Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 1)
gp: K = bnfinit(y^16 - 4*y^14 + 8*y^12 + 16*y^10 + 23*y^8 - 16*y^6 + 8*y^4 + 4*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 1)
\( x^{16} - 4x^{14} + 8x^{12} + 16x^{10} + 23x^{8} - 16x^{6} + 8x^{4} + 4x^{2} + 1 \)
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: |
\(109951162777600000000\)
\(\medspace = 2^{48}\cdot 5^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.89\) | 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: | $2^{3}5^{1/2}\approx 17.88854381999832$ | ||
| Ramified primes: |
\(2\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{3}$, $\frac{1}{27}a^{12}+\frac{1}{27}a^{10}-\frac{4}{27}a^{8}+\frac{2}{9}a^{6}+\frac{13}{27}a^{4}+\frac{1}{27}a^{2}+\frac{8}{27}$, $\frac{1}{27}a^{13}+\frac{1}{27}a^{11}-\frac{4}{27}a^{9}+\frac{2}{9}a^{7}+\frac{13}{27}a^{5}+\frac{1}{27}a^{3}+\frac{8}{27}a$, $\frac{1}{1107}a^{14}+\frac{5}{1107}a^{12}+\frac{5}{41}a^{10}-\frac{163}{1107}a^{8}+\frac{442}{1107}a^{6}+\frac{26}{1107}a^{4}+\frac{67}{369}a^{2}+\frac{419}{1107}$, $\frac{1}{1107}a^{15}+\frac{5}{1107}a^{13}+\frac{5}{41}a^{11}-\frac{163}{1107}a^{9}+\frac{442}{1107}a^{7}+\frac{26}{1107}a^{5}+\frac{67}{369}a^{3}+\frac{419}{1107}a$
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 $1$
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: |
\( \frac{68}{369} a^{14} - \frac{907}{1107} a^{12} + \frac{1997}{1107} a^{10} + \frac{2500}{1107} a^{8} + \frac{370}{123} a^{6} - \frac{5356}{1107} a^{4} + \frac{2177}{1107} a^{2} - \frac{419}{1107} \)
(order $8$)
| 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: |
$\frac{2}{123}a^{14}-\frac{115}{1107}a^{12}+\frac{380}{1107}a^{10}-\frac{269}{1107}a^{8}+\frac{28}{369}a^{6}+\frac{17}{1107}a^{4}+\frac{2675}{1107}a^{2}-\frac{2}{1107}$, $\frac{172}{1107}a^{15}-\frac{616}{1107}a^{13}+\frac{40}{41}a^{11}+\frac{3329}{1107}a^{9}+\frac{5176}{1107}a^{7}-\frac{1432}{1107}a^{5}-\frac{284}{369}a^{3}+\frac{1589}{1107}a$, $\frac{170}{1107}a^{15}-\frac{626}{1107}a^{13}+\frac{131}{123}a^{11}+\frac{2917}{1107}a^{9}+\frac{5399}{1107}a^{7}-\frac{1484}{1107}a^{5}-\frac{295}{369}a^{3}-\frac{1094}{1107}a$, $\frac{206}{1107}a^{15}+\frac{190}{1107}a^{14}-\frac{299}{369}a^{13}-\frac{731}{1107}a^{12}+\frac{1898}{1107}a^{11}+\frac{1460}{1107}a^{10}+\frac{2912}{1107}a^{9}+\frac{340}{123}a^{8}+\frac{3107}{1107}a^{7}+\frac{5260}{1107}a^{6}-\frac{1768}{369}a^{5}-\frac{1415}{1107}a^{4}+\frac{2210}{1107}a^{3}+\frac{2930}{1107}a^{2}+\frac{2264}{1107}a+\frac{160}{369}$, $\frac{688}{1107}a^{15}-\frac{172}{1107}a^{14}-\frac{2956}{1107}a^{13}+\frac{616}{1107}a^{12}+\frac{2137}{369}a^{11}-\frac{40}{41}a^{10}+\frac{9011}{1107}a^{9}-\frac{3329}{1107}a^{8}+\frac{13324}{1107}a^{7}-\frac{5176}{1107}a^{6}-\frac{14338}{1107}a^{5}+\frac{1432}{1107}a^{4}+\frac{3620}{369}a^{3}+\frac{284}{369}a^{2}+\frac{575}{1107}a-\frac{1589}{1107}$, $\frac{56}{369}a^{15}-\frac{2}{1107}a^{14}-\frac{677}{1107}a^{13}-\frac{17}{369}a^{12}+\frac{1237}{1107}a^{11}+\frac{58}{1107}a^{10}+\frac{3038}{1107}a^{9}+\frac{121}{1107}a^{8}+\frac{1054}{369}a^{7}-\frac{2237}{1107}a^{6}-\frac{5390}{1107}a^{5}-\frac{434}{123}a^{4}-\frac{4280}{1107}a^{3}-\frac{3395}{1107}a^{2}-\frac{415}{1107}a-\frac{428}{1107}$, $\frac{23}{123}a^{15}-\frac{44}{369}a^{14}-\frac{851}{1107}a^{13}+\frac{611}{1107}a^{12}+\frac{1705}{1107}a^{11}-\frac{1420}{1107}a^{10}+\frac{3323}{1107}a^{9}-\frac{1280}{1107}a^{8}+\frac{1265}{369}a^{7}-\frac{670}{369}a^{6}-\frac{3638}{1107}a^{5}+\frac{3128}{1107}a^{4}-\frac{131}{1107}a^{3}-\frac{169}{1107}a^{2}-\frac{679}{1107}a+\frac{247}{1107}$
| 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: | \( 9717.26549391 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 9717.26549391 \cdot 1}{8\cdot\sqrt{109951162777600000000}}\cr\approx \mathstrut & 0.281379928682 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 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 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 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 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 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 - 4*x^14 + 8*x^12 + 16*x^10 + 23*x^8 - 16*x^6 + 8*x^4 + 4*x^2 + 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
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 16 |
| The 10 conjugacy class representatives for $Q_8 : C_2$ |
| Character table for $Q_8 : C_2$ |
Intermediate fields
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 8 siblings: | 8.0.419430400.2, 8.4.2621440000.1, 8.0.2621440000.2 |
| Minimal sibling: | 8.0.419430400.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.1.0.1}{1} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $48$ | |||
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |