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Properties

Label 16.0.102...728.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.029\times 10^{34}$
Root discriminant \(133.59\)
Ramified primes $2,11,17$
Class number $2945032$ (GRH)
Class group [2, 2, 736258] (GRH)
Galois group $C_{16}$ (as 16T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112)
 
gp: K = bnfinit(y^16 + 374*y^14 + 57596*y^12 + 4706416*y^10 + 219029360*y^8 + 5782375104*y^6 + 80953251456*y^4 + 508849009152*y^2 + 932889850112, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112)
 

\( x^{16} + 374 x^{14} + 57596 x^{12} + 4706416 x^{10} + 219029360 x^{8} + 5782375104 x^{6} + \cdots + 932889850112 \) 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:   \(10294261539221265797567188438089728\) \(\medspace = 2^{24}\cdot 11^{8}\cdot 17^{15}\) 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:  \(133.59\)
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:  $2^{3/2}11^{1/2}17^{15/16}\approx 133.59407579823701$
Ramified primes:   \(2\), \(11\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1496=2^{3}\cdot 11\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(131,·)$, $\chi_{1496}(1409,·)$, $\chi_{1496}(1099,·)$, $\chi_{1496}(1233,·)$, $\chi_{1496}(483,·)$, $\chi_{1496}(89,·)$, $\chi_{1496}(923,·)$, $\chi_{1496}(353,·)$, $\chi_{1496}(1187,·)$, $\chi_{1496}(529,·)$, $\chi_{1496}(1451,·)$, $\chi_{1496}(1363,·)$, $\chi_{1496}(441,·)$, $\chi_{1496}(705,·)$, $\chi_{1496}(571,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $\frac{1}{22}a^{2}$, $\frac{1}{22}a^{3}$, $\frac{1}{484}a^{4}$, $\frac{1}{484}a^{5}$, $\frac{1}{10648}a^{6}$, $\frac{1}{10648}a^{7}$, $\frac{1}{234256}a^{8}$, $\frac{1}{234256}a^{9}$, $\frac{1}{5153632}a^{10}$, $\frac{1}{5153632}a^{11}$, $\frac{1}{113379904}a^{12}$, $\frac{1}{113379904}a^{13}$, $\frac{1}{2494357888}a^{14}$, $\frac{1}{2494357888}a^{15}$ 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

$C_{2}\times C_{2}\times C_{736258}$, which has order $2945032$ (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:   $\frac{1}{10648}a^{6}+\frac{3}{242}a^{4}+\frac{9}{22}a^{2}+2$, $\frac{1}{2494357888}a^{14}+\frac{15}{113379904}a^{12}+\frac{45}{2576816}a^{10}+\frac{137}{117128}a^{8}+\frac{441}{10648}a^{6}+\frac{351}{484}a^{4}+\frac{111}{22}a^{2}+9$, $\frac{1}{113379904}a^{12}+\frac{13}{5153632}a^{10}+\frac{65}{234256}a^{8}+\frac{39}{2662}a^{6}+\frac{91}{242}a^{4}+\frac{45}{11}a^{2}+10$, $\frac{1}{234256}a^{8}+\frac{1}{1331}a^{6}+\frac{5}{121}a^{4}+\frac{8}{11}a^{2}+3$, $\frac{1}{5153632}a^{10}+\frac{5}{117128}a^{8}+\frac{35}{10648}a^{6}+\frac{25}{242}a^{4}+\frac{25}{22}a^{2}+3$, $\frac{1}{113379904}a^{12}+\frac{3}{1288408}a^{10}+\frac{27}{117128}a^{8}+\frac{14}{1331}a^{6}+\frac{105}{484}a^{4}+\frac{18}{11}a^{2}+3$, $\frac{1}{113379904}a^{12}+\frac{3}{1288408}a^{10}+\frac{27}{117128}a^{8}+\frac{14}{1331}a^{6}+\frac{105}{484}a^{4}+\frac{18}{11}a^{2}+2$ 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:  \( 3640.012213375973 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.012213375973 \cdot 2945032}{2\cdot\sqrt{10294261539221265797567188438089728}}\cr\approx \mathstrut & 0.128322913420065 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112)
 
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 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112, 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 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112);
 
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 + 374*x^14 + 57596*x^12 + 4706416*x^10 + 219029360*x^8 + 5782375104*x^6 + 80953251456*x^4 + 508849009152*x^2 + 932889850112);
 
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

$C_{16}$ (as 16T1):

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 cyclic group of order 16
The 16 conjugacy class representatives for $C_{16}$
Character table for $C_{16}$

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $16$ $16$ $16$ R ${\href{/padicField/13.4.0.1}{4} }^{4}$ 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 2.8.12.6$x^{8} - 16 x^{6} + 72 x^{4} + 3664$$2$$4$$12$$C_8$$[3]^{4}$
2.8.12.6$x^{8} - 16 x^{6} + 72 x^{4} + 3664$$2$$4$$12$$C_8$$[3]^{4}$
\(11\) Copy content Toggle raw display 11.16.8.2$x^{16} + 102487 x^{8} - 1127357 x^{6} + 1771561 x^{4} - 136410197 x^{2} + 428717762$$2$$8$$8$$C_{16}$$[\ ]_{2}^{8}$
\(17\) Copy content Toggle raw display 17.16.15.1$x^{16} + 272$$16$$1$$15$$C_{16}$$[\ ]_{16}$