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Properties

Label 16.0.36005299687890625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.601\times 10^{16}$
Root discriminant \(10.83\)
Ramified primes $5,19,29$
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 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^16 - 3*y^15 + 3*y^14 - y^13 + 2*y^12 - 8*y^11 + 14*y^10 - 18*y^9 + 29*y^8 - 49*y^7 + 66*y^6 - 71*y^5 + 57*y^4 - 33*y^3 + 17*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*x + 1)
 

\( x^{16} - 3 x^{15} + 3 x^{14} - x^{13} + 2 x^{12} - 8 x^{11} + 14 x^{10} - 18 x^{9} + 29 x^{8} - 49 x^{7} + \cdots + 1 \) 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:   \(36005299687890625\) \(\medspace = 5^{8}\cdot 19^{4}\cdot 29^{4}\) 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:  \(10.83\)
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:  $5^{1/2}19^{1/2}29^{1/2}\approx 52.48809388804284$
Ramified primes:   \(5\), \(19\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{12651607}a^{15}+\frac{31251}{12651607}a^{14}+\frac{2545018}{12651607}a^{13}+\frac{1339362}{12651607}a^{12}-\frac{3747613}{12651607}a^{11}+\frac{680896}{12651607}a^{10}+\frac{720624}{12651607}a^{9}+\frac{2522018}{12651607}a^{8}+\frac{3638991}{12651607}a^{7}-\frac{4922265}{12651607}a^{6}+\frac{3070876}{12651607}a^{5}+\frac{2067731}{12651607}a^{4}+\frac{456175}{12651607}a^{3}-\frac{1067672}{12651607}a^{2}+\frac{5918595}{12651607}a+\frac{622177}{12651607}$ 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:   \( -1 \)  (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:   $\frac{12061026}{12651607}a^{15}-\frac{35507039}{12651607}a^{14}+\frac{34894177}{12651607}a^{13}-\frac{10628075}{12651607}a^{12}+\frac{22207787}{12651607}a^{11}-\frac{94124937}{12651607}a^{10}+\frac{165036220}{12651607}a^{9}-\frac{209949274}{12651607}a^{8}+\frac{338479101}{12651607}a^{7}-\frac{570831561}{12651607}a^{6}+\frac{765940914}{12651607}a^{5}-\frac{813933705}{12651607}a^{4}+\frac{640263740}{12651607}a^{3}-\frac{350888837}{12651607}a^{2}+\frac{166002370}{12651607}a-\frac{43247557}{12651607}$, $\frac{611945}{12651607}a^{15}-\frac{5336589}{12651607}a^{14}+\frac{10869917}{12651607}a^{13}-\frac{5828798}{12651607}a^{12}-\frac{1539609}{12651607}a^{11}-\frac{9773825}{12651607}a^{10}+\frac{35794909}{12651607}a^{9}-\frac{45884527}{12651607}a^{8}+\frac{52999425}{12651607}a^{7}-\frac{98815686}{12651607}a^{6}+\frac{165238966}{12651607}a^{5}-\frac{189949808}{12651607}a^{4}+\frac{160772811}{12651607}a^{3}-\frac{90814595}{12651607}a^{2}+\frac{28474957}{12651607}a-\frac{12008400}{12651607}$, $\frac{12008400}{12651607}a^{15}-\frac{35413255}{12651607}a^{14}+\frac{30688611}{12651607}a^{13}-\frac{1138483}{12651607}a^{12}+\frac{18188002}{12651607}a^{11}-\frac{97606809}{12651607}a^{10}+\frac{158343775}{12651607}a^{9}-\frac{180356291}{12651607}a^{8}+\frac{302359073}{12651607}a^{7}-\frac{535412175}{12651607}a^{6}+\frac{693738714}{12651607}a^{5}-\frac{687357434}{12651607}a^{4}+\frac{494528992}{12651607}a^{3}-\frac{235504389}{12651607}a^{2}+\frac{113328205}{12651607}a-\frac{43575443}{12651607}$, $\frac{3122743}{12651607}a^{15}-\frac{5654905}{12651607}a^{14}+\frac{1265542}{12651607}a^{13}+\frac{1203443}{12651607}a^{12}+\frac{5425397}{12651607}a^{11}-\frac{16461120}{12651607}a^{10}+\frac{20169363}{12651607}a^{9}-\frac{23953733}{12651607}a^{8}+\frac{48826162}{12651607}a^{7}-\frac{75770743}{12651607}a^{6}+\frac{81233113}{12651607}a^{5}-\frac{71416492}{12651607}a^{4}+\frac{22249467}{12651607}a^{3}+\frac{76807}{12651607}a^{2}-\frac{799149}{12651607}a+\frac{16887735}{12651607}$, $\frac{5158034}{12651607}a^{15}-\frac{13055860}{12651607}a^{14}+\frac{7254626}{12651607}a^{13}+\frac{1473923}{12651607}a^{12}+\frac{11804423}{12651607}a^{11}-\frac{39339557}{12651607}a^{10}+\frac{49517865}{12651607}a^{9}-\frac{56666570}{12651607}a^{8}+\frac{115158494}{12651607}a^{7}-\frac{193028336}{12651607}a^{6}+\frac{225098780}{12651607}a^{5}-\frac{219493535}{12651607}a^{4}+\frac{159457767}{12651607}a^{3}-\frac{84330281}{12651607}a^{2}+\frac{62460872}{12651607}a-\frac{9163209}{12651607}$, $\frac{3122743}{12651607}a^{15}-\frac{5654905}{12651607}a^{14}+\frac{1265542}{12651607}a^{13}+\frac{1203443}{12651607}a^{12}+\frac{5425397}{12651607}a^{11}-\frac{16461120}{12651607}a^{10}+\frac{20169363}{12651607}a^{9}-\frac{23953733}{12651607}a^{8}+\frac{48826162}{12651607}a^{7}-\frac{75770743}{12651607}a^{6}+\frac{81233113}{12651607}a^{5}-\frac{71416492}{12651607}a^{4}+\frac{22249467}{12651607}a^{3}+\frac{76807}{12651607}a^{2}-\frac{13450756}{12651607}a+\frac{16887735}{12651607}$, $\frac{611945}{12651607}a^{15}-\frac{5336589}{12651607}a^{14}+\frac{10869917}{12651607}a^{13}-\frac{5828798}{12651607}a^{12}-\frac{1539609}{12651607}a^{11}-\frac{9773825}{12651607}a^{10}+\frac{35794909}{12651607}a^{9}-\frac{45884527}{12651607}a^{8}+\frac{52999425}{12651607}a^{7}-\frac{98815686}{12651607}a^{6}+\frac{165238966}{12651607}a^{5}-\frac{189949808}{12651607}a^{4}+\frac{160772811}{12651607}a^{3}-\frac{90814595}{12651607}a^{2}+\frac{28474957}{12651607}a+\frac{643207}{12651607}$ 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:  \( 30.5819893814 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 30.5819893814 \cdot 1}{2\cdot\sqrt{36005299687890625}}\cr\approx \mathstrut & 0.195745356895 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*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 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*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 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*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 - 3*x^15 + 3*x^14 - x^13 + 2*x^12 - 8*x^11 + 14*x^10 - 18*x^9 + 29*x^8 - 49*x^7 + 66*x^6 - 71*x^5 + 57*x^4 - 33*x^3 + 17*x^2 - 6*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

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.2.475.1, 4.2.13775.1, 8.2.9986875.1 x2, 8.0.6543125.1 x2, 8.4.189750625.1

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: 8.0.6543125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(19\) Copy content Toggle raw display 19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
\(29\) Copy content Toggle raw display 29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$