Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 + 4*y^14 - 31*y^13 + 99*y^12 - 90*y^11 + 187*y^10 + 338*y^9 + 1263*y^8 + 1943*y^7 + 2252*y^6 + 1715*y^5 + 684*y^4 + 69*y^3 + 69*y^2 - 7*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*x + 1)
\( x^{16} - 2 x^{15} + 4 x^{14} - 31 x^{13} + 99 x^{12} - 90 x^{11} + 187 x^{10} + 338 x^{9} + 1263 x^{8} + \cdots + 1 \)
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: |
\(370387893043593994140625\)
\(\medspace = 5^{12}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.72\) | 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: | $5^{3/4}79^{1/2}\approx 29.719469226626792$ | ||
| Ramified primes: |
\(5\), \(79\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{5}-\frac{1}{10}$, $\frac{1}{50}a^{11}-\frac{1}{25}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}-\frac{8}{25}a^{7}+\frac{9}{50}a^{6}-\frac{8}{25}a^{5}+\frac{12}{25}a^{4}+\frac{12}{25}a^{3}+\frac{12}{25}a^{2}-\frac{17}{50}a-\frac{2}{25}$, $\frac{1}{50}a^{12}+\frac{1}{25}a^{9}+\frac{1}{25}a^{8}+\frac{17}{50}a^{7}+\frac{6}{25}a^{6}-\frac{9}{25}a^{5}-\frac{9}{25}a^{4}+\frac{6}{25}a^{3}-\frac{9}{50}a^{2}+\frac{1}{25}a+\frac{1}{25}$, $\frac{1}{50}a^{13}+\frac{1}{25}a^{10}+\frac{1}{25}a^{9}-\frac{3}{50}a^{8}+\frac{6}{25}a^{7}+\frac{6}{25}a^{6}-\frac{9}{25}a^{5}-\frac{4}{25}a^{4}-\frac{9}{50}a^{3}-\frac{9}{25}a^{2}+\frac{1}{25}a-\frac{2}{5}$, $\frac{1}{44750}a^{14}+\frac{1}{44750}a^{13}-\frac{439}{44750}a^{12}-\frac{2}{22375}a^{11}-\frac{1019}{44750}a^{10}+\frac{427}{44750}a^{9}+\frac{3967}{44750}a^{8}+\frac{2587}{44750}a^{7}+\frac{2341}{22375}a^{6}+\frac{9577}{44750}a^{5}+\frac{12051}{44750}a^{4}+\frac{15091}{44750}a^{3}-\frac{1749}{44750}a^{2}-\frac{2682}{22375}a+\frac{5071}{44750}$, $\frac{1}{1157015322250}a^{15}+\frac{2004528}{578507661125}a^{14}+\frac{9002686771}{1157015322250}a^{13}+\frac{6191238861}{1157015322250}a^{12}+\frac{1830657431}{1157015322250}a^{11}-\frac{20543327023}{1157015322250}a^{10}+\frac{46057622931}{578507661125}a^{9}-\frac{19236968693}{1157015322250}a^{8}+\frac{153497143177}{1157015322250}a^{7}-\frac{21361325103}{1157015322250}a^{6}-\frac{155651226879}{1157015322250}a^{5}+\frac{32042188803}{578507661125}a^{4}-\frac{131134847289}{1157015322250}a^{3}-\frac{33960784809}{1157015322250}a^{2}+\frac{457966303641}{1157015322250}a+\frac{19006767657}{115701532225}$
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
$C_{10}$, which has order $10$ (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: |
\( -1 \)
(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: |
$\frac{3765855786}{578507661125}a^{15}-\frac{11577614078}{578507661125}a^{14}+\frac{26338793962}{578507661125}a^{13}-\frac{140607189688}{578507661125}a^{12}+\frac{513111819092}{578507661125}a^{11}-\frac{841347235667}{578507661125}a^{10}+\frac{1419887609694}{578507661125}a^{9}+\frac{118259719054}{578507661125}a^{8}+\frac{4022686991744}{578507661125}a^{7}+\frac{3190688744084}{578507661125}a^{6}+\frac{4026060007918}{578507661125}a^{5}+\frac{1767236085922}{578507661125}a^{4}+\frac{94770408542}{578507661125}a^{3}+\frac{203784287132}{578507661125}a^{2}-\frac{21139074108}{578507661125}a-\frac{354732408254}{578507661125}$, $\frac{10433243407}{1157015322250}a^{15}-\frac{51587837}{46280612890}a^{14}-\frac{684286661}{231403064450}a^{13}-\frac{45901664847}{231403064450}a^{12}+\frac{78050494609}{231403064450}a^{11}+\frac{1211097493697}{1157015322250}a^{10}-\frac{107919525921}{231403064450}a^{9}+\frac{1646831967861}{231403064450}a^{8}+\frac{3565061534291}{231403064450}a^{7}+\frac{8855679326561}{231403064450}a^{6}+\frac{54158299156883}{1157015322250}a^{5}+\frac{10746734841929}{231403064450}a^{4}+\frac{5896840957157}{231403064450}a^{3}+\frac{1284947429273}{231403064450}a^{2}-\frac{142055326723}{231403064450}a-\frac{85502296866}{578507661125}$, $\frac{5673076468}{115701532225}a^{15}-\frac{108604750199}{1157015322250}a^{14}+\frac{210836859251}{1157015322250}a^{13}-\frac{1729037758609}{1157015322250}a^{12}+\frac{2723635881903}{578507661125}a^{11}-\frac{4441780963509}{1157015322250}a^{10}+\frac{9627175551027}{1157015322250}a^{9}+\frac{20345919409717}{1157015322250}a^{8}+\frac{72529083216587}{1157015322250}a^{7}+\frac{56774724220016}{578507661125}a^{6}+\frac{128581754445747}{1157015322250}a^{5}+\frac{93106437432201}{1157015322250}a^{4}+\frac{30078711967341}{1157015322250}a^{3}-\frac{6542416467679}{1157015322250}a^{2}-\frac{684041932187}{578507661125}a-\frac{451807195639}{1157015322250}$, $\frac{4045902506}{578507661125}a^{15}-\frac{11275370818}{578507661125}a^{14}+\frac{23865660322}{578507661125}a^{13}-\frac{140292096278}{578507661125}a^{12}+\frac{502420214927}{578507661125}a^{11}-\frac{715672577712}{578507661125}a^{10}+\frac{1154599536614}{578507661125}a^{9}+\frac{733588865974}{578507661125}a^{8}+\frac{4126369048114}{578507661125}a^{7}+\frac{4454647222154}{578507661125}a^{6}+\frac{4691206587068}{578507661125}a^{5}+\frac{2481074949082}{578507661125}a^{4}+\frac{56059762102}{578507661125}a^{3}+\frac{280983123342}{578507661125}a^{2}+\frac{906879078877}{578507661125}a+\frac{3765855786}{578507661125}$, $\frac{112116044721}{1157015322250}a^{15}-\frac{100763185879}{578507661125}a^{14}+\frac{202760886411}{578507661125}a^{13}-\frac{1693044430169}{578507661125}a^{12}+\frac{5197337850561}{578507661125}a^{11}-\frac{7896552503377}{1157015322250}a^{10}+\frac{9518777580432}{578507661125}a^{9}+\frac{21197776272687}{578507661125}a^{8}+\frac{74508776107782}{578507661125}a^{7}+\frac{124241339031827}{578507661125}a^{6}+\frac{300058121223903}{1157015322250}a^{5}+\frac{126111773148361}{578507661125}a^{4}+\frac{61592871488276}{578507661125}a^{3}+\frac{14979423697526}{578507661125}a^{2}+\frac{5387248479616}{578507661125}a+\frac{296551177163}{578507661125}$, $\frac{108889849658}{578507661125}a^{15}-\frac{470701856353}{1157015322250}a^{14}+\frac{457416893781}{578507661125}a^{13}-\frac{3416201513439}{578507661125}a^{12}+\frac{22534619012117}{1157015322250}a^{11}-\frac{11122907796156}{578507661125}a^{10}+\frac{41098625540189}{1157015322250}a^{9}+\frac{34948691716887}{578507661125}a^{8}+\frac{129124223344957}{578507661125}a^{7}+\frac{369770905833029}{1157015322250}a^{6}+\frac{196112618653179}{578507661125}a^{5}+\frac{250832858434817}{1157015322250}a^{4}+\frac{21407417370781}{578507661125}a^{3}-\frac{20632891667029}{578507661125}a^{2}+\frac{5345580133887}{1157015322250}a-\frac{315327781932}{578507661125}$, $\frac{45359048144}{578507661125}a^{15}-\frac{163223557289}{1157015322250}a^{14}+\frac{161016652268}{578507661125}a^{13}-\frac{1367285010807}{578507661125}a^{12}+\frac{8416151587791}{1157015322250}a^{11}-\frac{3133514239603}{578507661125}a^{10}+\frac{15062150237947}{1157015322250}a^{9}+\frac{16885248248151}{578507661125}a^{8}+\frac{60688292848586}{578507661125}a^{7}+\frac{195576435424417}{1157015322250}a^{6}+\frac{117238558681507}{578507661125}a^{5}+\frac{186709886517161}{1157015322250}a^{4}+\frac{41130222825533}{578507661125}a^{3}+\frac{6383908969943}{578507661125}a^{2}+\frac{5016048987001}{1157015322250}a-\frac{370918355671}{578507661125}$
| 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: | \( 24561.9499263 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 24561.9499263 \cdot 10}{2\cdot\sqrt{370387893043593994140625}}\cr\approx \mathstrut & 0.490166246298 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*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 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*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 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*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 - 2*x^15 + 4*x^14 - 31*x^13 + 99*x^12 - 90*x^11 + 187*x^10 + 338*x^9 + 1263*x^8 + 1943*x^7 + 2252*x^6 + 1715*x^5 + 684*x^4 + 69*x^3 + 69*x^2 - 7*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
$\SD_{16}$ (as 16T12):
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 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-395}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-79}) \), \(\Q(\sqrt{5}, \sqrt{-79})\), 4.2.1975.1 x2, 4.0.31205.1 x2, 8.0.24343800625.1, 8.2.7703734375.1 x4 |
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: | 8.2.7703734375.1 |
| Minimal sibling: | 8.2.7703734375.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(79\)
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |