Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*x + 1)
gp: K = bnfinit(y^16 - 3*y^15 + 3*y^14 - 2*y^13 + 2*y^12 + 5*y^11 - 19*y^10 + 28*y^9 - 27*y^8 + 12*y^7 + 19*y^6 - 58*y^5 + 78*y^4 - 62*y^3 + 32*y^2 - 9*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*x + 1)
\( x^{16} - 3 x^{15} + 3 x^{14} - 2 x^{13} + 2 x^{12} + 5 x^{11} - 19 x^{10} + 28 x^{9} - 27 x^{8} + \cdots + 1 \)
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: |
\(22028181368750000\)
\(\medspace = 2^{4}\cdot 5^{8}\cdot 59^{3}\cdot 131^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.51\) | 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: | $2^{3/2}5^{1/2}59^{3/4}131^{1/2}\approx 1541.007144503574$ | ||
| Ramified primes: |
\(2\), \(5\), \(59\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{59}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9852211}a^{15}+\frac{3570639}{9852211}a^{14}+\frac{3187049}{9852211}a^{13}+\frac{4847695}{9852211}a^{12}+\frac{4307659}{9852211}a^{11}+\frac{3821470}{9852211}a^{10}-\frac{4054848}{9852211}a^{9}+\frac{4624772}{9852211}a^{8}-\frac{3940035}{9852211}a^{7}+\frac{392781}{9852211}a^{6}-\frac{1604851}{9852211}a^{5}+\frac{2951741}{9852211}a^{4}+\frac{774119}{9852211}a^{3}+\frac{4905020}{9852211}a^{2}+\frac{1824603}{9852211}a+\frac{2980514}{9852211}$
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: |
$\frac{13510639}{9852211}a^{15}-\frac{32199509}{9852211}a^{14}+\frac{19747655}{9852211}a^{13}-\frac{12045429}{9852211}a^{12}+\frac{18264048}{9852211}a^{11}+\frac{78543729}{9852211}a^{10}-\frac{210626996}{9852211}a^{9}+\frac{242689171}{9852211}a^{8}-\frac{194992384}{9852211}a^{7}+\frac{25886129}{9852211}a^{6}+\frac{281981121}{9852211}a^{5}-\frac{611529126}{9852211}a^{4}+\frac{651410064}{9852211}a^{3}-\frac{381207693}{9852211}a^{2}+\frac{148116208}{9852211}a-\frac{17469946}{9852211}$, $\frac{40844818}{9852211}a^{15}-\frac{110368821}{9852211}a^{14}+\frac{88673538}{9852211}a^{13}-\frac{53633074}{9852211}a^{12}+\frac{64318415}{9852211}a^{11}+\frac{224418798}{9852211}a^{10}-\frac{708960144}{9852211}a^{9}+\frac{928143625}{9852211}a^{8}-\frac{815750477}{9852211}a^{7}+\frac{230529219}{9852211}a^{6}+\frac{853655282}{9852211}a^{5}-\frac{2113111051}{9852211}a^{4}+\frac{2541512425}{9852211}a^{3}-\frac{1736374510}{9852211}a^{2}+\frac{753947440}{9852211}a-\frac{125857461}{9852211}$, $\frac{19002470}{9852211}a^{15}-\frac{54117772}{9852211}a^{14}+\frac{45245388}{9852211}a^{13}-\frac{23599983}{9852211}a^{12}+\frac{31340908}{9852211}a^{11}+\frac{101058273}{9852211}a^{10}-\frac{349774200}{9852211}a^{9}+\frac{457736739}{9852211}a^{8}-\frac{391312040}{9852211}a^{7}+\frac{118942855}{9852211}a^{6}+\frac{410799641}{9852211}a^{5}-\frac{1035894021}{9852211}a^{4}+\frac{1245696003}{9852211}a^{3}-\frac{844302172}{9852211}a^{2}+\frac{348532329}{9852211}a-\frac{60757478}{9852211}$, $\frac{22759594}{9852211}a^{15}-\frac{62561374}{9852211}a^{14}+\frac{52588339}{9852211}a^{13}-\frac{30610274}{9852211}a^{12}+\frac{36004126}{9852211}a^{11}+\frac{121950354}{9852211}a^{10}-\frac{403233208}{9852211}a^{9}+\frac{536352216}{9852211}a^{8}-\frac{467004082}{9852211}a^{7}+\frac{142292275}{9852211}a^{6}+\frac{477654071}{9852211}a^{5}-\frac{1208923186}{9852211}a^{4}+\frac{1462161302}{9852211}a^{3}-\frac{1010999210}{9852211}a^{2}+\frac{429988090}{9852211}a-\frac{70881507}{9852211}$, $\frac{16013723}{9852211}a^{15}-\frac{43249492}{9852211}a^{14}+\frac{37337961}{9852211}a^{13}-\frac{24070291}{9852211}a^{12}+\frac{23378417}{9852211}a^{11}+\frac{86410575}{9852211}a^{10}-\frac{275198037}{9852211}a^{9}+\frac{383127349}{9852211}a^{8}-\frac{340792691}{9852211}a^{7}+\frac{96700309}{9852211}a^{6}+\frac{325929355}{9852211}a^{5}-\frac{841203754}{9852211}a^{4}+\frac{1046783075}{9852211}a^{3}-\frac{736370644}{9852211}a^{2}+\frac{319839443}{9852211}a-\frac{48526832}{9852211}$, $\frac{3702721}{9852211}a^{15}-\frac{15725043}{9852211}a^{14}+\frac{21229804}{9852211}a^{13}-\frac{12177328}{9852211}a^{12}+\frac{9633698}{9852211}a^{11}+\frac{12963982}{9852211}a^{10}-\frac{97828609}{9852211}a^{9}+\frac{171578778}{9852211}a^{8}-\frac{169192563}{9852211}a^{7}+\frac{93295813}{9852211}a^{6}+\frac{75121712}{9852211}a^{5}-\frac{313338653}{9852211}a^{4}+\frac{476428853}{9852211}a^{3}-\frac{413670438}{9852211}a^{2}+\frac{206683320}{9852211}a-\frac{50747377}{9852211}$, $\frac{28588219}{9852211}a^{15}-\frac{75562931}{9852211}a^{14}+\frac{58678528}{9852211}a^{13}-\frac{36573074}{9852211}a^{12}+\frac{45603216}{9852211}a^{11}+\frac{158900304}{9852211}a^{10}-\frac{487221115}{9852211}a^{9}+\frac{623631753}{9852211}a^{8}-\frac{550649828}{9852211}a^{7}+\frac{157178330}{9852211}a^{6}+\frac{594580357}{9852211}a^{5}-\frac{1444936462}{9852211}a^{4}+\frac{1711194649}{9852211}a^{3}-\frac{1166700904}{9852211}a^{2}+\frac{515563758}{9852211}a-\frac{87454658}{9852211}$
| 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: | \( 26.6804881826 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 26.6804881826 \cdot 1}{2\cdot\sqrt{22028181368750000}}\cr\approx \mathstrut & 0.218329993163 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*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 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*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 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*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 - 3*x^15 + 3*x^14 - 2*x^13 + 2*x^12 + 5*x^11 - 19*x^10 + 28*x^9 - 27*x^8 + 12*x^7 + 19*x^6 - 58*x^5 + 78*x^4 - 62*x^3 + 32*x^2 - 9*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
| \(\Q(\sqrt{5}) \), 8.0.4830625.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $16$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.12.0.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 59.2.0.1 | $x^{2} + 58 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 59.4.3.2 | $x^{4} + 118$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 59.4.0.1 | $x^{4} + 2 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 131.2.0.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.2.1.1 | $x^{2} + 262$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.6.0.1 | $x^{6} + 2 x^{4} + 66 x^{3} + 4 x^{2} + 22 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |