Properties

Label 16.4.499382553081640625.2
Degree $16$
Signature $[4, 6]$
Discriminant $4.994\times 10^{17}$
Root discriminant \(12.77\)
Ramified primes $5,89,119851$
Class number $1$
Class group trivial
Galois group $C_2^6.S_4^2:D_4$ (as 16T1905)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*x - 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + 4*y^14 - 5*y^13 - 2*y^12 - y^11 - 8*y^10 + 3*y^9 + 6*y^8 + 10*y^7 + 4*y^6 - y^5 - 16*y^4 - 24*y^3 - 18*y^2 - 9*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*x - 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*x - 1)
 

\( x^{16} - 2 x^{15} + 4 x^{14} - 5 x^{13} - 2 x^{12} - x^{11} - 8 x^{10} + 3 x^{9} + 6 x^{8} + 10 x^{7} + \cdots - 1 \) 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:   \(499382553081640625\) \(\medspace = 5^{8}\cdot 89\cdot 119851^{2}\) 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:  \(12.77\)
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:  $5^{1/2}89^{1/2}119851^{1/2}\approx 7302.99219498419$
Ramified primes:   \(5\), \(89\), \(119851\) 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:  \(\Q(\sqrt{89}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{268229993}a^{15}+\frac{114994542}{268229993}a^{14}+\frac{1567033}{4397213}a^{13}-\frac{92361020}{268229993}a^{12}+\frac{65728456}{268229993}a^{11}+\frac{14376041}{268229993}a^{10}-\frac{79197010}{268229993}a^{9}-\frac{101354724}{268229993}a^{8}-\frac{102032734}{268229993}a^{7}-\frac{67824888}{268229993}a^{6}-\frac{112514464}{268229993}a^{5}+\frac{71278165}{268229993}a^{4}+\frac{59318472}{268229993}a^{3}+\frac{14732274}{268229993}a^{2}-\frac{117025207}{268229993}a-\frac{92204831}{268229993}$ 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:   $\frac{20059440}{268229993}a^{15}-\frac{55834871}{268229993}a^{14}+\frac{1917258}{4397213}a^{13}-\frac{178278990}{268229993}a^{12}+\frac{72192916}{268229993}a^{11}-\frac{74747225}{268229993}a^{10}-\frac{13123349}{268229993}a^{9}-\frac{33052880}{268229993}a^{8}+\frac{336057778}{268229993}a^{7}+\frac{42421537}{268229993}a^{6}-\frac{121590505}{268229993}a^{5}+\frac{96441100}{268229993}a^{4}-\frac{546661599}{268229993}a^{3}-\frac{423731211}{268229993}a^{2}-\frac{3695763}{268229993}a+\frac{227506923}{268229993}$, $\frac{53947060}{268229993}a^{15}-\frac{188327627}{268229993}a^{14}+\frac{7598810}{4397213}a^{13}-\frac{812222269}{268229993}a^{12}+\frac{759815594}{268229993}a^{11}-\frac{570951118}{268229993}a^{10}-\frac{198628644}{268229993}a^{9}+\frac{707235604}{268229993}a^{8}-\frac{633529502}{268229993}a^{7}+\frac{732692971}{268229993}a^{6}-\frac{283740086}{268229993}a^{5}+\frac{168244345}{268229993}a^{4}-\frac{679875748}{268229993}a^{3}-\frac{185774623}{268229993}a^{2}-\frac{406826143}{268229993}a-\frac{247178052}{268229993}$, $\frac{21993941}{268229993}a^{15}-\frac{80495865}{268229993}a^{14}+\frac{3372230}{4397213}a^{13}-\frac{373812822}{268229993}a^{12}+\frac{358401645}{268229993}a^{11}-\frac{231190910}{268229993}a^{10}-\frac{173683591}{268229993}a^{9}+\frac{448757522}{268229993}a^{8}-\frac{221409032}{268229993}a^{7}+\frac{112136529}{268229993}a^{6}+\frac{54786769}{268229993}a^{5}+\frac{126560220}{268229993}a^{4}-\frac{644794450}{268229993}a^{3}+\frac{6527806}{268229993}a^{2}-\frac{74260449}{268229993}a-\frac{115452331}{268229993}$, $\frac{33969943}{268229993}a^{15}-\frac{77954639}{268229993}a^{14}+\frac{3104217}{4397213}a^{13}-\frac{286351175}{268229993}a^{12}+\frac{111197191}{268229993}a^{11}-\frac{179879773}{268229993}a^{10}-\frac{322749667}{268229993}a^{9}+\frac{209276897}{268229993}a^{8}+\frac{84802405}{268229993}a^{7}+\frac{349320905}{268229993}a^{6}+\frac{364208809}{268229993}a^{5}-\frac{163376321}{268229993}a^{4}-\frac{522126244}{268229993}a^{3}-\frac{965999228}{268229993}a^{2}-\frac{789047646}{268229993}a-\frac{412415481}{268229993}$, $\frac{14904067}{268229993}a^{15}-\frac{28120381}{268229993}a^{14}+\frac{1172383}{4397213}a^{13}-\frac{115572298}{268229993}a^{12}+\frac{46725735}{268229993}a^{11}-\frac{103689667}{268229993}a^{10}-\frac{193383464}{268229993}a^{9}+\frac{228295410}{268229993}a^{8}-\frac{117874964}{268229993}a^{7}+\frac{165709905}{268229993}a^{6}+\frac{379072228}{268229993}a^{5}-\frac{264989186}{268229993}a^{4}-\frac{112362362}{268229993}a^{3}-\frac{395901491}{268229993}a^{2}-\frac{480753991}{268229993}a-\frac{232360882}{268229993}$, $\frac{722548}{4397213}a^{15}-\frac{933034}{4397213}a^{14}+\frac{1169878}{4397213}a^{13}+\frac{202400}{4397213}a^{12}-\frac{7061435}{4397213}a^{11}+\frac{1702236}{4397213}a^{10}-\frac{4126485}{4397213}a^{9}-\frac{5375472}{4397213}a^{8}+\frac{10944230}{4397213}a^{7}+\frac{8233004}{4397213}a^{6}+\frac{2103953}{4397213}a^{5}+\frac{9285}{4397213}a^{4}-\frac{9685962}{4397213}a^{3}-\frac{24499526}{4397213}a^{2}-\frac{16350824}{4397213}a-\frac{4616886}{4397213}$, $\frac{66069369}{268229993}a^{15}-\frac{137410939}{268229993}a^{14}+\frac{4542108}{4397213}a^{13}-\frac{361466422}{268229993}a^{12}-\frac{72345890}{268229993}a^{11}-\frac{127354514}{268229993}a^{10}-\frac{425399176}{268229993}a^{9}+\frac{120853730}{268229993}a^{8}+\frac{446188998}{268229993}a^{7}+\frac{614349773}{268229993}a^{6}+\frac{183127902}{268229993}a^{5}+\frac{127126605}{268229993}a^{4}-\frac{1235638132}{268229993}a^{3}-\frac{1219713287}{268229993}a^{2}-\frac{1071660713}{268229993}a-\frac{397082377}{268229993}$, $\frac{79294920}{268229993}a^{15}-\frac{163618241}{268229993}a^{14}+\frac{5446099}{4397213}a^{13}-\frac{452706596}{268229993}a^{12}-\frac{75790635}{268229993}a^{11}-\frac{139249071}{268229993}a^{10}-\frac{613726315}{268229993}a^{9}+\frac{465538488}{268229993}a^{8}+\frac{395476243}{268229993}a^{7}+\frac{815606903}{268229993}a^{6}+\frac{370944170}{268229993}a^{5}-\frac{372188001}{268229993}a^{4}-\frac{1378946446}{268229993}a^{3}-\frac{1581955499}{268229993}a^{2}-\frac{1180925701}{268229993}a-\frac{441913505}{268229993}$, $\frac{50049863}{268229993}a^{15}-\frac{45961945}{268229993}a^{14}+\frac{992442}{4397213}a^{13}-\frac{10737756}{268229993}a^{12}-\frac{361857133}{268229993}a^{11}-\frac{295020320}{268229993}a^{10}+\frac{67497065}{268229993}a^{9}-\frac{550517435}{268229993}a^{8}+\frac{795894316}{268229993}a^{7}+\frac{972590043}{268229993}a^{6}+\frac{157380614}{268229993}a^{5}-\frac{748514}{268229993}a^{4}-\frac{1090381717}{268229993}a^{3}-\frac{1739188909}{268229993}a^{2}-\frac{1578279264}{268229993}a-\frac{335831634}{268229993}$ 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:  \( 239.201457412 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 239.201457412 \cdot 1}{2\cdot\sqrt{499382553081640625}}\cr\approx \mathstrut & 0.166615858761 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*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 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*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 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*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 - 2*x^15 + 4*x^14 - 5*x^13 - 2*x^12 - x^11 - 8*x^10 + 3*x^9 + 6*x^8 + 10*x^7 + 4*x^6 - x^5 - 16*x^4 - 24*x^3 - 18*x^2 - 9*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

$C_2^6.S_4^2:D_4$ (as 16T1905):

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 294912
The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$
Character table for $C_2^6.S_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.74906875.1

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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(5\) 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}$
\(89\) Copy content Toggle raw display $\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 86$$1$$1$$0$Trivial$[\ ]$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
\(119851\) Copy content Toggle raw display $\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{119851}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$