Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*x + 1)
gp: K = bnfinit(y^16 - 6*y^15 + 11*y^14 + 6*y^13 - 47*y^12 + 24*y^11 + 74*y^10 + 48*y^9 - 362*y^8 + 264*y^7 + 386*y^6 - 792*y^5 + 73*y^4 + 438*y^3 - 181*y^2 + 18*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*x + 1)
\( x^{16} - 6 x^{15} + 11 x^{14} + 6 x^{13} - 47 x^{12} + 24 x^{11} + 74 x^{10} + 48 x^{9} - 362 x^{8} + \cdots + 1 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1913187600000000000000\)
\(\medspace = 2^{16}\cdot 3^{14}\cdot 5^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.39\) | 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: | $2\cdot 3^{7/8}5^{7/8}\approx 21.385029187241088$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{12}-\frac{1}{4}$, $\frac{1}{76}a^{13}+\frac{3}{38}a^{12}-\frac{5}{76}a^{11}-\frac{1}{19}a^{10}+\frac{13}{76}a^{9}-\frac{1}{38}a^{8}-\frac{5}{76}a^{7}+\frac{1}{38}a^{6}-\frac{9}{76}a^{5}+\frac{4}{19}a^{4}+\frac{37}{76}a^{3}+\frac{17}{38}a^{2}-\frac{3}{38}a-\frac{3}{19}$, $\frac{1}{152}a^{14}+\frac{2}{19}a^{12}-\frac{3}{38}a^{11}-\frac{1}{152}a^{10}-\frac{1}{38}a^{9}+\frac{7}{152}a^{8}+\frac{4}{19}a^{7}-\frac{21}{152}a^{6}+\frac{4}{19}a^{5}+\frac{55}{152}a^{4}+\frac{5}{19}a^{3}-\frac{29}{76}a^{2}-\frac{13}{38}a-\frac{61}{152}$, $\frac{1}{105765317768}a^{15}+\frac{14311059}{52882658884}a^{14}+\frac{2387219}{695824459}a^{13}+\frac{6182297677}{52882658884}a^{12}-\frac{2101739}{506054152}a^{11}+\frac{2873640749}{52882658884}a^{10}-\frac{7846752119}{105765317768}a^{9}+\frac{783062259}{52882658884}a^{8}+\frac{5960797069}{105765317768}a^{7}-\frac{9746872097}{52882658884}a^{6}-\frac{20603449275}{105765317768}a^{5}+\frac{13179470499}{52882658884}a^{4}+\frac{230799299}{1201878611}a^{3}-\frac{4562763629}{26441329442}a^{2}+\frac{1817685903}{9615028888}a+\frac{3504545775}{13220664721}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$ (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: | $9$ | 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: |
$\frac{1405839}{87554071}a^{15}-\frac{58584879}{700432568}a^{14}+\frac{19006077}{175108142}a^{13}+\frac{33133827}{175108142}a^{12}-\frac{1017759}{1675676}a^{11}-\frac{84525873}{700432568}a^{10}+\frac{391729025}{350216284}a^{9}+\frac{1211337399}{700432568}a^{8}-\frac{1559328141}{350216284}a^{7}+\frac{258470967}{700432568}a^{6}+\frac{2324847939}{350216284}a^{5}-\frac{4880306697}{700432568}a^{4}-\frac{151074615}{31837844}a^{3}+\frac{1050426483}{350216284}a^{2}-\frac{11589495}{31837844}a+\frac{419605115}{700432568}$, $\frac{2636161443}{52882658884}a^{15}-\frac{31571195971}{105765317768}a^{14}+\frac{28726794333}{52882658884}a^{13}+\frac{865448121}{2783297836}a^{12}-\frac{5589551541}{2403757222}a^{11}+\frac{115660881183}{105765317768}a^{10}+\frac{49919295233}{13220664721}a^{9}+\frac{275372939051}{105765317768}a^{8}-\frac{241583872005}{13220664721}a^{7}+\frac{1330984667567}{105765317768}a^{6}+\frac{13492816464}{695824459}a^{5}-\frac{4000377165953}{105765317768}a^{4}+\frac{6290202181}{4807514444}a^{3}+\frac{61127023245}{2783297836}a^{2}-\frac{28149195101}{4807514444}a+\frac{53710267345}{105765317768}$, $\frac{5236624625}{52882658884}a^{15}-\frac{61194818273}{105765317768}a^{14}+\frac{53298307779}{52882658884}a^{13}+\frac{19138530835}{26441329442}a^{12}-\frac{21855650979}{4807514444}a^{11}+\frac{191256672545}{105765317768}a^{10}+\frac{394782392205}{52882658884}a^{9}+\frac{589455193353}{105765317768}a^{8}-\frac{1841419363117}{52882658884}a^{7}+\frac{2295304858449}{105765317768}a^{6}+\frac{2173287084021}{52882658884}a^{5}-\frac{7756253286871}{105765317768}a^{4}-\frac{2141392753}{2403757222}a^{3}+\frac{2296387313229}{52882658884}a^{2}-\frac{32575733929}{2403757222}a+\frac{95491329013}{105765317768}$, $\frac{387584407}{4807514444}a^{15}-\frac{990181869}{2403757222}a^{14}+\frac{598519649}{1201878611}a^{13}+\frac{5110778345}{4807514444}a^{12}-\frac{14974709859}{4807514444}a^{11}-\frac{1052346327}{1201878611}a^{10}+\frac{29363180667}{4807514444}a^{9}+\frac{1097014935}{126513538}a^{8}-\frac{5734410401}{253027076}a^{7}-\frac{228026096}{1201878611}a^{6}+\frac{186820518885}{4807514444}a^{5}-\frac{43957466897}{1201878611}a^{4}-\frac{37129942437}{1201878611}a^{3}+\frac{26093808297}{1201878611}a^{2}+\frac{3748081781}{4807514444}a-\frac{3991632789}{4807514444}$, $\frac{984862621}{52882658884}a^{15}-\frac{14909077593}{105765317768}a^{14}+\frac{5065192652}{13220664721}a^{13}-\frac{12818744867}{52882658884}a^{12}-\frac{2202385015}{2403757222}a^{11}+\frac{168771028505}{105765317768}a^{10}+\frac{17134033145}{26441329442}a^{9}-\frac{68595471063}{105765317768}a^{8}-\frac{227417172599}{26441329442}a^{7}+\frac{1601706954473}{105765317768}a^{6}-\frac{31309713416}{13220664721}a^{5}-\frac{2042418540047}{105765317768}a^{4}+\frac{82975370645}{4807514444}a^{3}+\frac{16070274323}{2783297836}a^{2}-\frac{9608483643}{1201878611}a+\frac{133770074687}{105765317768}$, $\frac{5686530065}{105765317768}a^{15}-\frac{4157744188}{13220664721}a^{14}+\frac{7276264599}{13220664721}a^{13}+\frac{10178867181}{26441329442}a^{12}-\frac{23741797665}{9615028888}a^{11}+\frac{13350030242}{13220664721}a^{10}+\frac{426202582145}{105765317768}a^{9}+\frac{39165129376}{13220664721}a^{8}-\frac{2003699051723}{105765317768}a^{7}+\frac{159848942490}{13220664721}a^{6}+\frac{2346391803093}{105765317768}a^{5}-\frac{1054797365847}{26441329442}a^{4}+\frac{175808936}{1201878611}a^{3}+\frac{626928083663}{26441329442}a^{2}-\frac{67352966537}{9615028888}a+\frac{3025468723}{13220664721}$, $\frac{838652230}{13220664721}a^{15}-\frac{40406479835}{105765317768}a^{14}+\frac{19149758147}{26441329442}a^{13}+\frac{7041477945}{26441329442}a^{12}-\frac{3378437678}{1201878611}a^{11}+\frac{171996891067}{105765317768}a^{10}+\frac{54213776358}{13220664721}a^{9}+\frac{339782711687}{105765317768}a^{8}-\frac{594764119373}{26441329442}a^{7}+\frac{1976370217123}{105765317768}a^{6}+\frac{508203631867}{26441329442}a^{5}-\frac{4878895098069}{105765317768}a^{4}+\frac{17731142559}{2403757222}a^{3}+\frac{1109264435449}{52882658884}a^{2}-\frac{23137300271}{2403757222}a-\frac{2600944425}{105765317768}$, $\frac{1658122743}{52882658884}a^{15}-\frac{20281758245}{105765317768}a^{14}+\frac{19638013883}{52882658884}a^{13}+\frac{1570473284}{13220664721}a^{12}-\frac{1710352040}{1201878611}a^{11}+\frac{92655632297}{105765317768}a^{10}+\frac{27660711844}{13220664721}a^{9}+\frac{153757621157}{105765317768}a^{8}-\frac{151636778878}{13220664721}a^{7}+\frac{1030169479049}{105765317768}a^{6}+\frac{248501380169}{26441329442}a^{5}-\frac{2474929771151}{105765317768}a^{4}+\frac{23048398349}{4807514444}a^{3}+\frac{643383103249}{52882658884}a^{2}-\frac{28026525575}{4807514444}a-\frac{2570573205}{5566595672}$, $\frac{1189267795}{9615028888}a^{15}-\frac{6847647051}{9615028888}a^{14}+\frac{1437270996}{1201878611}a^{13}+\frac{2362825817}{2403757222}a^{12}-\frac{52547174133}{9615028888}a^{11}+\frac{15526594995}{9615028888}a^{10}+\frac{88873841519}{9615028888}a^{9}+\frac{4310980953}{506054152}a^{8}-\frac{408531623921}{9615028888}a^{7}+\frac{218135357131}{9615028888}a^{6}+\frac{484760674839}{9615028888}a^{5}-\frac{790358098761}{9615028888}a^{4}-\frac{12639879659}{1201878611}a^{3}+\frac{226319744641}{4807514444}a^{2}-\frac{64676002533}{9615028888}a-\frac{2079164485}{9615028888}$
| 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: | \( 31121.0187165 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 31121.0187165 \cdot 2}{2\cdot\sqrt{1913187600000000000000}}\cr\approx \mathstrut & 0.700445289282 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*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 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*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 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*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 - 6*x^15 + 11*x^14 + 6*x^13 - 47*x^12 + 24*x^11 + 74*x^10 + 48*x^9 - 362*x^8 + 264*x^7 + 386*x^6 - 792*x^5 + 73*x^4 + 438*x^3 - 181*x^2 + 18*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
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 32 |
| The 11 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), 4.2.54000.2, 4.2.54000.1, \(\Q(\sqrt{3}, \sqrt{5})\), 8.4.2916000000.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.16.14.1 | $x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$ | $8$ | $2$ | $14$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
|
\(5\)
| 5.16.14.4 | $x^{16} + 10 x^{8} - 25$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |