Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361)
gp: K = bnfinit(y^16 - 5*y^15 + 21*y^14 - 60*y^13 + 167*y^12 - 390*y^11 + 863*y^10 - 1665*y^9 + 2915*y^8 - 4545*y^7 + 6502*y^6 - 8600*y^5 + 9652*y^4 - 8300*y^3 + 4994*y^2 - 1900*y + 361, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361)
\( x^{16} - 5 x^{15} + 21 x^{14} - 60 x^{13} + 167 x^{12} - 390 x^{11} + 863 x^{10} - 1665 x^{9} + \cdots + 361 \)
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: |
\(84480512701416015625\)
\(\medspace = 5^{14}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.60\) | 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: | $5^{7/8}7^{3/4}\approx 17.596337524163744$ | ||
| Ramified primes: |
\(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $16$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{1}{11}a^{7}-\frac{4}{11}a^{6}-\frac{2}{11}a^{5}-\frac{1}{11}a^{3}+\frac{1}{11}a^{2}+\frac{4}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{9}-\frac{5}{11}a^{7}+\frac{5}{11}a^{6}-\frac{2}{11}a^{5}-\frac{1}{11}a^{4}+\frac{5}{11}a^{2}-\frac{5}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{10}-\frac{1}{11}$, $\frac{1}{11}a^{11}-\frac{1}{11}a$, $\frac{1}{11}a^{12}-\frac{1}{11}a^{2}$, $\frac{1}{121}a^{13}+\frac{1}{121}a^{12}+\frac{2}{121}a^{11}-\frac{3}{121}a^{10}+\frac{4}{121}a^{9}+\frac{3}{121}a^{8}+\frac{21}{121}a^{7}+\frac{19}{121}a^{6}-\frac{36}{121}a^{5}-\frac{37}{121}a^{4}+\frac{18}{121}a^{3}-\frac{3}{11}a^{2}-\frac{21}{121}a+\frac{39}{121}$, $\frac{1}{121}a^{14}+\frac{1}{121}a^{12}-\frac{5}{121}a^{11}-\frac{4}{121}a^{10}-\frac{1}{121}a^{9}-\frac{4}{121}a^{8}+\frac{20}{121}a^{7}+\frac{3}{11}a^{6}+\frac{43}{121}a^{5}+\frac{5}{11}a^{4}-\frac{29}{121}a^{3}-\frac{10}{121}a^{2}-\frac{28}{121}a+\frac{49}{121}$, $\frac{1}{31927387789}a^{15}-\frac{130977284}{31927387789}a^{14}-\frac{5091029}{31927387789}a^{13}-\frac{964810959}{31927387789}a^{12}+\frac{1428416956}{31927387789}a^{11}+\frac{767808249}{31927387789}a^{10}+\frac{1227629566}{31927387789}a^{9}-\frac{707241553}{31927387789}a^{8}+\frac{7011204866}{31927387789}a^{7}+\frac{12823937573}{31927387789}a^{6}-\frac{10076014085}{31927387789}a^{5}+\frac{681230400}{2902489799}a^{4}+\frac{125545345}{1680388831}a^{3}+\frac{10631161944}{31927387789}a^{2}+\frac{5603726199}{31927387789}a-\frac{709468879}{1680388831}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $11$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: |
\( -\frac{636632551}{31927387789} a^{15} + \frac{2864587133}{31927387789} a^{14} - \frac{11934579568}{31927387789} a^{13} + \frac{32070288634}{31927387789} a^{12} - \frac{89756675916}{31927387789} a^{11} + \frac{201159205318}{31927387789} a^{10} - \frac{443855865113}{31927387789} a^{9} + \frac{823219991343}{31927387789} a^{8} - \frac{1415787238940}{31927387789} a^{7} + \frac{2121531755396}{31927387789} a^{6} - \frac{2976694803706}{31927387789} a^{5} + \frac{347074939690}{2902489799} a^{4} - \frac{211148594117}{1680388831} a^{3} + \frac{2986228431833}{31927387789} a^{2} - \frac{1342705929392}{31927387789} a + \frac{16111982569}{1680388831} \)
(order $10$)
| 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: |
$\frac{10328}{933031}a^{15}-\frac{61712}{933031}a^{14}+\frac{251590}{933031}a^{13}-\frac{761231}{933031}a^{12}+\frac{2044879}{933031}a^{11}-\frac{4949537}{933031}a^{10}+\frac{10748025}{933031}a^{9}-\frac{21191427}{933031}a^{8}+\frac{36711240}{933031}a^{7}-\frac{57461851}{933031}a^{6}+\frac{81282206}{933031}a^{5}-\frac{885397}{7711}a^{4}+\frac{121348090}{933031}a^{3}-\frac{100049286}{933031}a^{2}+\frac{51101298}{933031}a-\frac{12607688}{933031}$, $\frac{50396589}{31927387789}a^{15}-\frac{351350646}{31927387789}a^{14}+\frac{1158524831}{31927387789}a^{13}-\frac{3886328954}{31927387789}a^{12}+\frac{8735688136}{31927387789}a^{11}-\frac{23640808302}{31927387789}a^{10}+\frac{43260070588}{31927387789}a^{9}-\frac{92947489026}{31927387789}a^{8}+\frac{135212384423}{31927387789}a^{7}-\frac{221292641816}{31927387789}a^{6}+\frac{271625705577}{31927387789}a^{5}-\frac{32765740434}{2902489799}a^{4}+\frac{18838479178}{1680388831}a^{3}-\frac{196234255221}{31927387789}a^{2}+\frac{36471655368}{31927387789}a+\frac{1050804944}{1680388831}$, $\frac{359933022}{31927387789}a^{15}-\frac{1588911163}{31927387789}a^{14}+\frac{6556036175}{31927387789}a^{13}-\frac{17397555215}{31927387789}a^{12}+\frac{48523176143}{31927387789}a^{11}-\frac{108284537128}{31927387789}a^{10}+\frac{237134211527}{31927387789}a^{9}-\frac{437348636304}{31927387789}a^{8}+\frac{742271733994}{31927387789}a^{7}-\frac{1107326290084}{31927387789}a^{6}+\frac{1533181387143}{31927387789}a^{5}-\frac{178157389758}{2902489799}a^{4}+\frac{104999860852}{1680388831}a^{3}-\frac{1383943429553}{31927387789}a^{2}+\frac{551905330890}{31927387789}a-\frac{4236539883}{1680388831}$, $\frac{443505762}{31927387789}a^{15}-\frac{2128066740}{31927387789}a^{14}+\frac{8654095035}{31927387789}a^{13}-\frac{24014758342}{31927387789}a^{12}+\frac{65766864706}{31927387789}a^{11}-\frac{151313797411}{31927387789}a^{10}+\frac{327801054809}{31927387789}a^{9}-\frac{619971108960}{31927387789}a^{8}+\frac{1050339033159}{31927387789}a^{7}-\frac{1594107198019}{31927387789}a^{6}+\frac{2205658541447}{31927387789}a^{5}-\frac{258863581004}{2902489799}a^{4}+\frac{156828162791}{1680388831}a^{3}-\frac{2165340810056}{31927387789}a^{2}+\frac{963823469578}{31927387789}a-\frac{10873613418}{1680388831}$, $\frac{4691333}{31927387789}a^{15}+\frac{355082273}{31927387789}a^{14}-\frac{1234697358}{31927387789}a^{13}+\frac{5507356287}{31927387789}a^{12}-\frac{12588165951}{31927387789}a^{11}+\frac{37976158959}{31927387789}a^{10}-\frac{76369263098}{31927387789}a^{9}+\frac{174261819867}{31927387789}a^{8}-\frac{296583225434}{31927387789}a^{7}+\frac{501070863356}{31927387789}a^{6}-\frac{712030459383}{31927387789}a^{5}+\frac{87225555260}{2902489799}a^{4}-\frac{64550658242}{1680388831}a^{3}+\frac{1051841083224}{31927387789}a^{2}-\frac{571817667904}{31927387789}a+\frac{9099612518}{1680388831}$, $\frac{1335063501}{31927387789}a^{15}-\frac{5426273123}{31927387789}a^{14}+\frac{22864910006}{31927387789}a^{13}-\frac{58278801596}{31927387789}a^{12}+\frac{166718957293}{31927387789}a^{11}-\frac{359934511664}{31927387789}a^{10}+\frac{802262831762}{31927387789}a^{9}-\frac{1440665830992}{31927387789}a^{8}+\frac{2475202693932}{31927387789}a^{7}-\frac{3618493153657}{31927387789}a^{6}+\frac{5072172756344}{31927387789}a^{5}-\frac{580716347461}{2902489799}a^{4}+\frac{338340744174}{1680388831}a^{3}-\frac{4430311535856}{31927387789}a^{2}+\frac{1870405220529}{31927387789}a-\frac{19948879485}{1680388831}$, $\frac{287387475}{31927387789}a^{15}-\frac{932218373}{31927387789}a^{14}+\frac{4128592558}{31927387789}a^{13}-\frac{9278085885}{31927387789}a^{12}+\frac{28297037339}{31927387789}a^{11}-\frac{55355688411}{31927387789}a^{10}+\frac{128740655613}{31927387789}a^{9}-\frac{211788597357}{31927387789}a^{8}+\frac{369863951034}{31927387789}a^{7}-\frac{505377862004}{31927387789}a^{6}+\frac{723872157520}{31927387789}a^{5}-\frac{7098421098}{263862709}a^{4}+\frac{41100214751}{1680388831}a^{3}-\frac{469855832560}{31927387789}a^{2}+\frac{177706683940}{31927387789}a-\frac{2497734501}{1680388831}$
| 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: | \( 19929.8787165 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 19929.8787165 \cdot 1}{10\cdot\sqrt{84480512701416015625}}\cr\approx \mathstrut & 0.526702442375 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361)
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 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361, 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 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361);
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 - 5*x^15 + 21*x^14 - 60*x^13 + 167*x^12 - 390*x^11 + 863*x^10 - 1665*x^9 + 2915*x^8 - 4545*x^7 + 6502*x^6 - 8600*x^5 + 9652*x^4 - 8300*x^3 + 4994*x^2 - 1900*x + 361);
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
$\OD_{16}$ (as 16T6):
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 16 |
| The 10 conjugacy class representatives for $C_8: C_2$ |
| Character table for $C_8: C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.6125.1, \(\Q(\zeta_{5})\), 8.0.37515625.1, 8.4.9191328125.1 x2 |
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 sibling: | 8.4.9191328125.1 |
| Minimal sibling: | 8.4.9191328125.1 |
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 | R | ${\href{/padicField/11.1.0.1}{1} }^{16}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.14.1 | $x^{16} - 20 x^{8} - 100$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |
|
\(7\)
| 7.16.12.2 | $x^{16} - 70 x^{12} + 1519 x^{8} - 4802 x^{4} + 21609$ | $4$ | $4$ | $12$ | $C_8: C_2$ | $[\ ]_{4}^{4}$ |