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Properties

Label 16.2.195238297600000000.3
Degree $16$
Signature $[2, 7]$
Discriminant $-1.952\times 10^{17}$
Root discriminant \(12.04\)
Ramified primes $2,5,31$
Class number $1$
Class group trivial
Galois group $D_8\wr C_2$ (as 16T972)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^16 - 6*y^15 + 18*y^14 - 36*y^13 + 51*y^12 - 46*y^11 + 18*y^10 - 4*y^9 + 44*y^8 - 118*y^7 + 160*y^6 - 138*y^5 + 72*y^4 - 16*y^3 - 4*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*x - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*x - 1)
 

\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 51 x^{12} - 46 x^{11} + 18 x^{10} - 4 x^{9} + 44 x^{8} + \cdots - 1 \) 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:  $[2, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-195238297600000000\) \(\medspace = -\,2^{24}\cdot 5^{8}\cdot 31^{3}\) 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:  \(12.04\)
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:  $2^{15/8}5^{1/2}31^{1/2}\approx 45.66643307539152$
Ramified primes:   \(2\), \(5\), \(31\) 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:  \(\Q(\sqrt{-31}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}-\frac{3}{7}a^{13}-\frac{3}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{7}-\frac{2}{7}a^{6}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{2279459}a^{15}-\frac{3807}{175343}a^{14}+\frac{7349}{61607}a^{13}-\frac{294001}{2279459}a^{12}+\frac{115223}{325637}a^{11}-\frac{320952}{2279459}a^{10}+\frac{993248}{2279459}a^{9}+\frac{771496}{2279459}a^{8}-\frac{774547}{2279459}a^{7}+\frac{6366}{2279459}a^{6}+\frac{846540}{2279459}a^{5}+\frac{865464}{2279459}a^{4}-\frac{33365}{2279459}a^{3}+\frac{87419}{2279459}a^{2}-\frac{818585}{2279459}a+\frac{64114}{2279459}$ 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:  $8$
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:   $\frac{2308038}{2279459}a^{15}-\frac{989357}{175343}a^{14}+\frac{959316}{61607}a^{13}-\frac{9328015}{325637}a^{12}+\frac{83529209}{2279459}a^{11}-\frac{60512674}{2279459}a^{10}+\frac{5467494}{2279459}a^{9}-\frac{3879093}{2279459}a^{8}+\frac{105228387}{2279459}a^{7}-\frac{225437973}{2279459}a^{6}+\frac{249829324}{2279459}a^{5}-\frac{176483570}{2279459}a^{4}+\frac{65054201}{2279459}a^{3}-\frac{84391}{325637}a^{2}-\frac{640897}{325637}a+\frac{2885340}{2279459}$, $\frac{358850}{61607}a^{15}-\frac{150926}{4739}a^{14}+\frac{774078}{8801}a^{13}-\frac{10049261}{61607}a^{12}+\frac{12975387}{61607}a^{11}-\frac{1372781}{8801}a^{10}+\frac{1304077}{61607}a^{9}-\frac{617275}{61607}a^{8}+\frac{15316796}{61607}a^{7}-\frac{34210753}{61607}a^{6}+\frac{39394044}{61607}a^{5}-\frac{28611886}{61607}a^{4}+\frac{10367345}{61607}a^{3}+\frac{155765}{61607}a^{2}-\frac{1665038}{61607}a+\frac{623601}{61607}$, $\frac{18965735}{2279459}a^{15}-\frac{1114748}{25049}a^{14}+\frac{7439416}{61607}a^{13}-\frac{502965959}{2279459}a^{12}+\frac{637671325}{2279459}a^{11}-\frac{452455686}{2279459}a^{10}+\frac{39334909}{2279459}a^{9}-\frac{42616345}{2279459}a^{8}+\frac{803239244}{2279459}a^{7}-\frac{1716342833}{2279459}a^{6}+\frac{1914838369}{2279459}a^{5}-\frac{193474323}{325637}a^{4}+\frac{456615590}{2279459}a^{3}+\frac{16714891}{2279459}a^{2}-\frac{75377129}{2279459}a+\frac{27550310}{2279459}$, $\frac{12672936}{2279459}a^{15}-\frac{739365}{25049}a^{14}+\frac{4903482}{61607}a^{13}-\frac{329885015}{2279459}a^{12}+\frac{416105124}{2279459}a^{11}-\frac{292134165}{2279459}a^{10}+\frac{23476155}{2279459}a^{9}-\frac{35104192}{2279459}a^{8}+\frac{537269179}{2279459}a^{7}-\frac{1124103824}{2279459}a^{6}+\frac{1241133340}{2279459}a^{5}-\frac{125522190}{325637}a^{4}+\frac{300859434}{2279459}a^{3}+\frac{1195566}{2279459}a^{2}-\frac{43404169}{2279459}a+\frac{18670737}{2279459}$, $\frac{1105608}{325637}a^{15}-\frac{3243741}{175343}a^{14}+\frac{3140677}{61607}a^{13}-\frac{215113890}{2279459}a^{12}+\frac{277282249}{2279459}a^{11}-\frac{204788715}{2279459}a^{10}+\frac{27849516}{2279459}a^{9}-\frac{15719090}{2279459}a^{8}+\frac{331381431}{2279459}a^{7}-\frac{732480743}{2279459}a^{6}+\frac{119924254}{325637}a^{5}-\frac{609454537}{2279459}a^{4}+\frac{222831158}{2279459}a^{3}-\frac{2995334}{2279459}a^{2}-\frac{32257516}{2279459}a+\frac{12444448}{2279459}$, $\frac{13994124}{2279459}a^{15}-\frac{5675198}{175343}a^{14}+\frac{5341327}{61607}a^{13}-\frac{51038870}{325637}a^{12}+\frac{447328663}{2279459}a^{11}-\frac{308032694}{2279459}a^{10}+\frac{17014500}{2279459}a^{9}-\frac{39758864}{2279459}a^{8}+\frac{590885236}{2279459}a^{7}-\frac{1217756908}{2279459}a^{6}+\frac{1329168821}{2279459}a^{5}-\frac{928937085}{2279459}a^{4}+\frac{302856326}{2279459}a^{3}+\frac{1613275}{325637}a^{2}-\frac{6905745}{325637}a+\frac{19970455}{2279459}$, $\frac{11212121}{2279459}a^{15}-\frac{4580556}{175343}a^{14}+\frac{4349792}{61607}a^{13}-\frac{293679608}{2279459}a^{12}+\frac{372463438}{2279459}a^{11}-\frac{265437377}{2279459}a^{10}+\frac{27984874}{2279459}a^{9}-\frac{4858144}{325637}a^{8}+\frac{67611000}{325637}a^{7}-\frac{996135327}{2279459}a^{6}+\frac{1113573462}{2279459}a^{5}-\frac{802202028}{2279459}a^{4}+\frac{40660921}{325637}a^{3}-\frac{6172102}{2279459}a^{2}-\frac{38533506}{2279459}a+\frac{16761034}{2279459}$, $\frac{2201662}{2279459}a^{15}-\frac{782716}{175343}a^{14}+\frac{642882}{61607}a^{13}-\frac{36818427}{2279459}a^{12}+\frac{35438757}{2279459}a^{11}-\frac{5301334}{2279459}a^{10}-\frac{26059338}{2279459}a^{9}-\frac{5826486}{2279459}a^{8}+\frac{90135743}{2279459}a^{7}-\frac{19253649}{325637}a^{6}+\frac{88755949}{2279459}a^{5}-\frac{16124716}{2279459}a^{4}-\frac{38380788}{2279459}a^{3}+\frac{23438666}{2279459}a^{2}-\frac{3324667}{2279459}a-\frac{478757}{325637}$ 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:  \( 109.456398528 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 109.456398528 \cdot 1}{2\cdot\sqrt{195238297600000000}}\cr\approx \mathstrut & 0.191534813192 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*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 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*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 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*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 - 6*x^15 + 18*x^14 - 36*x^13 + 51*x^12 - 46*x^11 + 18*x^10 - 4*x^9 + 44*x^8 - 118*x^7 + 160*x^6 - 138*x^5 + 72*x^4 - 16*x^3 - 4*x^2 + 4*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

$D_8\wr C_2$ (as 16T972):

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 512
The 35 conjugacy class representatives for $D_8\wr C_2$
Character table for $D_8\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.400.1, 8.2.4960000.1

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 $16$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ R ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$4$$4$$24$
\(5\) 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}$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
31.2.1.1$x^{2} + 93$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} + 93$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} + 93$$2$$1$$1$$C_2$$[\ ]_{2}$
31.8.0.1$x^{8} + 25 x^{3} + 12 x^{2} + 24 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$