Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*x^2 - x - 1)
gp: K = bnfinit(y^16 - 3*y^15 + 5*y^14 - 4*y^13 - y^12 + 6*y^11 - 6*y^10 - 8*y^9 + 20*y^8 - 19*y^7 + 11*y^6 + 4*y^5 - 14*y^4 + 4*y^3 + 5*y^2 - y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*x^2 - x - 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*x^2 - x - 1)
\( x^{16} - 3 x^{15} + 5 x^{14} - 4 x^{13} - x^{12} + 6 x^{11} - 6 x^{10} - 8 x^{9} + 20 x^{8} - 19 x^{7} + \cdots - 1 \)
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: | $[2, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-19666546293359375\)
\(\medspace = -\,5^{8}\cdot 31^{2}\cdot 71\cdot 859^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.43\) | 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: | $5^{1/2}31^{1/2}71^{1/2}859^{1/2}\approx 3074.621114869278$ | ||
| Ramified primes: |
\(5\), \(31\), \(71\), \(859\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5173031}a^{15}-\frac{1243229}{5173031}a^{14}+\frac{895486}{5173031}a^{13}+\frac{1696701}{5173031}a^{12}-\frac{1811712}{5173031}a^{11}-\frac{1272668}{5173031}a^{10}+\frac{1031364}{5173031}a^{9}-\frac{38426}{5173031}a^{8}-\frac{738989}{5173031}a^{7}+\frac{32895}{5173031}a^{6}+\frac{2063827}{5173031}a^{5}-\frac{702022}{5173031}a^{4}-\frac{1095238}{5173031}a^{3}-\frac{1342935}{5173031}a^{2}+\frac{1818220}{5173031}a+\frac{978349}{5173031}$
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: | $8$ | 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: |
$\frac{7278181}{5173031}a^{15}-\frac{17820675}{5173031}a^{14}+\frac{28126190}{5173031}a^{13}-\frac{17777451}{5173031}a^{12}-\frac{10124461}{5173031}a^{11}+\frac{33310165}{5173031}a^{10}-\frac{27954596}{5173031}a^{9}-\frac{63884525}{5173031}a^{8}+\frac{101206869}{5173031}a^{7}-\frac{95571305}{5173031}a^{6}+\frac{53939452}{5173031}a^{5}+\frac{31959152}{5173031}a^{4}-\frac{72089310}{5173031}a^{3}+\frac{2518374}{5173031}a^{2}+\frac{17427604}{5173031}a+\frac{6700165}{5173031}$, $a$, $\frac{2184305}{5173031}a^{15}-\frac{3524364}{5173031}a^{14}+\frac{3584603}{5173031}a^{13}+\frac{3051506}{5173031}a^{12}-\frac{9422439}{5173031}a^{11}+\frac{9015164}{5173031}a^{10}-\frac{74232}{5173031}a^{9}-\frac{27541110}{5173031}a^{8}+\frac{15225595}{5173031}a^{7}-\frac{687615}{5173031}a^{6}-\frac{15402746}{5173031}a^{5}+\frac{26933713}{5173031}a^{4}-\frac{19096361}{5173031}a^{3}-\frac{12406625}{5173031}a^{2}+\frac{9390191}{5173031}a+\frac{7641190}{5173031}$, $\frac{74425}{5173031}a^{15}+\frac{2687172}{5173031}a^{14}-\frac{7958885}{5173031}a^{13}+\frac{13631277}{5173031}a^{12}-\frac{11958647}{5173031}a^{11}-\frac{118290}{5173031}a^{10}+\frac{12177784}{5173031}a^{9}-\frac{14688000}{5173031}a^{8}-\frac{20282857}{5173031}a^{7}+\frac{47923991}{5173031}a^{6}-\frac{49042287}{5173031}a^{5}+\frac{41009998}{5173031}a^{4}-\frac{1638683}{5173031}a^{3}-\frac{25670579}{5173031}a^{2}+\frac{4878602}{5173031}a+\frac{3213000}{5173031}$, $\frac{3396808}{5173031}a^{15}-\frac{6356182}{5173031}a^{14}+\frac{10396440}{5173031}a^{13}-\frac{4594281}{5173031}a^{12}-\frac{3562518}{5173031}a^{11}+\frac{10394460}{5173031}a^{10}-\frac{6990142}{5173031}a^{9}-\frac{30864202}{5173031}a^{8}+\frac{25064731}{5173031}a^{7}-\frac{35681657}{5173031}a^{6}+\frac{15221512}{5173031}a^{5}+\frac{9192480}{5173031}a^{4}-\frac{14149972}{5173031}a^{3}-\frac{10155060}{5173031}a^{2}-\frac{4718543}{5173031}a+\frac{5134972}{5173031}$, $\frac{3985756}{5173031}a^{15}-\frac{7608503}{5173031}a^{14}+\frac{9401687}{5173031}a^{13}+\frac{14059}{5173031}a^{12}-\frac{13174403}{5173031}a^{11}+\frac{16274910}{5173031}a^{10}-\frac{4525090}{5173031}a^{9}-\frac{45288518}{5173031}a^{8}+\frac{37107375}{5173031}a^{7}-\frac{14546168}{5173031}a^{6}-\frac{5088562}{5173031}a^{5}+\frac{40280485}{5173031}a^{4}-\frac{32317237}{5173031}a^{3}-\frac{24327850}{5173031}a^{2}+\frac{9723986}{5173031}a+\frac{14109951}{5173031}$, $\frac{236979}{5173031}a^{15}+\frac{469352}{5173031}a^{14}-\frac{1873919}{5173031}a^{13}+\frac{3498773}{5173031}a^{12}-\frac{1990203}{5173031}a^{11}-\frac{2709641}{5173031}a^{10}+\frac{6586730}{5173031}a^{9}-\frac{6793525}{5173031}a^{8}-\frac{7428819}{5173031}a^{7}+\frac{15185581}{5173031}a^{6}-\frac{10903324}{5173031}a^{5}+\frac{205422}{5173031}a^{4}+\frac{8424423}{5173031}a^{3}-\frac{18045338}{5173031}a^{2}+\frac{2686297}{5173031}a+\frac{3264313}{5173031}$, $\frac{1281339}{5173031}a^{15}-\frac{4291429}{5173031}a^{14}+\frac{6648737}{5173031}a^{13}-\frac{5056638}{5173031}a^{12}-\frac{4062025}{5173031}a^{11}+\frac{11630895}{5173031}a^{10}-\frac{11794081}{5173031}a^{9}-\frac{10169418}{5173031}a^{8}+\frac{33071310}{5173031}a^{7}-\frac{26075338}{5173031}a^{6}+\frac{13750184}{5173031}a^{5}+\frac{20539194}{5173031}a^{4}-\frac{31314002}{5173031}a^{3}+\frac{17560968}{5173031}a^{2}+\frac{4090265}{5173031}a-\frac{4565043}{5173031}$
| 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: | \( 28.6747805181 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 28.6747805181 \cdot 1}{2\cdot\sqrt{19666546293359375}}\cr\approx \mathstrut & 0.158097545411 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*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 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*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 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*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 - 3*x^15 + 5*x^14 - 4*x^13 - x^12 + 6*x^11 - 6*x^10 - 8*x^9 + 20*x^8 - 19*x^7 + 11*x^6 + 4*x^5 - 14*x^4 + 4*x^3 + 5*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
$C_2^6.S_4^2:D_4$ (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
| \(\Q(\sqrt{5}) \), 8.4.16643125.1 |
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 | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | $16$ | $16$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $16$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 71.4.0.1 | $x^{4} + 4 x^{2} + 41 x + 7$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(859\)
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |