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Properties

Label 16.0.36520347436056576.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.652\times 10^{16}$
Root discriminant \(10.84\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_4\times C_2$ (as 16T9)

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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 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^15 + 34*y^14 - 92*y^13 + 162*y^12 - 152*y^11 - 62*y^10 + 444*y^9 - 665*y^8 + 396*y^7 + 194*y^6 - 564*y^5 + 488*y^4 - 232*y^3 + 68*y^2 - 12*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*x + 1)
 

\( x^{16} - 8 x^{15} + 34 x^{14} - 92 x^{13} + 162 x^{12} - 152 x^{11} - 62 x^{10} + 444 x^{9} - 665 x^{8} + \cdots + 1 \) 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:   \(36520347436056576\) \(\medspace = 2^{36}\cdot 3^{12}\) 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:  \(10.84\)
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:  $2^{9/4}3^{3/4}\approx 10.843224043318138$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{33036003}a^{15}-\frac{1747868}{11012001}a^{14}+\frac{223114}{1738737}a^{13}-\frac{348820}{11012001}a^{12}-\frac{3855991}{11012001}a^{11}+\frac{8446423}{33036003}a^{10}+\frac{3493591}{11012001}a^{9}+\frac{392702}{847077}a^{8}-\frac{514474}{3003273}a^{7}+\frac{43843}{158067}a^{6}-\frac{162334}{1573143}a^{5}+\frac{2758169}{11012001}a^{4}+\frac{15117398}{33036003}a^{3}-\frac{4297658}{11012001}a^{2}+\frac{9635312}{33036003}a+\frac{4810093}{33036003}$ 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:   \( -\frac{2121347}{33036003} a^{15} + \frac{3125385}{3670667} a^{14} - \frac{8150315}{1738737} a^{13} + \frac{174692026}{11012001} a^{12} - \frac{388356308}{11012001} a^{11} + \frac{1605667480}{33036003} a^{10} - \frac{240330095}{11012001} a^{9} - \frac{57499898}{847077} a^{8} + \frac{510036689}{3003273} a^{7} - \frac{26570489}{158067} a^{6} + \frac{38777201}{1573143} a^{5} + \frac{485302385}{3670667} a^{4} - \frac{5372331385}{33036003} a^{3} + \frac{1019734853}{11012001} a^{2} - \frac{938665207}{33036003} a + \frac{152107357}{33036003} \)  (order $24$) 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:   $\frac{41413826}{33036003}a^{15}-\frac{98377612}{11012001}a^{14}+\frac{60950453}{1738737}a^{13}-\frac{949108234}{11012001}a^{12}+\frac{488104269}{3670667}a^{11}-\frac{2816093083}{33036003}a^{10}-\frac{1538314970}{11012001}a^{9}+\frac{369400708}{847077}a^{8}-\frac{1459427555}{3003273}a^{7}+\frac{19934024}{158067}a^{6}+\frac{171190254}{524381}a^{5}-\frac{4987504754}{11012001}a^{4}+\frac{8941880197}{33036003}a^{3}-\frac{303343889}{3670667}a^{2}+\frac{445223365}{33036003}a-\frac{12567934}{33036003}$, $\frac{54567251}{33036003}a^{15}-\frac{133817752}{11012001}a^{14}+\frac{84063509}{1738737}a^{13}-\frac{444218261}{3670667}a^{12}+\frac{2099910194}{11012001}a^{11}-\frac{4350657385}{33036003}a^{10}-\frac{660892711}{3670667}a^{9}+\frac{515811388}{847077}a^{8}-\frac{2119465454}{3003273}a^{7}+\frac{34557587}{158067}a^{6}+\frac{672333395}{1573143}a^{5}-\frac{6990264641}{11012001}a^{4}+\frac{13465515589}{33036003}a^{3}-\frac{1644835658}{11012001}a^{2}+\frac{1241070016}{33036003}a-\frac{156674899}{33036003}$, $\frac{45458846}{33036003}a^{15}-\frac{114272932}{11012001}a^{14}+\frac{73580357}{1738737}a^{13}-\frac{400073064}{3670667}a^{12}+\frac{657848816}{3670667}a^{11}-\frac{4627532398}{33036003}a^{10}-\frac{493313591}{3670667}a^{9}+\frac{466337737}{847077}a^{8}-\frac{2104790840}{3003273}a^{7}+\frac{45184193}{158067}a^{6}+\frac{192088763}{524381}a^{5}-\frac{7054512382}{11012001}a^{4}+\frac{14388937603}{33036003}a^{3}-\frac{555006748}{3670667}a^{2}+\frac{964118329}{33036003}a-\frac{84627964}{33036003}$, $\frac{8071904}{33036003}a^{15}-\frac{13983469}{11012001}a^{14}+\frac{6358775}{1738737}a^{13}-\frac{18810310}{3670667}a^{12}-\frac{2314573}{3670667}a^{11}+\frac{631372139}{33036003}a^{10}-\frac{143521426}{3670667}a^{9}+\frac{20403217}{847077}a^{8}+\frac{109130992}{3003273}a^{7}-\frac{12944167}{158067}a^{6}+\frac{25962291}{524381}a^{5}+\frac{213751364}{11012001}a^{4}-\frac{1563799550}{33036003}a^{3}+\frac{119258253}{3670667}a^{2}-\frac{357308762}{33036003}a+\frac{66333233}{33036003}$, $\frac{31286851}{33036003}a^{15}-\frac{77808370}{11012001}a^{14}+\frac{49643695}{1738737}a^{13}-\frac{266424502}{3670667}a^{12}+\frac{1287975928}{11012001}a^{11}-\frac{2821805918}{33036003}a^{10}-\frac{1121266010}{11012001}a^{9}+\frac{105258345}{282359}a^{8}-\frac{1352961976}{3003273}a^{7}+\frac{24320347}{158067}a^{6}+\frac{142206404}{524381}a^{5}-\frac{4636129000}{11012001}a^{4}+\frac{8683983548}{33036003}a^{3}-\frac{859907371}{11012001}a^{2}+\frac{286323080}{33036003}a-\frac{7583099}{33036003}$, $\frac{23756357}{33036003}a^{15}-\frac{16573615}{3670667}a^{14}+\frac{27931847}{1738737}a^{13}-\frac{126088531}{3670667}a^{12}+\frac{459423176}{11012001}a^{11}-\frac{37260589}{33036003}a^{10}-\frac{1076358221}{11012001}a^{9}+\frac{146137394}{847077}a^{8}-\frac{309647909}{3003273}a^{7}-\frac{12251284}{158067}a^{6}+\frac{268652992}{1573143}a^{5}-\frac{1146648011}{11012001}a^{4}+\frac{377690128}{33036003}a^{3}+\frac{93162172}{11012001}a^{2}-\frac{89064029}{33036003}a+\frac{29526317}{33036003}$, $\frac{28144}{33003}a^{15}-\frac{24024}{3667}a^{14}+\frac{47467}{1737}a^{13}-\frac{265365}{3667}a^{12}+\frac{1370252}{11001}a^{11}-\frac{3655013}{33003}a^{10}-\frac{668989}{11001}a^{9}+\frac{3875320}{11001}a^{8}-\frac{16656290}{33003}a^{7}+\frac{484814}{1737}a^{6}+\frac{638759}{3667}a^{5}-\frac{1618566}{3667}a^{4}+\frac{11980367}{33003}a^{3}-\frac{1758158}{11001}a^{2}+\frac{1262900}{33003}a-\frac{188348}{33003}$ 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:  \( 466.98215552 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 466.98215552 \cdot 1}{24\cdot\sqrt{36520347436056576}}\cr\approx \mathstrut & 0.24732074176 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*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 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*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 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*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 - 8*x^15 + 34*x^14 - 92*x^13 + 162*x^12 - 152*x^11 - 62*x^10 + 444*x^9 - 665*x^8 + 396*x^7 + 194*x^6 - 564*x^5 + 488*x^4 - 232*x^3 + 68*x^2 - 12*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

$C_2\times D_4$ (as 16T9):

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 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), 4.0.1728.1 x2, 4.0.432.1 x2, 4.2.6912.1 x2, 4.2.1728.1 x2, \(\Q(\zeta_{24})\), 8.0.47775744.1, 8.0.2985984.1, 8.0.47775744.4 x2, 8.4.191102976.1 x2, 8.0.47775744.3 x2, 8.0.191102976.4 x2

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 8 siblings: 8.4.191102976.1, 8.0.47775744.3, 8.0.47775744.4, 8.0.191102976.4
Minimal sibling: 8.0.47775744.3

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.16.36.1$x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$$8$$2$$36$$D_4\times C_2$$[2, 2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$