Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2)
gp: K = bnfinit(y^16 + 16*y^14 + 104*y^12 + 352*y^10 + 660*y^8 + 672*y^6 + 336*y^4 + 64*y^2 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2)
\( x^{16} + 16x^{14} + 104x^{12} + 352x^{10} + 660x^{8} + 672x^{6} + 336x^{4} + 64x^{2} + 2 \)
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: |
\(604462909807314587353088\)
\(\medspace = 2^{79}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.64\) | 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: | $2^{79/16}\approx 30.64330498235436$ | ||
| Ramified primes: |
\(2\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(64=2^{6}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{64}(1,·)$, $\chi_{64}(3,·)$, $\chi_{64}(9,·)$, $\chi_{64}(11,·)$, $\chi_{64}(17,·)$, $\chi_{64}(19,·)$, $\chi_{64}(25,·)$, $\chi_{64}(27,·)$, $\chi_{64}(33,·)$, $\chi_{64}(35,·)$, $\chi_{64}(41,·)$, $\chi_{64}(43,·)$, $\chi_{64}(49,·)$, $\chi_{64}(51,·)$, $\chi_{64}(57,·)$, $\chi_{64}(59,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{17}$, which has order $17$ (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 $2$)
| 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: |
$a^{8}+8a^{6}+20a^{4}+16a^{2}+1$, $a^{4}+4a^{2}+1$, $a^{8}+8a^{6}+21a^{4}+20a^{2}+5$, $a^{6}+6a^{4}+9a^{2}+1$, $a^{14}+14a^{12}+77a^{10}+210a^{8}+294a^{6}+196a^{4}+49a^{2}+3$, $a^{6}+7a^{4}+14a^{2}+7$, $a^{2}+1$
| 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: | \( 15753.9498624 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 15753.9498624 \cdot 17}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 0.418371895598 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2)
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 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2, 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 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2);
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 + 16*x^14 + 104*x^12 + 352*x^10 + 660*x^8 + 672*x^6 + 336*x^4 + 64*x^2 + 2);
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
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
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\) |
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 | $16$ | $16$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/31.1.0.1}{1} }^{16}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | $16$ | $16$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.2 | $x^{16} + 16 x^{14} + 56 x^{12} + 48 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 2$ | $16$ | $1$ | $79$ | $C_{16}$ | $[3, 4, 5, 6]$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.64.16t1.a.a | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.32.8t1.a.a | $1$ | $ 2^{5}$ | \(\Q(\zeta_{32})^+\) | $C_8$ (as 8T1) | $0$ | $1$ |
| * | 1.64.16t1.a.b | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.16.4t1.a.a | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.64.16t1.a.c | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.32.8t1.a.b | $1$ | $ 2^{5}$ | \(\Q(\zeta_{32})^+\) | $C_8$ (as 8T1) | $0$ | $1$ |
| * | 1.64.16t1.a.d | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.64.16t1.a.e | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.32.8t1.a.c | $1$ | $ 2^{5}$ | \(\Q(\zeta_{32})^+\) | $C_8$ (as 8T1) | $0$ | $1$ |
| * | 1.64.16t1.a.f | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.16.4t1.a.b | $1$ | $ 2^{4}$ | \(\Q(\zeta_{16})^+\) | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.64.16t1.a.g | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
| * | 1.32.8t1.a.d | $1$ | $ 2^{5}$ | \(\Q(\zeta_{32})^+\) | $C_8$ (as 8T1) | $0$ | $1$ |
| * | 1.64.16t1.a.h | $1$ | $ 2^{6}$ | 16.0.604462909807314587353088.1 | $C_{16}$ (as 16T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.