Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 3*y^14 - 8*y^13 + 9*y^12 - 5*y^11 + 12*y^10 - 7*y^9 - 2*y^8 - 11*y^7 + 7*y^6 + 5*y^5 + 6*y^4 - 7*y^3 - y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 1)
\( x^{16} - 2 x^{15} + 3 x^{14} - 8 x^{13} + 9 x^{12} - 5 x^{11} + 12 x^{10} - 7 x^{9} - 2 x^{8} - 11 x^{7} + \cdots + 1 \)
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: |
\(189837888276953125\)
\(\medspace = 5^{8}\cdot 11^{4}\cdot 29^{3}\cdot 1361\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.02\) | 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: | $5^{1/2}11^{1/2}29^{3/4}1361^{1/2}\approx 3419.077190924994$ | ||
| Ramified primes: |
\(5\), \(11\), \(29\), \(1361\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{39469}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{106039}a^{15}+\frac{44464}{106039}a^{14}+\frac{39072}{106039}a^{13}+\frac{32568}{106039}a^{12}-\frac{5926}{106039}a^{11}+\frac{1394}{106039}a^{10}-\frac{47199}{106039}a^{9}-\frac{26853}{106039}a^{8}-\frac{2440}{5581}a^{7}-\frac{45611}{106039}a^{6}-\frac{36805}{106039}a^{5}+\frac{34801}{106039}a^{4}+\frac{34145}{106039}a^{3}+\frac{25161}{106039}a^{2}-\frac{8464}{106039}a-\frac{27813}{106039}$
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: |
$\frac{194626}{106039}a^{15}-\frac{310443}{106039}a^{14}+\frac{476421}{106039}a^{13}-\frac{1386203}{106039}a^{12}+\frac{1198956}{106039}a^{11}-\frac{575352}{106039}a^{10}+\frac{2126776}{106039}a^{9}-\frac{371941}{106039}a^{8}-\frac{22474}{5581}a^{7}-\frac{2258420}{106039}a^{6}+\frac{148676}{106039}a^{5}+\frac{786613}{106039}a^{4}+\frac{1419147}{106039}a^{3}-\frac{426429}{106039}a^{2}-\frac{210677}{106039}a-\frac{160105}{106039}$, $\frac{54215}{106039}a^{15}-\frac{74866}{106039}a^{14}+\frac{159455}{106039}a^{13}-\frac{405425}{106039}a^{12}+\frac{338197}{106039}a^{11}-\frac{348214}{106039}a^{10}+\frac{675597}{106039}a^{9}-\frac{25964}{106039}a^{8}+\frac{13005}{5581}a^{7}-\frac{501080}{106039}a^{6}-\frac{47212}{106039}a^{5}-\frac{227790}{106039}a^{4}+\frac{366469}{106039}a^{3}+\frac{17919}{106039}a^{2}+\frac{167071}{106039}a-\frac{113254}{106039}$, $\frac{9452}{106039}a^{15}+\frac{41171}{106039}a^{14}-\frac{25293}{106039}a^{13}+\frac{1519}{106039}a^{12}-\frac{236038}{106039}a^{11}+\frac{27252}{106039}a^{10}+\frac{193203}{106039}a^{9}+\frac{466966}{106039}a^{8}+\frac{14555}{5581}a^{7}-\frac{278715}{106039}a^{6}-\frac{709174}{106039}a^{5}-\frac{312043}{106039}a^{4}+\frac{273941}{106039}a^{3}+\frac{506490}{106039}a^{2}+\frac{57717}{106039}a-\frac{123834}{106039}$, $\frac{194626}{106039}a^{15}-\frac{310443}{106039}a^{14}+\frac{476421}{106039}a^{13}-\frac{1386203}{106039}a^{12}+\frac{1198956}{106039}a^{11}-\frac{575352}{106039}a^{10}+\frac{2126776}{106039}a^{9}-\frac{371941}{106039}a^{8}-\frac{22474}{5581}a^{7}-\frac{2258420}{106039}a^{6}+\frac{148676}{106039}a^{5}+\frac{786613}{106039}a^{4}+\frac{1419147}{106039}a^{3}-\frac{426429}{106039}a^{2}-\frac{104638}{106039}a-\frac{160105}{106039}$, $\frac{136122}{106039}a^{15}-\frac{177512}{106039}a^{14}+\frac{278778}{106039}a^{13}-\frac{905528}{106039}a^{12}+\frac{615935}{106039}a^{11}-\frac{267820}{106039}a^{10}+\frac{1565278}{106039}a^{9}-\frac{13697}{106039}a^{8}-\frac{12370}{5581}a^{7}-\frac{1879755}{106039}a^{6}-\frac{263694}{106039}a^{5}+\frac{631670}{106039}a^{4}+\frac{1256710}{106039}a^{3}+\frac{11981}{106039}a^{2}-\frac{340990}{106039}a-\frac{156808}{106039}$, $\frac{179749}{106039}a^{15}-\frac{224050}{106039}a^{14}+\frac{402036}{106039}a^{13}-\frac{1196070}{106039}a^{12}+\frac{817493}{106039}a^{11}-\frac{530246}{106039}a^{10}+\frac{2010002}{106039}a^{9}+\frac{95383}{106039}a^{8}+\frac{906}{5581}a^{7}-\frac{2035056}{106039}a^{6}-\frac{312891}{106039}a^{5}+\frac{416417}{106039}a^{4}+\frac{1476831}{106039}a^{3}-\frac{110839}{106039}a^{2}-\frac{160042}{106039}a-\frac{256321}{106039}$, $\frac{21221}{106039}a^{15}+\frac{35522}{106039}a^{14}-\frac{78068}{106039}a^{13}+\frac{69365}{106039}a^{12}-\frac{417548}{106039}a^{11}+\frac{527388}{106039}a^{10}-\frac{177663}{106039}a^{9}+\frac{748346}{106039}a^{8}-\frac{21046}{5581}a^{7}-\frac{199117}{106039}a^{6}-\frac{591865}{106039}a^{5}+\frac{480581}{106039}a^{4}+\frac{238636}{106039}a^{3}+\frac{247294}{106039}a^{2}-\frac{408634}{106039}a+\frac{99440}{106039}$
| 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: | \( 77.7064932253 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 77.7064932253 \cdot 1}{2\cdot\sqrt{189837888276953125}}\cr\approx \mathstrut & 0.216608160616 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 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 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 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 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 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 - 2*x^15 + 3*x^14 - 8*x^13 + 9*x^12 - 5*x^11 + 12*x^10 - 7*x^9 - 2*x^8 - 11*x^7 + 7*x^6 + 5*x^5 + 6*x^4 - 7*x^3 - x^2 + 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
$C_4^4.C_2\wr D_4$ (as 16T1823):
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 32768 |
| The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$ |
| Character table for $C_4^4.C_2\wr D_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.275.1, 8.0.2193125.1 |
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 | $16$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(11\)
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.8.4.1 | $x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.3.3 | $x^{4} + 58$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.8.0.1 | $x^{8} + 3 x^{4} + 24 x^{3} + 26 x^{2} + 23 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(1361\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |