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Properties

Label 16.0.6926910400390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.927\times 10^{15}$
Root discriminant \(9.77\)
Ramified primes $5,61$
Class number $1$
Class group trivial
Galois group $C_4\wr C_4$ (as 16T1192)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 7*y^15 + 25*y^14 - 55*y^13 + 75*y^12 - 46*y^11 - 43*y^10 + 140*y^9 - 160*y^8 + 80*y^7 + 37*y^6 - 109*y^5 + 110*y^4 - 70*y^3 + 30*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*x + 1)
 

\( x^{16} - 7 x^{15} + 25 x^{14} - 55 x^{13} + 75 x^{12} - 46 x^{11} - 43 x^{10} + 140 x^{9} - 160 x^{8} + \cdots + 1 \) 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:   \(6926910400390625\) \(\medspace = 5^{15}\cdot 61^{3}\) 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:  \(9.77\)
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:  $5^{15/16}61^{3/4}\approx 98.69195574460448$
Ramified primes:   \(5\), \(61\) 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:  \(\Q(\sqrt{305}) \)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1231}a^{15}-\frac{252}{1231}a^{14}+\frac{215}{1231}a^{13}+\frac{203}{1231}a^{12}-\frac{420}{1231}a^{11}-\frac{550}{1231}a^{10}+\frac{528}{1231}a^{9}+\frac{35}{1231}a^{8}-\frac{118}{1231}a^{7}-\frac{554}{1231}a^{6}+\frac{357}{1231}a^{5}-\frac{173}{1231}a^{4}-\frac{590}{1231}a^{3}+\frac{453}{1231}a^{2}-\frac{165}{1231}a-\frac{206}{1231}$ 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:   \( \frac{6623}{1231} a^{15} - \frac{44076}{1231} a^{14} + \frac{149860}{1231} a^{13} - \frac{309995}{1231} a^{12} + \frac{383241}{1231} a^{11} - \frac{163844}{1231} a^{10} - \frac{343776}{1231} a^{9} + \frac{796834}{1231} a^{8} - \frac{764280}{1231} a^{7} + \frac{252824}{1231} a^{6} + \frac{327106}{1231} a^{5} - \frac{595522}{1231} a^{4} + \frac{512951}{1231} a^{3} - \frac{281627}{1231} a^{2} + \frac{100044}{1231} a - \frac{17624}{1231} \)  (order $10$) 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:   $\frac{11001}{1231}a^{15}-\frac{72669}{1231}a^{14}+\frac{246664}{1231}a^{13}-\frac{509465}{1231}a^{12}+\frac{628564}{1231}a^{11}-\frac{263619}{1231}a^{10}-\frac{577900}{1231}a^{9}+\frac{1326750}{1231}a^{8}-\frac{1258726}{1231}a^{7}+\frac{392816}{1231}a^{6}+\frac{577806}{1231}a^{5}-\frac{998388}{1231}a^{4}+\frac{830167}{1231}a^{3}-\frac{439102}{1231}a^{2}+\frac{147049}{1231}a-\frac{23324}{1231}$, $\frac{5015}{1231}a^{15}-\frac{31549}{1231}a^{14}+\frac{102042}{1231}a^{13}-\frac{196952}{1231}a^{12}+\frac{214135}{1231}a^{11}-\frac{32816}{1231}a^{10}-\frac{294170}{1231}a^{9}+\frac{511588}{1231}a^{8}-\frac{377576}{1231}a^{7}+\frac{7443}{1231}a^{6}+\frac{297152}{1231}a^{5}-\frac{350575}{1231}a^{4}+\frac{231902}{1231}a^{3}-\frac{94187}{1231}a^{2}+\frac{20684}{1231}a-\frac{281}{1231}$, $\frac{1635}{1231}a^{15}-\frac{8252}{1231}a^{14}+\frac{21617}{1231}a^{13}-\frac{28778}{1231}a^{12}+\frac{6353}{1231}a^{11}+\frac{51082}{1231}a^{10}-\frac{93207}{1231}a^{9}+\frac{65842}{1231}a^{8}+\frac{32343}{1231}a^{7}-\frac{98254}{1231}a^{6}+\frac{90064}{1231}a^{5}-\frac{20652}{1231}a^{4}-\frac{26628}{1231}a^{3}+\frac{37754}{1231}a^{2}-\frac{21113}{1231}a+\frac{6639}{1231}$, $\frac{11939}{1231}a^{15}-\frac{76386}{1231}a^{14}+\frac{251374}{1231}a^{13}-\frac{498777}{1231}a^{12}+\frac{575591}{1231}a^{11}-\frac{168943}{1231}a^{10}-\frac{643972}{1231}a^{9}+\frac{1273410}{1231}a^{8}-\frac{1082587}{1231}a^{7}+\frac{226461}{1231}a^{6}+\frac{613539}{1231}a^{5}-\frac{912000}{1231}a^{4}+\frac{716214}{1231}a^{3}-\frac{360099}{1231}a^{2}+\frac{114148}{1231}a-\frac{17130}{1231}$, $\frac{2116}{1231}a^{15}-\frac{14981}{1231}a^{14}+\frac{53634}{1231}a^{13}-\frac{117016}{1231}a^{12}+\frac{155168}{1231}a^{11}-\frac{82982}{1231}a^{10}-\frac{114983}{1231}a^{9}+\frac{311643}{1231}a^{8}-\frac{322317}{1231}a^{7}+\frac{123979}{1231}a^{6}+\frac{120216}{1231}a^{5}-\frac{236813}{1231}a^{4}+\frac{211526}{1231}a^{3}-\frac{118577}{1231}a^{2}+\frac{43549}{1231}a-\frac{7508}{1231}$, $\frac{3937}{1231}a^{15}-\frac{27020}{1231}a^{14}+\frac{93083}{1231}a^{13}-\frac{194206}{1231}a^{12}+\frac{239738}{1231}a^{11}-\frac{97270}{1231}a^{10}-\frac{229389}{1231}a^{9}+\frac{510788}{1231}a^{8}-\frac{471952}{1231}a^{7}+\frac{127027}{1231}a^{6}+\frac{232366}{1231}a^{5}-\frac{377044}{1231}a^{4}+\frac{302893}{1231}a^{3}-\frac{156595}{1231}a^{2}+\frac{50834}{1231}a-\frac{8410}{1231}$, $\frac{2520}{1231}a^{15}-\frac{15847}{1231}a^{14}+\frac{51862}{1231}a^{13}-\frac{101478}{1231}a^{12}+\frac{113512}{1231}a^{11}-\frac{23283}{1231}a^{10}-\frac{146640}{1231}a^{9}+\frac{266695}{1231}a^{8}-\frac{203804}{1231}a^{7}+\frac{13415}{1231}a^{6}+\frac{152423}{1231}a^{5}-\frac{181143}{1231}a^{4}+\frac{125810}{1231}a^{3}-\frac{53741}{1231}a^{2}+\frac{13819}{1231}a-\frac{869}{1231}$ 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:  \( 65.2827160145 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 65.2827160145 \cdot 1}{10\cdot\sqrt{6926910400390625}}\cr\approx \mathstrut & 0.190531758727 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*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 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*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 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*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 - 7*x^15 + 25*x^14 - 55*x^13 + 75*x^12 - 46*x^11 - 43*x^10 + 140*x^9 - 160*x^8 + 80*x^7 + 37*x^6 - 109*x^5 + 110*x^4 - 70*x^3 + 30*x^2 - 8*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

$C_4\wr C_4$ (as 16T1192):

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 1024
The 88 conjugacy class representatives for $C_4\wr C_4$
Character table for $C_4\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.4765625.1

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 ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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 5.16.15.2$x^{16} + 5$$16$$1$$15$16T125$[\ ]_{16}^{4}$
\(61\) Copy content Toggle raw display 61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.0.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.4.0.1$x^{4} + 3 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} + 3 x^{2} + 40 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.3.1$x^{4} + 183$$4$$1$$3$$C_4$$[\ ]_{4}$