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Properties

Label 16.0.4919707275390625.1
Degree $16$
Signature $[0, 8]$
Discriminant $4.920\times 10^{15}$
Root discriminant \(9.57\)
Ramified primes $5,67$
Class number $1$
Class group trivial
Galois group $C_4\times S_4$ (as 16T181)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 - 5*y^15 + 14*y^14 - 29*y^13 + 50*y^12 - 77*y^11 + 112*y^10 - 149*y^9 + 176*y^8 - 182*y^7 + 159*y^6 - 118*y^5 + 73*y^4 - 36*y^3 + 15*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*x^2 - 4*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*x^2 - 4*x + 1)
 

\( x^{16} - 5 x^{15} + 14 x^{14} - 29 x^{13} + 50 x^{12} - 77 x^{11} + 112 x^{10} - 149 x^{9} + 176 x^{8} + \cdots + 1 \) 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:   \(4919707275390625\) \(\medspace = 5^{12}\cdot 67^{4}\) 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:  \(9.57\)
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:  $5^{3/4}67^{1/2}\approx 27.369376545007917$
Ramified primes:   \(5\), \(67\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}-\frac{1}{19}a^{14}-\frac{9}{19}a^{13}-\frac{8}{19}a^{12}-\frac{1}{19}a^{11}-\frac{5}{19}a^{10}-\frac{3}{19}a^{9}-\frac{9}{19}a^{8}+\frac{7}{19}a^{7}-\frac{2}{19}a^{6}-\frac{1}{19}a^{5}-\frac{8}{19}a^{4}+\frac{3}{19}a^{3}-\frac{5}{19}a^{2}-\frac{5}{19}a-\frac{5}{19}$ 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:   \( \frac{217}{19} a^{15} - \frac{1015}{19} a^{14} + \frac{2626}{19} a^{13} - \frac{5042}{19} a^{12} + \frac{8162}{19} a^{11} - \frac{12010}{19} a^{10} + \frac{17057}{19} a^{9} - \frac{21846}{19} a^{8} + \frac{24053}{19} a^{7} - \frac{22588}{19} a^{6} + \frac{17035}{19} a^{5} - \frac{10419}{19} a^{4} + \frac{5040}{19} a^{3} - \frac{1541}{19} a^{2} + \frac{435}{19} a + \frac{36}{19} \)  (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:   $\frac{195}{19}a^{15}-\frac{974}{19}a^{14}+\frac{2653}{19}a^{13}-\frac{5303}{19}a^{12}+\frac{8830}{19}a^{11}-\frac{13211}{19}a^{10}+\frac{18871}{19}a^{9}-\frac{24650}{19}a^{8}+\frac{28060}{19}a^{7}-\frac{27427}{19}a^{6}+\frac{21959}{19}a^{5}-\frac{14290}{19}a^{4}+\frac{7463}{19}a^{3}-\frac{2742}{19}a^{2}+\frac{792}{19}a-\frac{82}{19}$, $\frac{183}{19}a^{15}-\frac{1000}{19}a^{14}+\frac{2894}{19}a^{13}-\frac{6024}{19}a^{12}+\frac{10305}{19}a^{11}-\frac{15697}{19}a^{10}+\frac{22612}{19}a^{9}-\frac{30128}{19}a^{8}+\frac{35329}{19}a^{7}-\frac{35725}{19}a^{6}+\frac{30160}{19}a^{5}-\frac{20901}{19}a^{4}+\frac{11816}{19}a^{3}-\frac{5038}{19}a^{2}+\frac{1593}{19}a-\frac{326}{19}$, $a$, $\frac{22}{19}a^{15}-\frac{41}{19}a^{14}-\frac{27}{19}a^{13}+\frac{261}{19}a^{12}-\frac{668}{19}a^{11}+\frac{1201}{19}a^{10}-\frac{1814}{19}a^{9}+\frac{2804}{19}a^{8}-\frac{4007}{19}a^{7}+\frac{4839}{19}a^{6}-\frac{4924}{19}a^{5}+\frac{3871}{19}a^{4}-\frac{2423}{19}a^{3}+\frac{1201}{19}a^{2}-\frac{357}{19}a+\frac{118}{19}$, $\frac{99}{19}a^{15}-\frac{422}{19}a^{14}+\frac{1009}{19}a^{13}-\frac{1837}{19}a^{12}+\frac{2884}{19}a^{11}-\frac{4181}{19}a^{10}+\frac{5916}{19}a^{9}-\frac{7332}{19}a^{8}+\frac{7666}{19}a^{7}-\frac{6886}{19}a^{6}+\frac{4803}{19}a^{5}-\frac{2749}{19}a^{4}+\frac{1209}{19}a^{3}-\frac{191}{19}a^{2}+\frac{56}{19}a+\frac{75}{19}$, $\frac{105}{19}a^{15}-\frac{542}{19}a^{14}+\frac{1506}{19}a^{13}-\frac{3025}{19}a^{12}+\frac{5025}{19}a^{11}-\frac{7498}{19}a^{10}+\frac{10686}{19}a^{9}-\frac{13998}{19}a^{8}+\frac{15954}{19}a^{7}-\frac{15467}{19}a^{6}+\frac{12245}{19}a^{5}-\frac{7813}{19}a^{4}+\frac{3906}{19}a^{3}-\frac{1380}{19}a^{2}+\frac{349}{19}a-\frac{31}{19}$, $\frac{46}{19}a^{15}-\frac{65}{19}a^{14}-\frac{167}{19}a^{13}+\frac{829}{19}a^{12}-\frac{1927}{19}a^{11}+\frac{3380}{19}a^{10}-\frac{5116}{19}a^{9}+\frac{7775}{19}a^{8}-\frac{10869}{19}a^{7}+\frac{12847}{19}a^{6}-\frac{12947}{19}a^{5}+\frac{10367}{19}a^{4}-\frac{6645}{19}a^{3}+\frac{3437}{19}a^{2}-\frac{1123}{19}a+\frac{359}{19}$ 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:  \( 48.4508350282 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 48.4508350282 \cdot 1}{10\cdot\sqrt{4919707275390625}}\cr\approx \mathstrut & 0.167791741570 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*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 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*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 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*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 - 5*x^15 + 14*x^14 - 29*x^13 + 50*x^12 - 77*x^11 + 112*x^10 - 149*x^9 + 176*x^8 - 182*x^7 + 159*x^6 - 118*x^5 + 73*x^4 - 36*x^3 + 15*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

$C_4\times S_4$ (as 16T181):

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 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.8375.1, 8.4.70140625.1

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 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: 12.0.1574306328125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.4.0.1}{4} }$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(67\) Copy content Toggle raw display 67.4.0.1$x^{4} + 8 x^{2} + 54 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} + 8 x^{2} + 54 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.8.4.1$x^{8} + 284 x^{6} + 108 x^{5} + 28074 x^{4} - 13608 x^{3} + 1141144 x^{2} - 1396332 x + 15837397$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$