Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*x^2 + 4*x + 1)
gp: K = bnfinit(y^16 - 4*y^13 + 7*y^12 - 8*y^11 + 8*y^10 - 20*y^9 + 49*y^8 - 36*y^7 - 28*y^6 + 44*y^5 + 5*y^4 - 20*y^3 - 2*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*x^2 + 4*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*x^2 + 4*x + 1)
\( x^{16} - 4 x^{13} + 7 x^{12} - 8 x^{11} + 8 x^{10} - 20 x^{9} + 49 x^{8} - 36 x^{7} - 28 x^{6} + \cdots + 1 \)
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: |
\(513114342400000000\)
\(\medspace = 2^{16}\cdot 5^{8}\cdot 11^{4}\cdot 37^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.79\) | 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: | $2^{15/8}5^{1/2}11^{1/2}37^{1/2}\approx 165.46778321069314$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(37\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{4}$, $\frac{1}{51822}a^{15}-\frac{10837}{51822}a^{14}+\frac{11917}{51822}a^{13}-\frac{4109}{51822}a^{12}-\frac{3923}{17274}a^{11}+\frac{6703}{51822}a^{10}-\frac{5935}{25911}a^{9}-\frac{4315}{17274}a^{8}+\frac{1430}{25911}a^{7}+\frac{21611}{51822}a^{6}-\frac{4939}{17274}a^{5}+\frac{703}{25911}a^{4}-\frac{383}{17274}a^{3}-\frac{5749}{25911}a^{2}-\frac{725}{17274}a+\frac{8690}{25911}$
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: |
$\frac{258353}{51822}a^{15}-\frac{87911}{51822}a^{14}-\frac{4141}{51822}a^{13}-\frac{1035247}{51822}a^{12}+\frac{357727}{8637}a^{11}-\frac{1332224}{25911}a^{10}+\frac{2741039}{51822}a^{9}-\frac{1933019}{17274}a^{8}+\frac{14288465}{51822}a^{7}-\frac{13405673}{51822}a^{6}-\frac{734644}{8637}a^{5}+\frac{7111574}{25911}a^{4}-\frac{524402}{8637}a^{3}-\frac{2514422}{25911}a^{2}+\frac{393359}{17274}a+\frac{861191}{51822}$, $\frac{10177}{25911}a^{15}-\frac{10933}{25911}a^{14}+\frac{15829}{25911}a^{13}-\frac{48761}{25911}a^{12}+\frac{39028}{8637}a^{11}-\frac{454951}{51822}a^{10}+\frac{307024}{25911}a^{9}-\frac{326063}{17274}a^{8}+\frac{940963}{25911}a^{7}-\frac{2663675}{51822}a^{6}+\frac{365891}{8637}a^{5}-\frac{583973}{51822}a^{4}-\frac{123422}{8637}a^{3}+\frac{697457}{51822}a^{2}-\frac{10964}{8637}a-\frac{69959}{25911}$, $\frac{87911}{51822}a^{15}+\frac{4141}{51822}a^{14}+\frac{1835}{51822}a^{13}-\frac{337891}{51822}a^{12}+\frac{99604}{8637}a^{11}-\frac{674215}{51822}a^{10}+\frac{631997}{51822}a^{9}-\frac{271528}{8637}a^{8}+\frac{4104965}{51822}a^{7}-\frac{1413010}{25911}a^{6}-\frac{475936}{8637}a^{5}+\frac{4438177}{51822}a^{4}-\frac{23036}{8637}a^{3}-\frac{1696783}{51822}a^{2}+\frac{57407}{17274}a+\frac{258353}{51822}$, $\frac{87911}{51822}a^{15}+\frac{4141}{51822}a^{14}+\frac{1835}{51822}a^{13}-\frac{337891}{51822}a^{12}+\frac{99604}{8637}a^{11}-\frac{674215}{51822}a^{10}+\frac{631997}{51822}a^{9}-\frac{271528}{8637}a^{8}+\frac{4104965}{51822}a^{7}-\frac{1413010}{25911}a^{6}-\frac{475936}{8637}a^{5}+\frac{4438177}{51822}a^{4}-\frac{23036}{8637}a^{3}-\frac{1696783}{51822}a^{2}+\frac{57407}{17274}a+\frac{206531}{51822}$, $\frac{43679}{17274}a^{15}-\frac{24449}{17274}a^{14}+\frac{22475}{17274}a^{13}-\frac{90764}{8637}a^{12}+\frac{137797}{5758}a^{11}-\frac{304958}{8637}a^{10}+\frac{367444}{8637}a^{9}-\frac{222258}{2879}a^{8}+\frac{1466476}{8637}a^{7}-\frac{1683964}{8637}a^{6}+\frac{332611}{5758}a^{5}+\frac{1238695}{17274}a^{4}-\frac{243903}{5758}a^{3}-\frac{119351}{8637}a^{2}+\frac{47789}{5758}a+\frac{17545}{8637}$, $\frac{818297}{51822}a^{15}-\frac{220396}{25911}a^{14}+\frac{195965}{51822}a^{13}-\frac{3383977}{51822}a^{12}+\frac{2508485}{17274}a^{11}-\frac{5222555}{25911}a^{10}+\frac{5943298}{25911}a^{9}-\frac{3730747}{8637}a^{8}+\frac{25857217}{25911}a^{7}-\frac{28172818}{25911}a^{6}+\frac{1773571}{17274}a^{5}+\frac{17450783}{25911}a^{4}-\frac{4825015}{17274}a^{3}-\frac{9537745}{51822}a^{2}+\frac{1184837}{17274}a+\frac{789331}{25911}$, $\frac{63263}{51822}a^{15}-\frac{27893}{51822}a^{14}+\frac{38224}{25911}a^{13}-\frac{267625}{51822}a^{12}+\frac{96730}{8637}a^{11}-\frac{509033}{25911}a^{10}+\frac{658171}{25911}a^{9}-\frac{776153}{17274}a^{8}+\frac{2290957}{25911}a^{7}-\frac{5588087}{51822}a^{6}+\frac{1188061}{17274}a^{5}-\frac{274408}{25911}a^{4}-\frac{165575}{8637}a^{3}+\frac{480118}{25911}a^{2}-\frac{14558}{8637}a-\frac{229633}{51822}$
| 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: | \( 136.008808013 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 136.008808013 \cdot 1}{2\cdot\sqrt{513114342400000000}}\cr\approx \mathstrut & 0.230605100432 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*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^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*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^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*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^13 + 7*x^12 - 8*x^11 + 8*x^10 - 20*x^9 + 49*x^8 - 36*x^7 - 28*x^6 + 44*x^5 + 5*x^4 - 20*x^3 - 2*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
$C_2^7.C_2\wr D_4$ (as 16T1772):
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 16384 |
| The 148 conjugacy class representatives for $C_2^7.C_2\wr D_4$ |
| Character table for $C_2^7.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.19360000.1 |
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 | R | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.8 | $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.8 | $x^{8} - 6 x^{7} + 40 x^{6} - 52 x^{5} + 192 x^{4} + 328 x^{3} + 1376 x^{2} + 1264 x + 1328$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(11\)
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(37\)
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 37.4.2.2 | $x^{4} - 1221 x^{2} + 2738$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} + 6 x^{2} + 24 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |