Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*x^2 + 1)
gp: K = bnfinit(y^16 - 9*y^14 + 19*y^12 + 2*y^10 - 27*y^8 + 2*y^6 + 19*y^4 - 9*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*x^2 + 1)
\( x^{16} - 9x^{14} + 19x^{12} + 2x^{10} - 27x^{8} + 2x^{6} + 19x^{4} - 9x^{2} + 1 \)
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: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(700762351958879698944\)
\(\medspace = 2^{16}\cdot 3^{6}\cdot 19^{4}\cdot 103^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.08\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(19\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{10}+\frac{1}{9}a^{8}+\frac{2}{9}a^{6}+\frac{1}{9}a^{4}+\frac{1}{9}a^{2}+\frac{1}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{2}{9}a^{7}+\frac{1}{9}a^{5}+\frac{1}{9}a^{3}+\frac{1}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{8}-\frac{1}{9}a^{6}-\frac{1}{9}$, $\frac{1}{9}a^{15}+\frac{1}{9}a^{9}-\frac{1}{9}a^{7}-\frac{1}{9}a$
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: | $13$ | 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: |
$a^{15}-9a^{13}+19a^{11}+2a^{9}-27a^{7}+2a^{5}+19a^{3}-9a$, $\frac{26}{9}a^{15}-\frac{223}{9}a^{13}+\frac{401}{9}a^{11}+\frac{211}{9}a^{9}-\frac{598}{9}a^{7}-\frac{187}{9}a^{5}+\frac{395}{9}a^{3}-8a$, $\frac{10}{9}a^{15}-10a^{13}+21a^{11}+\frac{28}{9}a^{9}-\frac{280}{9}a^{7}+23a^{3}-\frac{64}{9}a$, $\frac{10}{9}a^{14}-\frac{86}{9}a^{12}+\frac{157}{9}a^{10}+\frac{71}{9}a^{8}-\frac{221}{9}a^{6}-\frac{62}{9}a^{4}+\frac{154}{9}a^{2}-\frac{13}{3}$, $\frac{5}{9}a^{14}-5a^{12}+\frac{31}{3}a^{10}+\frac{26}{9}a^{8}-\frac{158}{9}a^{6}-a^{4}+\frac{38}{3}a^{2}-\frac{26}{9}$, $a^{15}-\frac{82}{9}a^{13}+\frac{179}{9}a^{11}+\frac{8}{9}a^{9}-\frac{263}{9}a^{7}+\frac{35}{9}a^{5}+\frac{206}{9}a^{3}-\frac{91}{9}a$, $\frac{16}{9}a^{15}+\frac{2}{3}a^{14}-\frac{139}{9}a^{13}-\frac{53}{9}a^{12}+\frac{260}{9}a^{11}+\frac{103}{9}a^{10}+13a^{9}+\frac{46}{9}a^{8}-44a^{7}-\frac{175}{9}a^{6}-\frac{106}{9}a^{5}-\frac{44}{9}a^{4}+\frac{278}{9}a^{3}+\frac{124}{9}a^{2}-\frac{56}{9}a-\frac{17}{9}$, $\frac{4}{9}a^{15}+\frac{5}{9}a^{14}-4a^{13}-\frac{41}{9}a^{12}+\frac{25}{3}a^{11}+\frac{61}{9}a^{10}+\frac{16}{9}a^{9}+\frac{23}{3}a^{8}-\frac{121}{9}a^{7}-11a^{6}+a^{5}-\frac{71}{9}a^{4}+\frac{26}{3}a^{3}+\frac{61}{9}a^{2}-\frac{34}{9}a-\frac{1}{9}$, $\frac{7}{3}a^{15}-\frac{10}{9}a^{14}-\frac{61}{3}a^{13}+\frac{85}{9}a^{12}+\frac{115}{3}a^{11}-\frac{149}{9}a^{10}+17a^{9}-\frac{28}{3}a^{8}-\frac{178}{3}a^{7}+\frac{74}{3}a^{6}-\frac{44}{3}a^{5}+\frac{64}{9}a^{4}+\frac{119}{3}a^{3}-\frac{152}{9}a^{2}-8a+\frac{26}{9}$, $\frac{26}{9}a^{15}+\frac{10}{9}a^{14}-\frac{224}{9}a^{13}-\frac{85}{9}a^{12}+\frac{409}{9}a^{11}+\frac{149}{9}a^{10}+\frac{67}{3}a^{9}+\frac{28}{3}a^{8}-\frac{206}{3}a^{7}-\frac{74}{3}a^{6}-\frac{170}{9}a^{5}-\frac{64}{9}a^{4}+\frac{430}{9}a^{3}+\frac{152}{9}a^{2}-\frac{82}{9}a-\frac{26}{9}$, $\frac{1}{9}a^{15}-\frac{16}{9}a^{14}-\frac{4}{9}a^{13}+\frac{139}{9}a^{12}-\frac{22}{9}a^{11}-\frac{260}{9}a^{10}+7a^{9}-13a^{8}+\frac{14}{3}a^{7}+44a^{6}-\frac{97}{9}a^{5}+\frac{106}{9}a^{4}-\frac{52}{9}a^{3}-\frac{278}{9}a^{2}+\frac{52}{9}a+\frac{56}{9}$, $\frac{26}{9}a^{15}+\frac{16}{9}a^{14}-\frac{73}{3}a^{13}-\frac{133}{9}a^{12}+41a^{11}+\frac{212}{9}a^{10}+\frac{254}{9}a^{9}+\frac{61}{3}a^{8}-\frac{539}{9}a^{7}-\frac{106}{3}a^{6}-\frac{83}{3}a^{5}-\frac{187}{9}a^{4}+38a^{3}+\frac{188}{9}a^{2}-\frac{56}{9}a-\frac{17}{9}$, $a^{15}-\frac{10}{9}a^{14}-9a^{13}+\frac{86}{9}a^{12}+19a^{11}-\frac{157}{9}a^{10}+2a^{9}-\frac{71}{9}a^{8}-27a^{7}+\frac{221}{9}a^{6}+2a^{5}+\frac{62}{9}a^{4}+19a^{3}-\frac{154}{9}a^{2}-8a+\frac{13}{3}$
| 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: | \( 56100.6192446 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 56100.6192446 \cdot 1}{2\cdot\sqrt{700762351958879698944}}\cr\approx \mathstrut & 0.171345208100 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*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 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*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 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*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 - 9*x^14 + 19*x^12 + 2*x^10 - 27*x^8 + 2*x^6 + 19*x^4 - 9*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
$C_2^8.S_4$ (as 16T1664):
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 6144 |
| The 105 conjugacy class representatives for $C_2^8.S_4$ are not computed |
| Character table for $C_2^8.S_4$ is not computed |
Intermediate fields
| 4.4.1957.1, 8.8.26471916288.1, 8.6.980441344.2, 8.6.103405923.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.8651387061220737024.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 $16$ | $2$ | $8$ | $16$ | |||
|
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(103\)
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 103.4.2.1 | $x^{4} + 204 x^{3} + 10620 x^{2} + 22032 x + 1081216$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |