Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^2 - x + 1)
gp: K = bnfinit(y^16 - y^15 - 2*y^14 + 6*y^13 + y^12 - 16*y^11 + 16*y^10 + 5*y^9 - 19*y^8 + 5*y^7 + 16*y^6 - 16*y^5 + y^4 + 6*y^3 - 2*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^2 - x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^2 - x + 1)
\( x^{16} - x^{15} - 2 x^{14} + 6 x^{13} + x^{12} - 16 x^{11} + 16 x^{10} + 5 x^{9} - 19 x^{8} + 5 x^{7} + \cdots + 1 \)
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: |
\(5850424087444225\)
\(\medspace = 5^{2}\cdot 37^{2}\cdot 643^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.67\) | 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: | $5^{1/2}37^{1/2}643^{1/2}\approx 344.8985358043725$ | ||
| Ramified primes: |
\(5\), \(37\), \(643\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{635}a^{14}+\frac{77}{635}a^{13}+\frac{288}{635}a^{12}+\frac{168}{635}a^{11}+\frac{117}{635}a^{10}+\frac{52}{635}a^{9}+\frac{29}{127}a^{8}-\frac{167}{635}a^{7}+\frac{29}{127}a^{6}+\frac{52}{635}a^{5}+\frac{117}{635}a^{4}+\frac{168}{635}a^{3}+\frac{288}{635}a^{2}+\frac{77}{635}a+\frac{1}{635}$, $\frac{1}{635}a^{15}+\frac{74}{635}a^{13}+\frac{217}{635}a^{12}-\frac{119}{635}a^{11}-\frac{67}{635}a^{10}-\frac{49}{635}a^{9}+\frac{98}{635}a^{8}+\frac{304}{635}a^{7}+\frac{317}{635}a^{6}-\frac{77}{635}a^{5}+\frac{49}{635}a^{4}+\frac{52}{635}a^{3}+\frac{126}{635}a^{2}-\frac{213}{635}a-\frac{77}{635}$
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: |
\( -1 \)
(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: |
$\frac{1}{635}a^{15}-\frac{2}{635}a^{14}-\frac{16}{127}a^{13}+\frac{276}{635}a^{12}+\frac{36}{127}a^{11}-\frac{936}{635}a^{10}+\frac{482}{635}a^{9}+\frac{2348}{635}a^{8}-\frac{3172}{635}a^{7}-\frac{608}{635}a^{6}+\frac{2994}{635}a^{5}-\frac{164}{127}a^{4}-\frac{2824}{635}a^{3}+\frac{418}{127}a^{2}+\frac{268}{635}a-\frac{714}{635}$, $\frac{344}{635}a^{15}+\frac{257}{635}a^{14}-\frac{222}{127}a^{13}+\frac{709}{635}a^{12}+\frac{702}{127}a^{11}-\frac{3774}{635}a^{10}-\frac{3492}{635}a^{9}+\frac{8112}{635}a^{8}-\frac{1208}{635}a^{7}-\frac{6722}{635}a^{6}+\frac{4656}{635}a^{5}+\frac{622}{127}a^{4}-\frac{4341}{635}a^{3}+\frac{104}{127}a^{2}+\frac{1127}{635}a-\frac{196}{635}$, $a$, $\frac{193}{635}a^{15}-\frac{30}{127}a^{14}-\frac{443}{635}a^{13}+\frac{1221}{635}a^{12}+\frac{728}{635}a^{11}-\frac{3176}{635}a^{10}+\frac{2428}{635}a^{9}+\frac{2879}{635}a^{8}-\frac{3273}{635}a^{7}-\frac{574}{635}a^{6}+\frac{3374}{635}a^{5}-\frac{1743}{635}a^{4}-\frac{559}{635}a^{3}+\frac{803}{635}a^{2}+\frac{46}{635}a+\frac{229}{635}$, $\frac{398}{635}a^{15}-\frac{518}{635}a^{14}-\frac{909}{635}a^{13}+\frac{2587}{635}a^{12}+\frac{234}{635}a^{11}-\frac{7262}{635}a^{10}+\frac{6902}{635}a^{9}+\frac{2629}{635}a^{8}-\frac{8402}{635}a^{7}+\frac{1526}{635}a^{6}+\frac{7823}{635}a^{5}-\frac{6814}{635}a^{4}-\frac{288}{635}a^{3}+\frac{2564}{635}a^{2}+\frac{87}{127}a-\frac{684}{635}$, $\frac{361}{635}a^{15}-\frac{169}{635}a^{14}-\frac{904}{635}a^{13}+\frac{345}{127}a^{12}+\frac{1674}{635}a^{11}-\frac{1045}{127}a^{10}+\frac{2098}{635}a^{9}+\frac{4523}{635}a^{8}-\frac{3638}{635}a^{7}-\frac{2778}{635}a^{6}+\frac{811}{127}a^{5}-\frac{814}{635}a^{4}-\frac{273}{127}a^{3}-\frac{11}{635}a^{2}+\frac{899}{635}a-\frac{26}{635}$, $\frac{1046}{635}a^{15}-\frac{699}{635}a^{14}-\frac{2454}{635}a^{13}+\frac{1070}{127}a^{12}+\frac{3204}{635}a^{11}-\frac{3195}{127}a^{10}+\frac{10188}{635}a^{9}+\frac{10043}{635}a^{8}-\frac{15498}{635}a^{7}-\frac{2818}{635}a^{6}+\frac{3419}{127}a^{5}-\frac{8939}{635}a^{4}-\frac{797}{127}a^{3}+\frac{4144}{635}a^{2}+\frac{874}{635}a-\frac{1231}{635}$
| 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: | \( 10.6388599051 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10.6388599051 \cdot 1}{2\cdot\sqrt{5850424087444225}}\cr\approx \mathstrut & 0.168931394456 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^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 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^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 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^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 - x^15 - 2*x^14 + 6*x^13 + x^12 - 16*x^11 + 16*x^10 + 5*x^9 - 19*x^8 + 5*x^7 + 16*x^6 - 16*x^5 + x^4 + 6*x^3 - 2*x^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
$C_2^8.S_4$ (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 $C_2^8.S_4$ |
| Character table for $C_2^8.S_4$ |
Intermediate fields
| 4.2.643.1, 8.0.2067245.1, 8.0.76488065.1, 8.4.15297613.1 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(643\)
| $\Q_{643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{643}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |