Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16)
gp: K = bnfinit(y^16 - 2*y^15 - 8*y^14 + 10*y^13 + 63*y^12 - 98*y^11 - 86*y^10 + 48*y^9 + 757*y^8 - 1896*y^7 + 2378*y^6 - 1668*y^5 + 404*y^4 + 296*y^3 - 184*y^2 + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16)
\( x^{16} - 2 x^{15} - 8 x^{14} + 10 x^{13} + 63 x^{12} - 98 x^{11} - 86 x^{10} + 48 x^{9} + 757 x^{8} + \cdots + 16 \)
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: |
\(136048896000000000000\)
\(\medspace = 2^{20}\cdot 3^{12}\cdot 5^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.13\) | 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: | $2^{3/2}3^{3/4}5^{3/4}\approx 21.558246717785053$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{10}+\frac{1}{6}a^{9}+\frac{1}{4}a^{8}+\frac{1}{3}a^{6}+\frac{1}{12}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{12}a^{13}-\frac{1}{6}a^{11}+\frac{1}{6}a^{10}+\frac{1}{4}a^{9}+\frac{1}{3}a^{7}+\frac{1}{12}a^{5}-\frac{1}{6}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{264}a^{14}-\frac{1}{132}a^{13}-\frac{1}{44}a^{12}+\frac{1}{44}a^{11}+\frac{7}{264}a^{10}-\frac{43}{132}a^{9}+\frac{2}{33}a^{8}+\frac{13}{66}a^{7}+\frac{43}{88}a^{6}+\frac{7}{33}a^{5}+\frac{7}{22}a^{4}+\frac{7}{66}a^{3}-\frac{1}{3}a+\frac{10}{33}$, $\frac{1}{25\!\cdots\!44}a^{15}+\frac{1478822526283}{12\!\cdots\!72}a^{14}-\frac{3937395561113}{114928206983052}a^{13}-\frac{28876355234803}{12\!\cdots\!72}a^{12}-\frac{14390182995283}{229856413966104}a^{11}+\frac{27703853102921}{421403425604524}a^{10}+\frac{1595424807629}{316052569203393}a^{9}+\frac{73866875854327}{316052569203393}a^{8}+\frac{10\!\cdots\!41}{25\!\cdots\!44}a^{7}-\frac{1049731126357}{210701712802262}a^{6}+\frac{244501857498713}{632105138406786}a^{5}+\frac{931449643933}{2899564855077}a^{4}-\frac{72533459078195}{316052569203393}a^{3}+\frac{11001523455493}{57464103491526}a^{2}-\frac{46473533853553}{105350856401131}a+\frac{18569914885655}{105350856401131}$
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
$C_{2}$, which has order $2$
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: |
\( -\frac{113228593}{1491719064} a^{15} + \frac{223667255}{1491719064} a^{14} + \frac{151524195}{248619844} a^{13} - \frac{540949465}{745859532} a^{12} - \frac{7154801365}{1491719064} a^{11} + \frac{3563010075}{497239688} a^{10} + \frac{1223706185}{186464883} a^{9} - \frac{1784062055}{745859532} a^{8} - \frac{28606393535}{497239688} a^{7} + \frac{209896644725}{1491719064} a^{6} - \frac{44764569493}{248619844} a^{5} + \frac{307129985}{2280916} a^{4} - \frac{8559260455}{186464883} a^{3} - \frac{120169955}{16951353} a^{2} + \frac{430292905}{62154961} a - \frac{46449853}{186464883} \)
(order $6$)
| 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: |
$\frac{67565166403613}{12\!\cdots\!72}a^{15}-\frac{21853164850923}{210701712802262}a^{14}-\frac{134323817389889}{316052569203393}a^{13}+\frac{206145176145833}{421403425604524}a^{12}+\frac{42\!\cdots\!49}{12\!\cdots\!72}a^{11}-\frac{519988925483793}{105350856401131}a^{10}-\frac{455227013442594}{105350856401131}a^{9}+\frac{572711552063105}{421403425604524}a^{8}+\frac{50\!\cdots\!83}{12\!\cdots\!72}a^{7}-\frac{31\!\cdots\!94}{316052569203393}a^{6}+\frac{80\!\cdots\!41}{632105138406786}a^{5}-\frac{104509398845549}{1054387220028}a^{4}+\frac{12\!\cdots\!58}{316052569203393}a^{3}+\frac{10229110830725}{28732051745763}a^{2}-\frac{347887476721590}{105350856401131}a+\frac{304906651759526}{316052569203393}$, $\frac{45576014980417}{25\!\cdots\!44}a^{15}-\frac{29402227648261}{12\!\cdots\!72}a^{14}-\frac{17216875516937}{114928206983052}a^{13}+\frac{65534011289063}{12\!\cdots\!72}a^{12}+\frac{249765819436825}{229856413966104}a^{11}-\frac{11\!\cdots\!05}{12\!\cdots\!72}a^{10}-\frac{158705330248013}{105350856401131}a^{9}-\frac{787793638028399}{632105138406786}a^{8}+\frac{30\!\cdots\!33}{25\!\cdots\!44}a^{7}-\frac{53\!\cdots\!09}{210701712802262}a^{6}+\frac{20\!\cdots\!59}{632105138406786}a^{5}-\frac{52785747067985}{1933043236718}a^{4}+\frac{10\!\cdots\!31}{632105138406786}a^{3}-\frac{338063476456279}{57464103491526}a^{2}+\frac{505114233382849}{316052569203393}a+\frac{83165753046590}{316052569203393}$, $\frac{10675785}{855692728}a^{15}-\frac{18006749}{2567078184}a^{14}-\frac{155525947}{1283539092}a^{13}-\frac{56144777}{1283539092}a^{12}+\frac{713512205}{855692728}a^{11}+\frac{16767943}{855692728}a^{10}-\frac{579829511}{320884773}a^{9}-\frac{2620663627}{1283539092}a^{8}+\frac{1923474413}{233370744}a^{7}-\frac{25618213303}{2567078184}a^{6}+\frac{9769514053}{1283539092}a^{5}-\frac{13407665}{11775588}a^{4}-\frac{174368273}{106961591}a^{3}+\frac{6373387}{9723781}a^{2}+\frac{9711021}{106961591}a+\frac{184700909}{320884773}$, $\frac{9158307372475}{12\!\cdots\!72}a^{15}+\frac{26905728400435}{25\!\cdots\!44}a^{14}-\frac{63072157345421}{632105138406786}a^{13}-\frac{91382884536409}{632105138406786}a^{12}+\frac{197093584767437}{316052569203393}a^{11}+\frac{24\!\cdots\!49}{25\!\cdots\!44}a^{10}-\frac{138606323008457}{57464103491526}a^{9}-\frac{31\!\cdots\!77}{12\!\cdots\!72}a^{8}+\frac{63\!\cdots\!77}{12\!\cdots\!72}a^{7}+\frac{45\!\cdots\!65}{842806851209048}a^{6}-\frac{94\!\cdots\!19}{421403425604524}a^{5}+\frac{134386633492479}{3866086473436}a^{4}-\frac{90\!\cdots\!58}{316052569203393}a^{3}+\frac{658157497071361}{57464103491526}a^{2}-\frac{393254132131976}{316052569203393}a+\frac{2533905001315}{105350856401131}$, $\frac{146859374267657}{25\!\cdots\!44}a^{15}-\frac{123728810439955}{25\!\cdots\!44}a^{14}-\frac{317221617889427}{632105138406786}a^{13}-\frac{3997074522010}{316052569203393}a^{12}+\frac{29\!\cdots\!27}{842806851209048}a^{11}-\frac{42\!\cdots\!37}{25\!\cdots\!44}a^{10}-\frac{74\!\cdots\!85}{12\!\cdots\!72}a^{9}-\frac{14\!\cdots\!18}{316052569203393}a^{8}+\frac{94\!\cdots\!09}{25\!\cdots\!44}a^{7}-\frac{17\!\cdots\!85}{25\!\cdots\!44}a^{6}+\frac{44\!\cdots\!99}{632105138406786}a^{5}-\frac{217344670222193}{5799129710154}a^{4}+\frac{22\!\cdots\!49}{632105138406786}a^{3}+\frac{129546797634049}{19154701163842}a^{2}-\frac{58175904128041}{316052569203393}a-\frac{42857933841676}{316052569203393}$, $\frac{11012752477862}{316052569203393}a^{15}-\frac{60892508134015}{842806851209048}a^{14}-\frac{55827969701025}{210701712802262}a^{13}+\frac{10577867714653}{28732051745763}a^{12}+\frac{26\!\cdots\!51}{12\!\cdots\!72}a^{11}-\frac{30\!\cdots\!37}{842806851209048}a^{10}-\frac{730632726150647}{316052569203393}a^{9}+\frac{831561334055317}{421403425604524}a^{8}+\frac{54\!\cdots\!47}{210701712802262}a^{7}-\frac{58\!\cdots\!99}{842806851209048}a^{6}+\frac{11\!\cdots\!93}{12\!\cdots\!72}a^{5}-\frac{276088046018617}{3866086473436}a^{4}+\frac{28\!\cdots\!50}{105350856401131}a^{3}+\frac{205576209309881}{57464103491526}a^{2}-\frac{639120261360831}{105350856401131}a+\frac{497674001853334}{316052569203393}$, $\frac{14132911095145}{632105138406786}a^{15}-\frac{20097136157863}{25\!\cdots\!44}a^{14}-\frac{137135990981683}{632105138406786}a^{13}-\frac{3063215833214}{28732051745763}a^{12}+\frac{18\!\cdots\!69}{12\!\cdots\!72}a^{11}+\frac{466763567404991}{25\!\cdots\!44}a^{10}-\frac{10\!\cdots\!00}{316052569203393}a^{9}-\frac{44\!\cdots\!51}{12\!\cdots\!72}a^{8}+\frac{91\!\cdots\!71}{632105138406786}a^{7}-\frac{13\!\cdots\!01}{842806851209048}a^{6}+\frac{35\!\cdots\!13}{421403425604524}a^{5}+\frac{22760224416697}{3866086473436}a^{4}-\frac{65\!\cdots\!25}{632105138406786}a^{3}+\frac{82892254803391}{28732051745763}a^{2}+\frac{246260907767363}{316052569203393}a+\frac{15948901525654}{105350856401131}$
| 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: | \( 5165.64894396 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5165.64894396 \cdot 2}{6\cdot\sqrt{136048896000000000000}}\cr\approx \mathstrut & 0.358587436473 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16)
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 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16, 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 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16);
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 - 8*x^14 + 10*x^13 + 63*x^12 - 98*x^11 - 86*x^10 + 48*x^9 + 757*x^8 - 1896*x^7 + 2378*x^6 - 1668*x^5 + 404*x^4 + 296*x^3 - 184*x^2 + 16);
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
$C_4^2:C_2$ (as 16T30):
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 14 conjugacy class representatives for $C_4^2:C_2$ |
| Character table for $C_4^2:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.2.2000.1, 4.2.18000.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.324000000.4, 8.0.7290000.1, 8.0.2916000000.5 |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.4.2176782336000000000000.2 |
| 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 | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.2 | $x^{8} + 8 x^{7} + 56 x^{6} + 240 x^{5} + 816 x^{4} + 2048 x^{3} + 3776 x^{2} + 4928 x + 3760$ | $2$ | $4$ | $8$ | $C_2^2:C_4$ | $[2, 2]^{4}$ |
| 2.8.12.15 | $x^{8} + 4 x^{7} + 4 x^{6} + 12$ | $4$ | $2$ | $12$ | $C_2^2:C_4$ | $[2, 2]^{4}$ | |
|
\(3\)
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |