Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + y^14 - y^13 + 4*y^12 - 4*y^11 + 5*y^10 - 8*y^9 + 6*y^8 - 5*y^7 + 8*y^6 - 7*y^5 + 7*y^4 - 7*y^3 + 4*y^2 - 2*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*x + 1)
\( x^{16} - 2 x^{15} + x^{14} - x^{13} + 4 x^{12} - 4 x^{11} + 5 x^{10} - 8 x^{9} + 6 x^{8} - 5 x^{7} + \cdots + 1 \)
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: |
\(3914091521024229\)
\(\medspace = 3^{8}\cdot 61\cdot 98893^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.43\) | 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: | $3^{1/2}61^{1/2}98893^{1/2}\approx 4254.1061340779925$ | ||
| Ramified primes: |
\(3\), \(61\), \(98893\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{61}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
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
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: |
\( 2 a^{15} - 6 a^{14} + 3 a^{13} + 2 a^{12} + 8 a^{11} - 16 a^{10} + 8 a^{9} - 13 a^{8} + 16 a^{7} - 6 a^{6} + 13 a^{5} - 20 a^{4} + 12 a^{3} - 10 a^{2} + 6 a \)
(order $6$)
| 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: |
$2a^{15}-a^{14}-4a^{13}+8a^{11}+2a^{10}-3a^{9}-4a^{8}-4a^{7}+3a^{6}+6a^{5}+2a^{4}-4a^{3}+2a^{2}-3a+2$, $3a^{15}-4a^{14}-2a^{13}+a^{12}+11a^{11}-6a^{10}+2a^{9}-10a^{8}+5a^{7}+11a^{5}-9a^{4}+3a^{3}-3a^{2}+a+2$, $a^{15}-a^{14}-a^{13}-a^{12}+4a^{11}+a^{10}+a^{9}-7a^{8}-a^{7}+a^{6}+6a^{5}-a^{4}-3a^{2}-a+1$, $a^{14}-2a^{13}+a^{12}+a^{11}+2a^{10}-6a^{9}+5a^{8}-a^{7}+4a^{6}-4a^{5}+3a^{4}-6a^{3}+5a^{2}-a+1$, $a^{15}+a^{14}-4a^{13}-a^{12}+4a^{11}+7a^{10}-4a^{9}-a^{8}-8a^{7}+4a^{6}+2a^{5}+8a^{4}-5a^{3}+3a^{2}-4a+1$, $2a^{15}-3a^{14}-a^{13}+9a^{11}-5a^{10}+a^{9}-10a^{8}+7a^{7}-a^{6}+10a^{5}-9a^{4}+3a^{3}-5a^{2}+3a$, $5a^{15}-9a^{14}+a^{13}+a^{12}+18a^{11}-19a^{10}+13a^{9}-22a^{8}+19a^{7}-11a^{6}+23a^{5}-25a^{4}+19a^{3}-13a^{2}+7a-1$
| 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: | \( 25.217952399 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 25.217952399 \cdot 1}{6\cdot\sqrt{3914091521024229}}\cr\approx \mathstrut & 0.16318567377 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*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 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*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 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*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 - 2*x^15 + x^14 - x^13 + 4*x^12 - 4*x^11 + 5*x^10 - 8*x^9 + 6*x^8 - 5*x^7 + 8*x^6 - 7*x^5 + 7*x^4 - 7*x^3 + 4*x^2 - 2*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
$C_2^6.S_4^2:D_4$ (as 16T1905):
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 294912 |
| The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$ |
| Character table for $C_2^6.S_4^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 8.0.8010333.1 |
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: | 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 | $16$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(61\)
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.6.0.1 | $x^{6} + 49 x^{3} + 3 x^{2} + 29 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(98893\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |