Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*x + 1)
gp: K = bnfinit(y^16 - 2*y^15 - 16*y^14 + 8*y^13 + 113*y^12 + 108*y^11 - 294*y^10 - 776*y^9 - 297*y^8 + 1148*y^7 + 1764*y^6 + 894*y^5 + 13*y^4 - 124*y^3 - 24*y^2 + 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*x + 1)
\( x^{16} - 2 x^{15} - 16 x^{14} + 8 x^{13} + 113 x^{12} + 108 x^{11} - 294 x^{10} - 776 x^{9} - 297 x^{8} + \cdots + 1 \)
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: | $[12, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(671846400000000000000\)
\(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(20.03\) | 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: | $2^{15/8}3^{1/2}5^{7/8}\approx 25.977097417310098$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1532246861}a^{15}-\frac{580171248}{1532246861}a^{14}-\frac{746831060}{1532246861}a^{13}-\frac{713543816}{1532246861}a^{12}+\frac{525143636}{1532246861}a^{11}-\frac{20955718}{1532246861}a^{10}-\frac{354870617}{1532246861}a^{9}+\frac{368930896}{1532246861}a^{8}-\frac{245865247}{1532246861}a^{7}-\frac{232133215}{1532246861}a^{6}-\frac{8785903}{25118801}a^{5}-\frac{264523654}{1532246861}a^{4}-\frac{246802163}{1532246861}a^{3}-\frac{744191206}{1532246861}a^{2}-\frac{22341355}{1532246861}a-\frac{715922016}{1532246861}$
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 $1$ (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: | $13$ | 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: |
$\frac{87004364}{15170761}a^{15}-\frac{221568360}{15170761}a^{14}-\frac{1274714950}{15170761}a^{13}+\frac{1400095069}{15170761}a^{12}+\frac{9127326434}{15170761}a^{11}+\frac{4379292048}{15170761}a^{10}-\frac{28403399030}{15170761}a^{9}-\frac{52409809967}{15170761}a^{8}+\frac{3907353386}{15170761}a^{7}+\frac{100704014486}{15170761}a^{6}+\frac{1633336606}{248701}a^{5}+\frac{19059490502}{15170761}a^{4}-\frac{15971307572}{15170761}a^{3}-\frac{5654025554}{15170761}a^{2}+\frac{605850404}{15170761}a+\frac{265540779}{15170761}$, $\frac{2529856819}{1532246861}a^{15}-\frac{7126496070}{1532246861}a^{14}-\frac{34479322913}{1532246861}a^{13}+\frac{48369145810}{1532246861}a^{12}+\frac{242683315514}{1532246861}a^{11}+\frac{72009949759}{1532246861}a^{10}-\frac{777283655466}{1532246861}a^{9}-\frac{1278136822651}{1532246861}a^{8}+\frac{254290163790}{1532246861}a^{7}+\frac{2463242467964}{1532246861}a^{6}+\frac{36458608983}{25118801}a^{5}+\frac{659191736099}{1532246861}a^{4}+\frac{127096462243}{1532246861}a^{3}+\frac{68123679739}{1532246861}a^{2}-\frac{11833868526}{1532246861}a-\frac{8608999857}{1532246861}$, $\frac{1750289257}{1532246861}a^{15}-\frac{5092798014}{1532246861}a^{14}-\frac{23747456408}{1532246861}a^{13}+\frac{36775652501}{1532246861}a^{12}+\frac{169188692810}{1532246861}a^{11}+\frac{26305174498}{1532246861}a^{10}-\frac{573426670728}{1532246861}a^{9}-\frac{835693057179}{1532246861}a^{8}+\frac{365952376360}{1532246861}a^{7}+\frac{1835413017770}{1532246861}a^{6}+\frac{21359274419}{25118801}a^{5}-\frac{14011140244}{1532246861}a^{4}-\frac{181656500928}{1532246861}a^{3}+\frac{23052136929}{1532246861}a^{2}+\frac{11807291616}{1532246861}a-\frac{2377246021}{1532246861}$, $\frac{6305013575}{1532246861}a^{15}-\frac{14082847174}{1532246861}a^{14}-\frac{97581688546}{1532246861}a^{13}+\frac{73420201136}{1532246861}a^{12}+\frac{694485827716}{1532246861}a^{11}+\frac{516482199414}{1532246861}a^{10}-\frac{1968526278702}{1532246861}a^{9}-\frac{4415365692535}{1532246861}a^{8}-\frac{844536154630}{1532246861}a^{7}+\frac{7368735177972}{1532246861}a^{6}+\frac{152811891351}{25118801}a^{5}+\frac{3515593391376}{1532246861}a^{4}-\frac{550184672440}{1532246861}a^{3}-\frac{523233579639}{1532246861}a^{2}-\frac{14343220376}{1532246861}a+\frac{18079463750}{1532246861}$, $\frac{294608072}{1532246861}a^{15}+\frac{501895794}{1532246861}a^{14}-\frac{6977422670}{1532246861}a^{13}-\frac{14818136190}{1532246861}a^{12}+\frac{43089977276}{1532246861}a^{11}+\frac{152470684160}{1532246861}a^{10}+\frac{23087948612}{1532246861}a^{9}-\frac{543931196081}{1532246861}a^{8}-\frac{898346290523}{1532246861}a^{7}+\frac{38457555328}{1532246861}a^{6}+\frac{27934618334}{25118801}a^{5}+\frac{2071062139959}{1532246861}a^{4}+\frac{985314996817}{1532246861}a^{3}+\frac{100037921318}{1532246861}a^{2}-\frac{63744724346}{1532246861}a-\frac{15003058869}{1532246861}$, $\frac{6463083332}{1532246861}a^{15}-\frac{14918473484}{1532246861}a^{14}-\frac{98555806509}{1532246861}a^{13}+\frac{81601860406}{1532246861}a^{12}+\frac{700948455268}{1532246861}a^{11}+\frac{483945744742}{1532246861}a^{10}-\frac{2019421543177}{1532246861}a^{9}-\frac{4365270464025}{1532246861}a^{8}-\frac{652502827536}{1532246861}a^{7}+\frac{7419553685953}{1532246861}a^{6}+\frac{148289694688}{25118801}a^{5}+\frac{3297171989863}{1532246861}a^{4}-\frac{491753168571}{1532246861}a^{3}-\frac{446367208926}{1532246861}a^{2}-\frac{11914331809}{1532246861}a+\frac{13951928472}{1532246861}$, $\frac{2653040647}{1532246861}a^{15}-\frac{5393761184}{1532246861}a^{14}-\frac{42843596500}{1532246861}a^{13}+\frac{24400430962}{1532246861}a^{12}+\frac{306430455398}{1532246861}a^{11}+\frac{263378495939}{1532246861}a^{10}-\frac{841590328649}{1532246861}a^{9}-\frac{2032815286356}{1532246861}a^{8}-\frac{535823039991}{1532246861}a^{7}+\frac{3311608517752}{1532246861}a^{6}+\frac{72559398234}{25118801}a^{5}+\frac{1653341895495}{1532246861}a^{4}-\frac{385149143042}{1532246861}a^{3}-\frac{286426366491}{1532246861}a^{2}+\frac{9088972464}{1532246861}a+\frac{10951789837}{1532246861}$, $\frac{3813987691}{1532246861}a^{15}-\frac{11685196509}{1532246861}a^{14}-\frac{50831925434}{1532246861}a^{13}+\frac{89886732698}{1532246861}a^{12}+\frac{368977060806}{1532246861}a^{11}-\frac{10388133122}{1532246861}a^{10}-\frac{1348166661792}{1532246861}a^{9}-\frac{1681977814438}{1532246861}a^{8}+\frac{1343520321004}{1532246861}a^{7}+\frac{4408595264510}{1532246861}a^{6}+\frac{36759733545}{25118801}a^{5}-\frac{1431341115506}{1532246861}a^{4}-\frac{1415533807571}{1532246861}a^{3}-\frac{161631590397}{1532246861}a^{2}+\frac{84313897587}{1532246861}a+\frac{14168246538}{1532246861}$, $\frac{158069757}{1532246861}a^{15}-\frac{835626310}{1532246861}a^{14}-\frac{974117963}{1532246861}a^{13}+\frac{8181659270}{1532246861}a^{12}+\frac{6462627552}{1532246861}a^{11}-\frac{32536454672}{1532246861}a^{10}-\frac{50895264475}{1532246861}a^{9}+\frac{50095228510}{1532246861}a^{8}+\frac{192033327094}{1532246861}a^{7}+\frac{50818507981}{1532246861}a^{6}-\frac{4522196663}{25118801}a^{5}-\frac{218421401513}{1532246861}a^{4}+\frac{58431503869}{1532246861}a^{3}+\frac{76866370713}{1532246861}a^{2}+\frac{2428888567}{1532246861}a-\frac{4127535278}{1532246861}$, $\frac{10285669026}{1532246861}a^{15}-\frac{26804924882}{1532246861}a^{14}-\frac{149210603678}{1532246861}a^{13}+\frac{174759563216}{1532246861}a^{12}+\frac{1069739451707}{1532246861}a^{11}+\frac{451688309402}{1532246861}a^{10}-\frac{3392236954686}{1532246861}a^{9}-\frac{5994378654728}{1532246861}a^{8}+\frac{844702434728}{1532246861}a^{7}+\frac{11892201784665}{1532246861}a^{6}+\frac{181288394132}{25118801}a^{5}+\frac{1525807064993}{1532246861}a^{4}-\frac{2048378262665}{1532246861}a^{3}-\frac{584538108574}{1532246861}a^{2}+\frac{97538177615}{1532246861}a+\frac{26858029984}{1532246861}$, $\frac{7249892820}{1532246861}a^{15}-\frac{19999176281}{1532246861}a^{14}-\frac{102572439689}{1532246861}a^{13}+\frac{139704896823}{1532246861}a^{12}+\frac{740053459356}{1532246861}a^{11}+\frac{201848759765}{1532246861}a^{10}-\frac{2474057510960}{1532246861}a^{9}-\frac{3895437446040}{1532246861}a^{8}+\frac{1327698523321}{1532246861}a^{7}+\frac{8529329049673}{1532246861}a^{6}+\frac{108371814521}{25118801}a^{5}-\frac{468278779685}{1532246861}a^{4}-\frac{2138087841582}{1532246861}a^{3}-\frac{442296851375}{1532246861}a^{2}+\frac{119304584853}{1532246861}a+\frac{27887825547}{1532246861}$, $\frac{2861836954}{1532246861}a^{15}-\frac{5946846910}{1532246861}a^{14}-\frac{45297567222}{1532246861}a^{13}+\frac{26786075063}{1532246861}a^{12}+\frac{319353576794}{1532246861}a^{11}+\frac{280179315279}{1532246861}a^{10}-\frac{849995761043}{1532246861}a^{9}-\frac{2121032346139}{1532246861}a^{8}-\frac{697014940845}{1532246861}a^{7}+\frac{3204595025800}{1532246861}a^{6}+\frac{76294052877}{25118801}a^{5}+\frac{2321214644738}{1532246861}a^{4}+\frac{217138361262}{1532246861}a^{3}-\frac{131391505160}{1532246861}a^{2}-\frac{26307243731}{1532246861}a-\frac{3057604080}{1532246861}$, $\frac{4145967826}{1532246861}a^{15}-\frac{10505547349}{1532246861}a^{14}-\frac{61650169743}{1532246861}a^{13}+\frac{68303661951}{1532246861}a^{12}+\frac{445647322086}{1532246861}a^{11}+\frac{197781232398}{1532246861}a^{10}-\frac{1420878767369}{1532246861}a^{9}-\frac{2524873337926}{1532246861}a^{8}+\frac{392215216369}{1532246861}a^{7}+\frac{5149947822346}{1532246861}a^{6}+\frac{76595177439}{25118801}a^{5}+\frac{230681793133}{1532246861}a^{4}-\frac{1325491908552}{1532246861}a^{3}-\frac{361146775296}{1532246861}a^{2}+\frac{68308275521}{1532246861}a+\frac{21251889176}{1532246861}$
| 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: | \( 50423.9439199 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 50423.9439199 \cdot 1}{2\cdot\sqrt{671846400000000000000}}\cr\approx \mathstrut & 0.157286519730 \end{aligned}\] (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 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*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 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*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 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*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 - 16*x^14 + 8*x^13 + 113*x^12 + 108*x^11 - 294*x^10 - 776*x^9 - 297*x^8 + 1148*x^7 + 1764*x^6 + 894*x^5 + 13*x^4 - 124*x^3 - 24*x^2 + 4*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
$C_2^5:C_4$ (as 16T261):
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 128 |
| The 26 conjugacy class representatives for $C_2^5:C_4$ |
| Character table for $C_2^5:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.6.1620000000.1, 8.6.1620000000.2, \(\Q(\zeta_{60})^+\) |
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: | 16.0.8294400000000000000.3 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $4$ | $4$ | $24$ | |||
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.14.1 | $x^{16} - 20 x^{8} - 100$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |