Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 1)
gp: K = bnfinit(y^16 - y^15 + y^14 + y^13 + 3*y^12 + 4*y^11 + 4*y^10 + 8*y^9 + 8*y^8 + 9*y^7 + 6*y^6 + 8*y^5 + 8*y^4 + 3*y^3 + 4*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 1)
\( x^{16} - x^{15} + x^{14} + x^{13} + 3 x^{12} + 4 x^{11} + 4 x^{10} + 8 x^{9} + 8 x^{8} + 9 x^{7} + \cdots + 1 \)
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: |
\(6472705322265625\)
\(\medspace = 5^{12}\cdot 19^{2}\cdot 271^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.73\) | 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: | $5^{3/4}19^{1/2}271^{1/2}\approx 239.93242815578742$ | ||
| Ramified primes: |
\(5\), \(19\), \(271\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{9}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{5}-\frac{2}{5}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{6}-\frac{2}{5}a$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{7}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{5}+\frac{1}{5}$
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: |
\( -\frac{3}{5} a^{15} + a^{14} - a^{13} - \frac{2}{5} a^{12} - \frac{7}{5} a^{11} - \frac{6}{5} a^{10} - a^{9} - 4 a^{8} - \frac{16}{5} a^{7} - \frac{16}{5} a^{6} - 2 a^{5} - 4 a^{4} - 4 a^{3} - \frac{1}{5} a^{2} - \frac{11}{5} a - \frac{6}{5} \)
(order $10$)
| 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: |
$\frac{1}{5}a^{15}-\frac{2}{5}a^{14}+\frac{3}{5}a^{13}-\frac{2}{5}a^{12}+a^{11}-\frac{1}{5}a^{10}+\frac{4}{5}a^{9}+\frac{4}{5}a^{8}+\frac{4}{5}a^{7}+a^{6}-\frac{4}{5}a^{5}+\frac{4}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{4}{5}a^{10}-\frac{3}{5}a^{9}-\frac{1}{5}a^{8}-\frac{4}{5}a^{7}+\frac{1}{5}a^{6}-a^{5}-\frac{7}{5}a^{4}-\frac{4}{5}a^{3}-\frac{3}{5}a^{2}-\frac{3}{5}a-\frac{6}{5}$, $\frac{1}{5}a^{15}-\frac{3}{5}a^{14}+\frac{3}{5}a^{13}-a^{9}+a^{8}-a^{7}+\frac{2}{5}a^{6}-\frac{7}{5}a^{5}+a^{4}+\frac{4}{5}a^{3}-\frac{7}{5}a^{2}+\frac{6}{5}a-\frac{1}{5}$, $\frac{2}{5}a^{14}-\frac{4}{5}a^{13}+\frac{4}{5}a^{12}+\frac{4}{5}a^{10}+\frac{2}{5}a^{9}+\frac{8}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{6}{5}a^{4}+\frac{7}{5}a^{3}+\frac{1}{5}a^{2}-\frac{4}{5}a+\frac{2}{5}$, $\frac{2}{5}a^{15}-\frac{3}{5}a^{14}+\frac{3}{5}a^{13}+\frac{1}{5}a^{12}+a^{11}+a^{10}+a^{9}+\frac{11}{5}a^{8}+2a^{7}+\frac{8}{5}a^{6}+\frac{8}{5}a^{5}+\frac{14}{5}a^{4}+\frac{12}{5}a^{3}+a^{2}+a+\frac{6}{5}$, $\frac{4}{5}a^{15}-a^{14}+\frac{3}{5}a^{13}+\frac{4}{5}a^{12}+2a^{11}+\frac{11}{5}a^{10}+a^{9}+\frac{17}{5}a^{8}+\frac{13}{5}a^{7}+\frac{13}{5}a^{6}-\frac{3}{5}a^{5}+\frac{7}{5}a^{4}+\frac{13}{5}a^{3}-\frac{6}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{15}+\frac{1}{5}a^{14}-\frac{4}{5}a^{13}+a^{12}+\frac{3}{5}a^{11}+\frac{8}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{8}+\frac{9}{5}a^{7}+a^{6}+\frac{3}{5}a^{5}-\frac{6}{5}a^{4}+\frac{9}{5}a^{3}+\frac{4}{5}a^{2}-\frac{6}{5}a+\frac{3}{5}$
| 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: | \( 56.3298547938 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 56.3298547938 \cdot 1}{10\cdot\sqrt{6472705322265625}}\cr\approx \mathstrut & 0.170072721267 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 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 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 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 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 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 - x^15 + x^14 + x^13 + 3*x^12 + 4*x^11 + 4*x^10 + 8*x^9 + 8*x^8 + 9*x^7 + 6*x^6 + 8*x^5 + 8*x^4 + 3*x^3 + 4*x^2 + 3*x + 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
$S_4^2:C_4$ (as 16T1497):
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 2304 |
| The 40 conjugacy class representatives for $S_4^2:C_4$ |
| Character table for $S_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.4.80453125.1 |
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 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.2071265703125.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | R | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(271\)
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{271}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |