}{238064703}a^{6}-\frac{11562380833}{238064703}a^{5}+\frac{1005412262}{238064703}a^{4}+\frac{4237809836}{238064703}a^{3}-\frac{1296257243}{238064703}a^{2}-\frac{45125230}{238064703}a+\frac{385210354}{238064703}$ 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:  \( 278.178633436 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 278.178633436 \cdot 1}{2\cdot\sqrt{513114342400000000}}\cr\approx \mathstrut & 0.191155099982 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*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 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*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 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*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 - 4*x^15 + 6*x^14 - 12*x^13 + 25*x^12 - 10*x^11 - 15*x^10 - 36*x^9 + 83*x^8 + 2*x^7 - 81*x^6 + 32*x^5 + 20*x^4 - 12*x^3 + 3*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

$C_2^7.C_2\wr D_4$ (as 16T1772):

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 16384
The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$
Character table for $C_2^7.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.19360000.2

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: 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 ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ R ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 Deg $16$$2$$8$$16$
\(5\) 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}$
\(11\) Copy content Toggle raw display 11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(37\) Copy content Toggle raw display 37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.0.1$x^{4} + 6 x^{2} + 24 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.2.1$x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} + 6 x^{2} + 24 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$