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Properties

Label 16.0.5575888916015625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.576\times 10^{15}$
Root discriminant \(9.64\)
Ramified primes $3,5,59$
Class number $1$
Class group trivial
Galois group $C_2^5:C_4$ (as 16T261)

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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 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^16 - 5*y^15 + 15*y^14 - 30*y^13 + 49*y^12 - 70*y^11 + 90*y^10 - 105*y^9 + 111*y^8 - 105*y^7 + 90*y^6 - 70*y^5 + 49*y^4 - 30*y^3 + 15*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*x + 1)
 

\( x^{16} - 5 x^{15} + 15 x^{14} - 30 x^{13} + 49 x^{12} - 70 x^{11} + 90 x^{10} - 105 x^{9} + 111 x^{8} + \cdots + 1 \) 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:   \(5575888916015625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 59^{2}\) 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:  \(9.64\)
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:  $3^{1/2}5^{3/4}59^{1/2}\approx 44.48505546908213$
Ramified primes:   \(3\), \(5\), \(59\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{29}a^{15}-\frac{6}{29}a^{14}-\frac{8}{29}a^{13}+\frac{7}{29}a^{12}+\frac{13}{29}a^{11}+\frac{4}{29}a^{10}-\frac{1}{29}a^{9}+\frac{12}{29}a^{8}+\frac{12}{29}a^{7}-\frac{1}{29}a^{6}+\frac{4}{29}a^{5}+\frac{13}{29}a^{4}+\frac{7}{29}a^{3}-\frac{8}{29}a^{2}-\frac{6}{29}a+\frac{1}{29}$ 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:   \( -\frac{15}{29} a^{15} + \frac{32}{29} a^{14} - \frac{54}{29} a^{13} + \frac{11}{29} a^{12} - \frac{50}{29} a^{11} + \frac{114}{29} a^{10} - \frac{217}{29} a^{9} + \frac{284}{29} a^{8} - \frac{296}{29} a^{7} + \frac{276}{29} a^{6} - \frac{263}{29} a^{5} + \frac{182}{29} a^{4} - \frac{105}{29} a^{3} + \frac{4}{29} a^{2} + \frac{61}{29} a - \frac{44}{29} \)  (order $30$) 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:   $\frac{4}{29}a^{15}+\frac{5}{29}a^{14}-\frac{32}{29}a^{13}+\frac{115}{29}a^{12}-\frac{180}{29}a^{11}+\frac{306}{29}a^{10}-\frac{439}{29}a^{9}+\frac{541}{29}a^{8}-\frac{590}{29}a^{7}+\frac{576}{29}a^{6}-\frac{506}{29}a^{5}+\frac{400}{29}a^{4}-\frac{291}{29}a^{3}+\frac{171}{29}a^{2}-\frac{82}{29}a+\frac{33}{29}$, $\frac{28}{29}a^{15}-\frac{110}{29}a^{14}+\frac{269}{29}a^{13}-\frac{413}{29}a^{12}+\frac{567}{29}a^{11}-\frac{758}{29}a^{10}+\frac{900}{29}a^{9}-\frac{1027}{29}a^{8}+\frac{1061}{29}a^{7}-\frac{927}{29}a^{6}+\frac{808}{29}a^{5}-\frac{622}{29}a^{4}+\frac{428}{29}a^{3}-\frac{282}{29}a^{2}+\frac{93}{29}a-\frac{1}{29}$, $\frac{7}{29}a^{15}-\frac{42}{29}a^{14}+\frac{147}{29}a^{13}-\frac{357}{29}a^{12}+\frac{700}{29}a^{11}-\frac{1161}{29}a^{10}+\frac{1646}{29}a^{9}-\frac{2004}{29}a^{8}+\frac{2143}{29}a^{7}-\frac{2095}{29}a^{6}+\frac{1797}{29}a^{5}-\frac{1388}{29}a^{4}+\frac{890}{29}a^{3}-\frac{462}{29}a^{2}+\frac{190}{29}a-\frac{51}{29}$, $\frac{12}{29}a^{15}-\frac{72}{29}a^{14}+\frac{252}{29}a^{13}-\frac{583}{29}a^{12}+\frac{1055}{29}a^{11}-\frac{1576}{29}a^{10}+\frac{2076}{29}a^{9}-\frac{2408}{29}a^{8}+\frac{2493}{29}a^{7}-\frac{2361}{29}a^{6}+\frac{1962}{29}a^{5}-\frac{1439}{29}a^{4}+\frac{925}{29}a^{3}-\frac{502}{29}a^{2}+\frac{218}{29}a-\frac{46}{29}$, $\frac{15}{29}a^{15}-\frac{90}{29}a^{14}+\frac{315}{29}a^{13}-\frac{736}{29}a^{12}+\frac{1355}{29}a^{11}-\frac{2057}{29}a^{10}+\frac{2740}{29}a^{9}-\frac{3184}{29}a^{8}+\frac{3341}{29}a^{7}-\frac{3147}{29}a^{6}+\frac{2612}{29}a^{5}-\frac{1951}{29}a^{4}+\frac{1236}{29}a^{3}-\frac{642}{29}a^{2}+\frac{258}{29}a-\frac{72}{29}$, $\frac{55}{29}a^{15}-\frac{243}{29}a^{14}+\frac{633}{29}a^{13}-\frac{1036}{29}a^{12}+\frac{1382}{29}a^{11}-\frac{1723}{29}a^{10}+\frac{2004}{29}a^{9}-\frac{2211}{29}a^{8}+\frac{2226}{29}a^{7}-\frac{1969}{29}a^{6}+\frac{1612}{29}a^{5}-\frac{1286}{29}a^{4}+\frac{965}{29}a^{3}-\frac{643}{29}a^{2}+\frac{308}{29}a-\frac{61}{29}$, $\frac{39}{29}a^{15}-\frac{176}{29}a^{14}+\frac{471}{29}a^{13}-\frac{800}{29}a^{12}+\frac{1087}{29}a^{11}-\frac{1294}{29}a^{10}+\frac{1353}{29}a^{9}-\frac{1301}{29}a^{8}+\frac{1164}{29}a^{7}-\frac{851}{29}a^{6}+\frac{533}{29}a^{5}-\frac{334}{29}a^{4}+\frac{186}{29}a^{3}-\frac{138}{29}a^{2}+\frac{85}{29}a-\frac{19}{29}$ 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:  \( 155.628338249 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 155.628338249 \cdot 1}{30\cdot\sqrt{5575888916015625}}\cr\approx \mathstrut & 0.168752150436 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*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 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*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 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*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 + 15*x^14 - 30*x^13 + 49*x^12 - 70*x^11 + 90*x^10 - 105*x^9 + 111*x^8 - 105*x^7 + 90*x^6 - 70*x^5 + 49*x^4 - 30*x^3 + 15*x^2 - 5*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

$C_2^5:C_4$ (as 16T261):

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 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.6.74671875.1, 8.2.74671875.1, \(\Q(\zeta_{15})\)

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: 16.0.239625547119140625.1

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ R

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 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(59\) Copy content Toggle raw display 59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.1$x^{4} + 116 x^{3} + 3486 x^{2} + 7076 x + 201725$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$