Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818)
gp: K = bnfinit(y^16 - 48*y^14 + 936*y^12 - 9504*y^10 + 53604*y^8 - 166752*y^6 + 270864*y^4 - 202176*y^2 + 46818, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818)
\( x^{16} - 48x^{14} + 936x^{12} - 9504x^{10} + 53604x^{8} - 166752x^{6} + 270864x^{4} - 202176x^{2} + 46818 \)
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: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(321236373250909071617512439808\)
\(\medspace = 2^{79}\cdot 3^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.85\) | 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: | $2^{5}3^{3/4}\approx 72.94422582255288$ | ||
| Ramified primes: |
\(2\), \(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{63}a^{8}-\frac{1}{21}a^{6}-\frac{1}{7}a^{4}+\frac{1}{7}a^{2}-\frac{2}{7}$, $\frac{1}{63}a^{9}-\frac{1}{21}a^{7}-\frac{1}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a$, $\frac{1}{63}a^{10}+\frac{1}{21}a^{6}+\frac{1}{21}a^{4}+\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{63}a^{11}+\frac{1}{21}a^{7}+\frac{1}{21}a^{5}+\frac{1}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{189}a^{12}-\frac{1}{21}a^{6}-\frac{1}{7}a^{4}-\frac{3}{7}a^{2}+\frac{2}{7}$, $\frac{1}{3213}a^{13}+\frac{1}{357}a^{11}+\frac{1}{153}a^{9}-\frac{5}{357}a^{7}-\frac{1}{51}a^{5}+\frac{7}{17}a^{3}-\frac{44}{119}a$, $\frac{1}{3213}a^{14}-\frac{8}{3213}a^{12}+\frac{1}{153}a^{10}+\frac{2}{1071}a^{8}-\frac{1}{51}a^{6}+\frac{4}{51}a^{4}+\frac{24}{119}a^{2}+\frac{3}{7}$, $\frac{1}{3213}a^{15}-\frac{1}{357}a^{11}+\frac{1}{153}a^{9}-\frac{10}{119}a^{7}-\frac{4}{51}a^{5}-\frac{26}{119}a^{3}+\frac{5}{119}a$
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 $1$ (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: | $15$ | 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: |
$\frac{1}{63}a^{8}-\frac{8}{21}a^{6}+\frac{20}{7}a^{4}-\frac{48}{7}a^{2}+\frac{19}{7}$, $\frac{2}{189}a^{12}-\frac{8}{21}a^{10}+\frac{36}{7}a^{8}-32a^{6}+\frac{1903}{21}a^{4}-100a^{2}+\frac{193}{7}$, $\frac{1}{189}a^{12}-\frac{4}{21}a^{10}+\frac{18}{7}a^{8}-16a^{6}+\frac{955}{21}a^{4}-52a^{2}+\frac{107}{7}$, $\frac{11}{3213}a^{14}-\frac{394}{3213}a^{12}+\frac{1760}{1071}a^{10}-\frac{10807}{1071}a^{8}+\frac{3295}{119}a^{6}-\frac{9824}{357}a^{4}+\frac{366}{119}a^{2}+\frac{19}{7}$, $\frac{5}{3213}a^{14}-\frac{10}{153}a^{12}+\frac{1174}{1071}a^{10}-\frac{3340}{357}a^{8}+\frac{5077}{119}a^{6}-\frac{35237}{357}a^{4}+\frac{11612}{119}a^{2}-\frac{179}{7}$, $\frac{11}{3213}a^{14}-\frac{1}{459}a^{13}-\frac{496}{3213}a^{12}+\frac{9}{119}a^{11}+\frac{2984}{1071}a^{10}-\frac{115}{119}a^{9}-\frac{27331}{1071}a^{8}+\frac{652}{119}a^{7}+\frac{14719}{119}a^{6}-\frac{1542}{119}a^{5}-\frac{106877}{357}a^{4}+\frac{1153}{119}a^{3}+\frac{36185}{119}a^{2}-\frac{236}{119}a-81$, $\frac{4}{3213}a^{15}+\frac{1}{1071}a^{14}-\frac{190}{3213}a^{13}-\frac{160}{3213}a^{12}+\frac{401}{357}a^{11}+\frac{1109}{1071}a^{10}-\frac{11570}{1071}a^{9}-\frac{11435}{1071}a^{8}+\frac{19547}{357}a^{7}+\frac{6793}{119}a^{6}-\frac{49136}{357}a^{5}-\frac{7655}{51}a^{4}+\frac{17089}{119}a^{3}+\frac{19639}{119}a^{2}-\frac{4404}{119}a-\frac{333}{7}$, $\frac{11}{3213}a^{14}+\frac{1}{459}a^{13}-\frac{479}{3213}a^{12}-\frac{9}{119}a^{11}+\frac{2780}{1071}a^{10}+\frac{115}{119}a^{9}-\frac{8198}{357}a^{8}-\frac{652}{119}a^{7}+\frac{38581}{357}a^{6}+\frac{1542}{119}a^{5}-\frac{30554}{119}a^{4}-\frac{1153}{119}a^{3}+\frac{30813}{119}a^{2}+\frac{236}{119}a-\frac{479}{7}$, $\frac{1}{459}a^{15}+\frac{11}{3213}a^{14}-\frac{299}{3213}a^{13}-\frac{445}{3213}a^{12}+\frac{100}{63}a^{11}+\frac{2372}{1071}a^{10}-\frac{14743}{1071}a^{9}-\frac{6362}{357}a^{8}+\frac{22790}{357}a^{7}+\frac{27157}{357}a^{6}-\frac{54254}{357}a^{5}-\frac{2830}{17}a^{4}+\frac{19235}{119}a^{3}+\frac{19389}{119}a^{2}-\frac{6512}{119}a-\frac{349}{7}$, $\frac{1}{459}a^{15}-\frac{11}{3213}a^{14}-\frac{299}{3213}a^{13}+\frac{428}{3213}a^{12}+\frac{100}{63}a^{11}-\frac{2168}{1071}a^{10}-\frac{14743}{1071}a^{9}+\frac{16349}{1071}a^{8}+\frac{22790}{357}a^{7}-\frac{3083}{51}a^{6}-\frac{54254}{357}a^{5}+\frac{44215}{357}a^{4}+\frac{19235}{119}a^{3}-\frac{14017}{119}a^{2}-\frac{6512}{119}a+\frac{261}{7}$, $\frac{11}{3213}a^{14}-\frac{137}{1071}a^{12}+\frac{1964}{1071}a^{10}-\frac{13544}{1071}a^{8}+\frac{15461}{357}a^{6}-\frac{3560}{51}a^{4}+\frac{5262}{119}a^{2}-\frac{27}{7}$, $\frac{1}{459}a^{15}-\frac{11}{3213}a^{14}-\frac{37}{357}a^{13}+\frac{479}{3213}a^{12}+\frac{124}{63}a^{11}-\frac{2780}{1071}a^{10}-\frac{2893}{153}a^{9}+\frac{8198}{357}a^{8}+\frac{34214}{357}a^{7}-\frac{38581}{357}a^{6}-\frac{86605}{357}a^{5}+\frac{91781}{357}a^{4}+\frac{31135}{119}a^{3}-\frac{31289}{119}a^{2}-\frac{1382}{17}a+\frac{535}{7}$, $\frac{4}{3213}a^{15}-\frac{1}{1071}a^{14}-\frac{173}{3213}a^{13}+\frac{143}{3213}a^{12}+\frac{111}{119}a^{11}-\frac{905}{1071}a^{10}-\frac{8833}{1071}a^{9}+\frac{2888}{357}a^{8}+\frac{4657}{119}a^{7}-\frac{14531}{357}a^{6}-\frac{11307}{119}a^{5}+\frac{5207}{51}a^{4}+\frac{11717}{119}a^{3}-\frac{1873}{17}a^{2}-\frac{2908}{119}a+\frac{221}{7}$, $\frac{11}{3213}a^{14}-\frac{10}{3213}a^{13}-\frac{445}{3213}a^{12}+\frac{41}{357}a^{11}+\frac{2372}{1071}a^{10}-\frac{191}{119}a^{9}-\frac{6362}{357}a^{8}+\frac{1252}{119}a^{7}+\frac{27157}{357}a^{6}-\frac{11609}{357}a^{5}-\frac{8473}{51}a^{4}+\frac{5035}{119}a^{3}+\frac{18913}{119}a^{2}-\frac{1940}{119}a-\frac{293}{7}$, $\frac{4}{3213}a^{15}-\frac{1}{1071}a^{14}-\frac{164}{3213}a^{13}+\frac{109}{3213}a^{12}+\frac{890}{1071}a^{11}-\frac{71}{153}a^{10}-\frac{7325}{1071}a^{9}+\frac{1052}{357}a^{8}+\frac{10679}{357}a^{7}-\frac{3107}{357}a^{6}-\frac{23699}{357}a^{5}+\frac{3979}{357}a^{4}+\frac{7517}{119}a^{3}-\frac{105}{17}a^{2}-\frac{2046}{119}a+1$
| 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: | \( 8601040814.68 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 8601040814.68 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.497265794585 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818)
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 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818, 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 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818);
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 - 48*x^14 + 936*x^12 - 9504*x^10 + 53604*x^8 - 166752*x^6 + 270864*x^4 - 202176*x^2 + 46818);
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
$\OD_{32}$ (as 16T22):
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 32 |
| The 20 conjugacy class representatives for $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.173946175488.1 |
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
| Galois closure: | deg 32 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | $16$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | $16$ | $16$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ | $16$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | $16$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | $16$ | $16$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.79.9 | $x^{16} + 16 x^{14} + 40 x^{12} + 32 x^{11} + 16 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 34$ | $16$ | $1$ | $79$ | $C_{16} : C_2$ | $[2, 3, 4, 5, 6]$ |
|
\(3\)
| 3.16.12.3 | $x^{16} - 6 x^{12} + 162$ | $4$ | $4$ | $12$ | $C_{16} : C_2$ | $[\ ]_{4}^{8}$ |