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Properties

Label 16.0.905...832.1
Degree $16$
Signature $[0, 8]$
Discriminant $9.055\times 10^{29}$
Root discriminant \(74.53\)
Ramified primes $2,13,17$
Class number $41760$ (GRH)
Class group [2, 2, 2, 6, 870] (GRH)
Galois group $C_2^4.C_8$ (as 16T306)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753)
 
gp: K = bnfinit(y^16 + 136*y^14 + 7174*y^12 + 186524*y^10 + 2536859*y^8 + 18002218*y^6 + 66518569*y^4 + 119927639*y^2 + 82055753, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753)
 

\( x^{16} + 136 x^{14} + 7174 x^{12} + 186524 x^{10} + 2536859 x^{8} + 18002218 x^{6} + 66518569 x^{4} + \cdots + 82055753 \) 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:   \(905469581514181342527129976832\) \(\medspace = 2^{16}\cdot 13^{6}\cdot 17^{15}\) 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:  \(74.53\)
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:  $2^{3/2}13^{1/2}17^{15/16}\approx 145.23207201172457$
Ramified primes:   \(2\), \(13\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}+\frac{6}{13}a^{4}-\frac{2}{13}a^{2}$, $\frac{1}{13}a^{7}+\frac{6}{13}a^{5}-\frac{2}{13}a^{3}$, $\frac{1}{169}a^{8}+\frac{6}{169}a^{6}-\frac{28}{169}a^{4}+\frac{3}{13}a^{2}$, $\frac{1}{169}a^{9}+\frac{6}{169}a^{7}-\frac{28}{169}a^{5}+\frac{3}{13}a^{3}$, $\frac{1}{2197}a^{10}+\frac{6}{2197}a^{8}-\frac{28}{2197}a^{6}+\frac{3}{169}a^{4}+\frac{3}{13}a^{2}$, $\frac{1}{2197}a^{11}+\frac{6}{2197}a^{9}-\frac{28}{2197}a^{7}+\frac{3}{169}a^{5}+\frac{3}{13}a^{3}$, $\frac{1}{28561}a^{12}+\frac{6}{28561}a^{10}-\frac{28}{28561}a^{8}+\frac{3}{2197}a^{6}+\frac{16}{169}a^{4}-\frac{2}{13}a^{2}$, $\frac{1}{28561}a^{13}+\frac{6}{28561}a^{11}-\frac{28}{28561}a^{9}+\frac{3}{2197}a^{7}+\frac{16}{169}a^{5}-\frac{2}{13}a^{3}$, $\frac{1}{65722530488719}a^{14}+\frac{839039025}{65722530488719}a^{12}+\frac{9540465803}{65722530488719}a^{10}+\frac{2617297649}{5055579268363}a^{8}-\frac{4729657454}{388890712951}a^{6}-\frac{14627075999}{29914670227}a^{4}+\frac{895106165}{2301128479}a^{2}+\frac{71505139}{177009883}$, $\frac{1}{65722530488719}a^{15}+\frac{839039025}{65722530488719}a^{13}+\frac{9540465803}{65722530488719}a^{11}+\frac{2617297649}{5055579268363}a^{9}-\frac{4729657454}{388890712951}a^{7}-\frac{14627075999}{29914670227}a^{5}+\frac{895106165}{2301128479}a^{3}+\frac{71505139}{177009883}a$ 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

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{870}$, which has order $41760$ (assuming GRH)

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:   $\frac{17986174}{5055579268363}a^{14}+\frac{2361465963}{5055579268363}a^{12}+\frac{117963257066}{5055579268363}a^{10}+\frac{2805225650101}{5055579268363}a^{8}+\frac{2513271925063}{388890712951}a^{6}+\frac{79545196507}{2301128479}a^{4}+\frac{188221550099}{2301128479}a^{2}+\frac{12115835306}{177009883}$, $\frac{157629275}{65722530488719}a^{14}+\frac{20807835439}{65722530488719}a^{12}+\frac{1048117950702}{65722530488719}a^{10}+\frac{1942703801799}{5055579268363}a^{8}+\frac{1777960941581}{388890712951}a^{6}+\frac{756776914479}{29914670227}a^{4}+\frac{141437583583}{2301128479}a^{2}+\frac{8987714766}{177009883}$, $\frac{7261955}{5055579268363}a^{14}+\frac{951576712}{5055579268363}a^{12}+\frac{47359149300}{5055579268363}a^{10}+\frac{1117757303050}{5055579268363}a^{8}+\frac{984725440194}{388890712951}a^{6}+\frac{389190702964}{29914670227}a^{4}+\frac{65703429979}{2301128479}a^{2}+\frac{3775742672}{177009883}$, $\frac{7261955}{5055579268363}a^{14}+\frac{951576712}{5055579268363}a^{12}+\frac{47359149300}{5055579268363}a^{10}+\frac{1117757303050}{5055579268363}a^{8}+\frac{984725440194}{388890712951}a^{6}+\frac{389190702964}{29914670227}a^{4}+\frac{65703429979}{2301128479}a^{2}+\frac{3952752555}{177009883}$, $\frac{179714765}{65722530488719}a^{14}+\frac{23860179514}{65722530488719}a^{12}+\frac{1211518600859}{65722530488719}a^{10}+\frac{2271803296652}{5055579268363}a^{8}+\frac{2115861964076}{388890712951}a^{6}+\frac{924928466990}{29914670227}a^{4}+\frac{177869196124}{2301128479}a^{2}+\frac{11965222798}{177009883}$, $\frac{59503261}{65722530488719}a^{14}+\frac{7682472961}{65722530488719}a^{12}+\frac{373439971608}{65722530488719}a^{10}+\frac{651073624443}{5055579268363}a^{8}+\frac{530618650686}{388890712951}a^{6}+\frac{13453715947}{2301128479}a^{4}+\frac{18837095983}{2301128479}a^{2}+\frac{266968683}{177009883}$, $\frac{79911586}{65722530488719}a^{14}+\frac{10646087302}{65722530488719}a^{12}+\frac{544413429350}{65722530488719}a^{10}+\frac{1036394722608}{5055579268363}a^{8}+\frac{997927834183}{388890712951}a^{6}+\frac{469998544316}{29914670227}a^{4}+\frac{103681024137}{2301128479}a^{2}+\frac{8427143717}{177009883}$ Copy content Toggle raw display (assuming GRH)
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:  \( 3640.01221338 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 41760}{2\cdot\sqrt{905469581514181342527129976832}}\cr\approx \mathstrut & 0.194015018481 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753)
 
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 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753, 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 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753);
 
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 + 136*x^14 + 7174*x^12 + 186524*x^10 + 2536859*x^8 + 18002218*x^6 + 66518569*x^4 + 119927639*x^2 + 82055753);
 
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

$C_2^4.C_8$ (as 16T306):

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 $C_2^4.C_8$
Character table for $C_2^4.C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 sibling: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.31703006950533291639898112.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ $16$ $16$ $16$ R R ${\href{/padicField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(2\) Copy content Toggle raw display 2.8.8.5$x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.8.5$x^{8} + 8 x^{7} + 56 x^{6} + 184 x^{5} + 576 x^{4} + 960 x^{3} + 1632 x^{2} + 1120 x + 304$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
\(13\) Copy content Toggle raw display 13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 156 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(17\) Copy content Toggle raw display 17.16.15.1$x^{16} + 272$$16$$1$$15$$C_{16}$$[\ ]_{16}$