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Properties

Label 16.0.8148125331379089.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.148\times 10^{15}$
Root discriminant \(9.87\)
Ramified primes $3,7,19$
Class number $1$
Class group trivial
Galois group $C_2\wr D_4$ (as 16T388)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + x + 1)
 
gp: K = bnfinit(y^16 - y^15 - 6*y^14 + 4*y^13 + 15*y^12 - 6*y^11 - 20*y^10 + 7*y^9 + 21*y^8 - 7*y^7 - 20*y^6 + 6*y^5 + 15*y^4 - 4*y^3 - 6*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + x + 1)
 

\( x^{16} - x^{15} - 6 x^{14} + 4 x^{13} + 15 x^{12} - 6 x^{11} - 20 x^{10} + 7 x^{9} + 21 x^{8} - 7 x^{7} + \cdots + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(8148125331379089\) \(\medspace = 3^{12}\cdot 7^{6}\cdot 19^{4}\) 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:  \(9.87\)
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:  $3^{3/4}7^{3/4}19^{1/2}\approx 42.76035198975074$
Ramified primes:   \(3\), \(7\), \(19\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7}a^{14}+\frac{2}{7}a^{13}+\frac{1}{7}a^{12}+\frac{2}{7}a^{11}+\frac{1}{7}a^{10}-\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{3}{7}a^{7}+\frac{1}{7}a^{6}-\frac{1}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{259}a^{15}-\frac{1}{37}a^{14}-\frac{38}{259}a^{13}+\frac{12}{37}a^{12}-\frac{45}{259}a^{11}+\frac{116}{259}a^{10}-\frac{13}{259}a^{9}-\frac{100}{259}a^{8}-\frac{82}{259}a^{7}+\frac{4}{259}a^{6}-\frac{118}{259}a^{5}+\frac{11}{259}a^{4}+\frac{23}{259}a^{3}-\frac{31}{259}a^{2}-\frac{5}{259}a-\frac{117}{259}$ 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:   \( -\frac{1}{7} a^{15} + \frac{19}{7} a^{14} - \frac{8}{7} a^{13} - \frac{107}{7} a^{12} + \frac{27}{7} a^{11} + \frac{239}{7} a^{10} - \frac{13}{7} a^{9} - \frac{269}{7} a^{8} + \frac{27}{7} a^{7} + \frac{274}{7} a^{6} - \frac{27}{7} a^{5} - \frac{254}{7} a^{4} + \frac{15}{7} a^{3} + \frac{166}{7} a^{2} - \frac{13}{7} a - 5 \)  (order $6$) 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:   $\frac{198}{259}a^{15}-\frac{239}{259}a^{14}-\frac{1086}{259}a^{13}+\frac{944}{259}a^{12}+\frac{2449}{259}a^{11}-\frac{181}{37}a^{10}-\frac{2944}{259}a^{9}+\frac{1068}{259}a^{8}+\frac{3004}{259}a^{7}-\frac{130}{37}a^{6}-\frac{2755}{259}a^{5}+\frac{772}{259}a^{4}+\frac{1927}{259}a^{3}-\frac{292}{259}a^{2}-\frac{509}{259}a-\frac{226}{259}$, $\frac{880}{259}a^{15}-\frac{1054}{259}a^{14}-\frac{4839}{259}a^{13}+\frac{4175}{259}a^{12}+\frac{11016}{259}a^{11}-\frac{6256}{259}a^{10}-\frac{12920}{259}a^{9}+\frac{7127}{259}a^{8}+\frac{12570}{259}a^{7}-\frac{7432}{259}a^{6}-\frac{12080}{259}a^{5}+\frac{6128}{259}a^{4}+\frac{7919}{259}a^{3}-\frac{4155}{259}a^{2}-\frac{1699}{259}a+\frac{102}{37}$, $\frac{393}{259}a^{15}-\frac{716}{259}a^{14}-\frac{257}{37}a^{13}+\frac{3190}{259}a^{12}+\frac{497}{37}a^{11}-\frac{5990}{259}a^{10}-\frac{3777}{259}a^{9}+\frac{1051}{37}a^{8}+\frac{3923}{259}a^{7}-\frac{7271}{259}a^{6}-\frac{3343}{259}a^{5}+\frac{6691}{259}a^{4}+\frac{1972}{259}a^{3}-\frac{4376}{259}a^{2}+\frac{33}{259}a+\frac{935}{259}$, $6a^{15}-\frac{43}{7}a^{14}-\frac{233}{7}a^{13}+\frac{160}{7}a^{12}+\frac{523}{7}a^{11}-\frac{225}{7}a^{10}-\frac{601}{7}a^{9}+\frac{281}{7}a^{8}+\frac{613}{7}a^{7}-\frac{267}{7}a^{6}-\frac{566}{7}a^{5}+\frac{225}{7}a^{4}+\frac{376}{7}a^{3}-\frac{153}{7}a^{2}-\frac{86}{7}a+\frac{29}{7}$, $\frac{71}{259}a^{15}-\frac{90}{259}a^{14}-\frac{330}{259}a^{13}+\frac{414}{259}a^{12}+\frac{468}{259}a^{11}-\frac{940}{259}a^{10}-\frac{5}{37}a^{9}+\frac{1558}{259}a^{8}+\frac{61}{259}a^{7}-\frac{1381}{259}a^{6}-\frac{34}{37}a^{5}+\frac{1151}{259}a^{4}-\frac{143}{259}a^{3}-\frac{1054}{259}a^{2}+\frac{459}{259}a+\frac{351}{259}$, $\frac{762}{259}a^{15}-\frac{709}{259}a^{14}-\frac{4425}{259}a^{13}+\frac{2588}{259}a^{12}+\frac{10443}{259}a^{11}-\frac{3331}{259}a^{10}-\frac{12718}{259}a^{9}+\frac{3868}{259}a^{8}+\frac{13033}{259}a^{7}-\frac{3723}{259}a^{6}-\frac{12179}{259}a^{5}+\frac{3239}{259}a^{4}+\frac{8387}{259}a^{3}-\frac{2088}{259}a^{2}-\frac{2330}{259}a+\frac{34}{37}$, $\frac{1075}{259}a^{15}-\frac{1013}{259}a^{14}-\frac{6070}{259}a^{13}+\frac{3572}{259}a^{12}+\frac{13859}{259}a^{11}-\frac{4504}{259}a^{10}-\frac{16084}{259}a^{9}+\frac{5646}{259}a^{8}+\frac{2334}{37}a^{7}-\frac{5505}{259}a^{6}-\frac{15258}{259}a^{5}+\frac{648}{37}a^{4}+\frac{10295}{259}a^{3}-\frac{400}{37}a^{2}-\frac{2452}{259}a+\frac{321}{259}$ 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:  \( 41.5444168882 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 41.5444168882 \cdot 1}{6\cdot\sqrt{8148125331379089}}\cr\approx \mathstrut & 0.186325164731 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + 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 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + 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 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + 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 - x^15 - 6*x^14 + 4*x^13 + 15*x^12 - 6*x^11 - 20*x^10 + 7*x^9 + 21*x^8 - 7*x^7 - 20*x^6 + 6*x^5 + 15*x^4 - 4*x^3 - 6*x^2 + 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

$C_2\wr D_4$ (as 16T388):

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 128
The 20 conjugacy class representatives for $C_2\wr D_4$
Character table for $C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.1197.2, 4.0.513.1, 8.0.4750893.1 x2, 8.0.30088989.1 x2, 8.0.12895281.1

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 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 8.0.4750893.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) 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}$
\(7\) Copy content Toggle raw display 7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.2$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(19\) Copy content Toggle raw display 19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$