Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1)
gp: K = bnfinit(y^16 - 4*y^15 + 6*y^14 - 14*y^12 + 24*y^11 - 18*y^10 + 15*y^8 - 20*y^7 + 22*y^6 - 24*y^5 + 24*y^4 - 20*y^3 + 12*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*x + 1)
\( x^{16} - 4 x^{15} + 6 x^{14} - 14 x^{12} + 24 x^{11} - 18 x^{10} + 15 x^{8} - 20 x^{7} + 22 x^{6} + \cdots + 1 \)
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: |
\(8084777718513664\)
\(\medspace = 2^{36}\cdot 7^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.87\) | 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: | $2^{9/4}7^{3/4}\approx 20.471092479852693$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5549}a^{15}+\frac{2615}{5549}a^{14}+\frac{1225}{5549}a^{13}+\frac{953}{5549}a^{12}-\frac{1157}{5549}a^{11}-\frac{405}{5549}a^{10}-\frac{854}{5549}a^{9}-\frac{379}{5549}a^{8}+\frac{685}{5549}a^{7}+\frac{1668}{5549}a^{6}+\frac{1451}{5549}a^{5}-\frac{920}{5549}a^{4}-\frac{1190}{5549}a^{3}+\frac{1908}{5549}a^{2}-\frac{2585}{5549}a-\frac{339}{5549}$
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: |
\( -\frac{6779}{5549} a^{15} + \frac{24166}{5549} a^{14} - \frac{30716}{5549} a^{13} - \frac{12449}{5549} a^{12} + \frac{91350}{5549} a^{11} - \frac{128887}{5549} a^{10} + \frac{68247}{5549} a^{9} + \frac{38897}{5549} a^{8} - \frac{98984}{5549} a^{7} + \frac{95823}{5549} a^{6} - \frac{97834}{5549} a^{5} + \frac{110584}{5549} a^{4} - \frac{112216}{5549} a^{3} + \frac{83622}{5549} a^{2} - \frac{38870}{5549} a + \frac{6344}{5549} \)
(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: |
$\frac{2266}{5549}a^{15}-\frac{6291}{5549}a^{14}+\frac{6899}{5549}a^{13}+\frac{6486}{5549}a^{12}-\frac{24830}{5549}a^{11}+\frac{31149}{5549}a^{10}-\frac{9661}{5549}a^{9}-\frac{15366}{5549}a^{8}+\frac{26235}{5549}a^{7}-\frac{21377}{5549}a^{6}+\frac{25154}{5549}a^{5}-\frac{26041}{5549}a^{4}+\frac{22470}{5549}a^{3}-\frac{10241}{5549}a^{2}+\frac{7683}{5549}a+\frac{3137}{5549}$, $\frac{2568}{5549}a^{15}-\frac{10068}{5549}a^{14}+\frac{10615}{5549}a^{13}+\frac{11293}{5549}a^{12}-\frac{41304}{5549}a^{11}+\frac{42015}{5549}a^{10}-\frac{6766}{5549}a^{9}-\frac{29942}{5549}a^{8}+\frac{38890}{5549}a^{7}-\frac{28149}{5549}a^{6}+\frac{30534}{5549}a^{5}-\frac{31980}{5549}a^{4}+\frac{23775}{5549}a^{3}-\frac{16670}{5549}a^{2}+\frac{3873}{5549}a+\frac{6190}{5549}$, $\frac{5171}{5549}a^{15}-\frac{17395}{5549}a^{14}+\frac{19713}{5549}a^{13}+\frac{11549}{5549}a^{12}-\frac{62064}{5549}a^{11}+\frac{80953}{5549}a^{10}-\frac{43422}{5549}a^{9}-\frac{17659}{5549}a^{8}+\frac{51814}{5549}a^{7}-\frac{64506}{5549}a^{6}+\frac{78559}{5549}a^{5}-\frac{85062}{5549}a^{4}+\frac{78037}{5549}a^{3}-\frac{60893}{5549}a^{2}+\frac{28251}{5549}a-\frac{10583}{5549}$, $\frac{1528}{5549}a^{15}-\frac{5109}{5549}a^{14}+\frac{1787}{5549}a^{13}+\frac{13444}{5549}a^{12}-\frac{25510}{5549}a^{11}+\frac{8197}{5549}a^{10}+\frac{26848}{5549}a^{9}-\frac{40859}{5549}a^{8}+\frac{14566}{5549}a^{7}+\frac{12811}{5549}a^{6}-\frac{8021}{5549}a^{5}-\frac{1863}{5549}a^{4}-\frac{3797}{5549}a^{3}+\frac{13297}{5549}a^{2}-\frac{21188}{5549}a+\frac{9163}{5549}$, $\frac{2721}{5549}a^{15}-\frac{3952}{5549}a^{14}-\frac{1724}{5549}a^{13}+\frac{12828}{5549}a^{12}-\frac{13012}{5549}a^{11}+\frac{2246}{5549}a^{10}+\frac{12395}{5549}a^{9}-\frac{15792}{5549}a^{8}+\frac{10519}{5549}a^{7}-\frac{6003}{5549}a^{6}+\frac{2832}{5549}a^{5}+\frac{4828}{5549}a^{4}+\frac{2626}{5549}a^{3}+\frac{3353}{5549}a^{2}-\frac{8751}{5549}a-\frac{1285}{5549}$, $\frac{754}{5549}a^{15}-\frac{3734}{5549}a^{14}+\frac{2516}{5549}a^{13}+\frac{8290}{5549}a^{12}-\frac{17832}{5549}a^{11}+\frac{5374}{5549}a^{10}+\frac{16415}{5549}a^{9}-\frac{19414}{5549}a^{8}+\frac{433}{5549}a^{7}+\frac{9147}{5549}a^{6}+\frac{6450}{5549}a^{5}-\frac{16702}{5549}a^{4}+\frac{1678}{5549}a^{3}+\frac{6990}{5549}a^{2}-\frac{6940}{5549}a+\frac{5197}{5549}$, $\frac{2868}{5549}a^{15}-\frac{7977}{5549}a^{14}+\frac{6332}{5549}a^{13}+\frac{14194}{5549}a^{12}-\frac{33268}{5549}a^{11}+\frac{25946}{5549}a^{10}+\frac{8935}{5549}a^{9}-\frac{27113}{5549}a^{8}+\frac{22430}{5549}a^{7}-\frac{10512}{5549}a^{6}+\frac{21914}{5549}a^{5}-\frac{24981}{5549}a^{4}+\frac{16362}{5549}a^{3}-\frac{4719}{5549}a^{2}+\frac{5233}{5549}a+\frac{4372}{5549}$
| 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: | \( 60.0447869363 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 60.0447869363 \cdot 1}{8\cdot\sqrt{8084777718513664}}\cr\approx \mathstrut & 0.202763708602 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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 - 4*x^15 + 6*x^14 - 14*x^12 + 24*x^11 - 18*x^10 + 15*x^8 - 20*x^7 + 22*x^6 - 24*x^5 + 24*x^4 - 20*x^3 + 12*x^2 - 4*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
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 64 |
| The 16 conjugacy class representatives for $D_4:D_4$ |
| Character table for $D_4:D_4$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.89915392.1 x2, 8.2.22478848.1 x2, 8.0.3211264.1 |
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 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.22478848.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.16.36.1 | $x^{16} + 8 x^{12} + 8 x^{11} + 24 x^{10} + 44 x^{8} - 16 x^{7} + 16 x^{6} - 56 x^{4} + 112 x^{3} + 196$ | $8$ | $2$ | $36$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.8.6.2 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2660 x^{4} + 3408 x^{3} + 3312 x^{2} + 5184 x + 6304$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |